\begin{equation}
		\label{eq:orth_decomp}
		\{f\in\ell^2(E)\colon \partial f=0\}= \ell_{0,c}^2(E)\oplus_{\ell^2(E)} \{du\mid u\colon G\to \R, \, \Delta_0 u = 0, \, du\in \ell^2(E)\}.
	\end{equation}

\label{sec:Intro}

Let $G=(V,E)$ be countable oriented graph with degrees of vertices bounded from the above. Let $\tilde G$ be the non-oriented graph obtained from $G$ by forgetting the orientation of edges. Pick $D\in\mathbb N$ large enough. Consider all cycles in $\tilde G$ having lengths $\le D$. In $G$, glue each such cycle with a polygon. Choose any orientation of the latter polygons. We arrive to oriented $2$-dimensional complex, denote it by $G^{(2)}$ with implicit dependence on~$D$. Denote by $F$ the set of $2$-dimensional faces in $G^{(2)}$ which are polygons. Sometimes we write $F=FG$ and also $E=EG$ to indicate the dependence of these sets on $G$. Any of sets $V, E, F$ is endowed with counting measure which we denote by $\card$. In graph $G$, we define graph metric $\dist_G$ at $V\cup E$ along edges in $E$ so that any edge has length $1$.

If $\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\Cay(\Gamma,S)$, Cayley graph of $\Gamma$; then $V=\Gamma$. If $\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\Cay^{(2)}(\Gamma,S)$ is $\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that
\begin{equation*}\mbox{$G^{(2)}$ is simply connected}
\end{equation*}
(and that such $D$ does exist).

A \emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\mathbb R}$. We often understand  cochains as chains.
Discrete differentials (coboundaries)
$$
\{0\mbox{-cochains}\}\xrightarrow{d}\{1\mbox{-cochains}\}\xrightarrow{d}\{2\mbox{-cochains}\}
$$
and boundary operators
$$
\{2\mbox{-cochains}\}\xrightarrow{\partial}\{1\mbox{-cochains}\}\xrightarrow{\partial}\{0\mbox{-cochains}\}
$$
are introduced in the standard way with respect to the orientation of edges and faces.  Since valencies of vertices are bounded, all these operators are also bounded with respect to $\ell^2$-norms on cochains. We have $(d|_{\ell^2(V)})^*=\partial|_{\ell^2(E)}$, $(d|_{\ell^2(E)})^*=\partial|_{\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for  $\ell^2$-cochains by $\ell^2$-approximation.
If $\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\gamma$: for $e\in E$, let $f_\gamma(e)$ be the number of passes of $\gamma$ through $e$ in its direction minus number of passes of $\gamma$ over $e$ in its reversed direction. Then we have $\partial f_\gamma=0$.

Our space of interest is
\begin{equation*}
	\ell_{0,c}^2(E) := \clos_{\ell^2(E)}\{f\colon E\to {\mathbb R}\mid \partial f=0, \, \supp f
	\,\mbox{is finite}\}.\end{equation*}
Any of $1$-cochains at the right-hand side can be (convexly) decomposed into simple finite loops. Thus, $\ell_{0,c}^2(E)$ is $\ell^2$-closed linear span of (co)chains of the form $f_\gamma$ with $\gamma$ a finite loop in $G$.

Laplace operator on $0$-cochains is
$$
-\Delta_0=\partial d\colon \left(\mbox{functions on }V\right)\to \left(\mbox{functions on }V\right).
$$
A discrete integration by parts leads to the following Hodge-type decomposition:

\begin{predl}

	We have
	\begin{equation}
		\label{eq:orth_decomp}
		\{f\in\ell^2(E)\colon \partial f=0\}= \ell_{0,c}^2(E)\oplus_{\ell^2(E)} \{du\mid u\colon G\to {\mathbb R}, \, \Delta_0 u = 0, \, du\in \ell^2(E)\}.
	\end{equation}
\end{predl}

The second summand in the right-hand side of the latter relation is \emph{$\ell^2$-cohomology} of $G$. It is known to be invariant with respect to change of generating system in a group: the factorspace nature of cohomology allows to implement "discrete change of variables" from one to another set of generators. Cohomology is invariant with respect to more general quasiisometries.
Now, we pass to spectral estimates for $1$-cochains. Define non-negative Laplacian operator $\Delta_1:=\partial d+d\partial \colon \ell^2(E)\to \ell^2(E)$. On $\ell^2_{0,c}(E)$, our space of interest, this reduces to $\partial d$.

We have one more Hodge-type decomposition:
$$
\ell^2(E) = \clos_{\ell^2(E)}\{du\mid u \colon V\to {\mathbb R}, \, \supp u \mbox{ is finite}\} \oplus_{\ell^2(E)} \{f\in\ell^2(E)\colon \partial f=0\}.$$
Spectral questions for $\Delta_1$ on the first summand are generally reduced to the same for $\Delta_0$ on $\ell^2(V)$.   What concerns decomposition (\ref{eq:orth_decomp}) for $\{f\in\ell^2(E)\colon \partial f=0\}$, operator $\Delta_1$ vanishes at the second summand of its right-hand side, $\ell^2$-cohomology. Also, by the definition of $\ell^2_{0,c}(E)$ and by $\ell^2$-approximation, we see that $\Delta_1(\ell^2_{0,c}(E))\subset\ell^2_{0,c}(E)$.

\begin{define}
	We say that $\Delta_1$ \emph{has a spectral gap at $\ell^2_{0,c}(E)$} \emph{(}or just that graph $G$ has \emph{{coexact $1$-Laplacian spectral gap}}\emph{)} if $$\spec\left(\Delta_1|_{\ell^2_{0,c}(E)}\right)\cap [0,\varepsilon)=\varnothing$$ for some $\varepsilon>0$ small enough.
\end{define}

Applying discrete integration by parts, we conclude that this is equivalent to the estimate
\begin{equation}
	\label{eq:rot_estim}
\langle f, f \rangle_{\ell^2(E)}	\le 1/\varepsilon\cdot \langle df, df\rangle_{\ell^2(F)}
\end{equation}
for $f\in \ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner.
 We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way.
 \begin{predl}
 	\label{predl:general_invariance}
 	Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\subset E_2$.  	Assume the following:
 	\begin{enumerate}
 		\item $G_1$ is connected;

 		\item degrees of vertices in $G_2$ are bounded from the above;

 		\item $\sup\limits_{e\in E_2\setminus E_1} \dist_{G_1}(\beg e, \ennd e) < +\infty$ with the obvious notation.
 	\end{enumerate}
	\emph{(}In other words, metrics $\dist_{G_1}$ and $\dist_{G_2}$ on $V$ are bilipshitz equivalent.\emph{)} Then, $G_1$ has a coexact $1$-Laplacian spectral gap  \emph{(}with some $D$ implied at the construction of $G_1^{(2)}$\emph{)}
	if and only if  $G_2$ has such a spectral gap \emph{(}with some $D'$ for $G_2^{(2)}$\emph{)}.
 \end{predl}

 \noindent Proof of this quasiinvariance  is similar to the proof of quasiiinvariance of $\ell^2$-cohomology; both are based on orthogonal projection. We give a detailed argument in the Appendix. Notice also that the proof is constructive: we may estimate $D'$ via $D$ and the supremum from the third assumption of Proposition \ref{predl:general_invariance}, and vice versa.

 \begin{sled}
 	For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them  are equivalent.
 \end{sled}

Our main result is

If $\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\Cay(\Gamma,S)$, Cayley graph of $\Gamma$; then $V=\Gamma$. If $\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\Cay^{(2)}(\Gamma,S)$ is $\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that
\begin{equation*}\mbox{$G^{(2)}$ is simply connected}
\end{equation*}
(and that such $D$ does exist).

A \emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\mathbb R}$. We often understand cochains as chains.
Discrete differentials (coboundaries)
$$
\{0\mbox{-cochains}\}\xrightarrow{d}\{1\mbox{-cochains}\}\xrightarrow{d}\{2\mbox{-cochains}\}
$$
and boundary operators
$$
\{2\mbox{-cochains}\}\xrightarrow{\partial}\{1\mbox{-cochains}\}\xrightarrow{\partial}\{0\mbox{-cochains}\}
$$
are introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\ell^2$-norms on cochains. We have $(d|_{\ell^2(V)})^*=\partial|_{\ell^2(E)}$, $(d|_{\ell^2(E)})^*=\partial|_{\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\ell^2$-cochains by $\ell^2$-approximation.
If $\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\gamma$: for $e\in E$, let $f_\gamma(e)$ be the number of passes of $\gamma$ through $e$ in its direction minus number of passes of $\gamma$ over $e$ in its reversed direction. Then we have $\partial f_\gamma=0$.

\begin{define}
 We say that $\Delta_1$ \emph{has a spectral gap at $\ell^2_{0,c}(E)$} \emph{(}or just that graph $G$ has \emph{{coexact $1$-Laplacian spectral gap}}\emph{)} if $$\spec\left(\Delta_1|_{\ell^2_{0,c}(E)}\right)\cap [0,\varepsilon)=\varnothing$$ for some $\varepsilon>0$ small enough.
\end{define}

Applying discrete integration by parts, we conclude that this is equivalent to the estimate
\begin{equation}
 \label{eq:rot_estim}
\langle f, f \rangle_{\ell^2(E)} \le 1/\varepsilon\cdot \langle df, df\rangle_{\ell^2(F)}
\end{equation}
for $f\in \ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner.
 We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way.
 \begin{predl}
 \label{predl:general_invariance}
 Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\subset E_2$. Assume the following:
 \begin{enumerate}
 \item $G_1$ is connected;

\begin{sled}
 For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.
 \end{sled}

Our main result is

\renewcommand{\abstractname}{}
\begin{abstract}
Let $\Gamma$ be a discrete finitely presented group. Pick any system $S$ of generators in $\Gamma$. In Cayley graph $\Cay(\Gamma)=\Cay(\Gamma, S)$ with edge set~$E$, glue with oriented polygons all the group relations translated to all the points of $\Gamma$; denote the obtained simply connected complex by $\Cay^{(2)}(\Gamma)$.
We study non-negative \emph{Hodge--Laplace operator $\Delta_1$} on edge functions which is defined via complex $\Cay^{(2)}(\Gamma)$; $\Delta_1$ acts on
$$
\ell^2_{0,c}(E):= \clos_{\ell^2(E)}\left\{\mbox{finitely supported closed $1$-(co)chains in }\Cay^{}(\Gamma)\right\}.
$$

We prove the following implication in the spirit of Kesten Theorem: \emph{if $\Delta_1|_{\ell_{0,c}^2(E)}$ has a spectral gap then $\Gamma$ either has exponential growth or is virtually $\mathbb Z$}. \end{abstract}

Our main result is

Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:

\begin{theorem}[\cite{K59}] \label{th:Kesten}
 Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$, $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

If we assume the contrary to Theorem \ref{th:exp_growth}, then, first, $\Gamma$ cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic, see, e.g., \cite{Me08}. Second, $\Gamma$ cannot have infinitely many ends because then $\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end.

Now, let us briefly recall the proof of Kesten Theorem \ref{th:Kesten} to compare it to our argument. Non-amenability of $\Gamma$ means that
\begin{equation}
\label{eq:isoperimetr}
\|du\|_{\ell^1(E)} \gtrsim \|u\|_{\ell^1(\Gamma)}
\end{equation}
for $u=\mathds 1_E$, $E$ ranges all finite subsets in $\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\colon \Gamma\to\R$.
The spectral gap condition $0\notin \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$ means that
$$
\langle u, u\rangle_{\ell^2(\Gamma)} \lesssim \langle du, du\rangle_{\ell^2(E)}, ~~~ u\colon\Gamma\to\R \mbox{ is finitely supported}.
$$
To obtain this from (\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.

This paper is organized as follows. In Section \ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \ref{th:exp_growth}. In Section~\ref{sec:loop}, we prove Lemma \ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \ref{sec:spectra_homology}, we conclude the proof of Theorem~\ref{th:exp_growth}. This is done by approximating the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$ provided that $0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\gamma$ from Lemma \ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\Cay(\Gamma)$ and also make use of homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$. Finally, in Section \ref{sec:examples} we check the most natural examples of Cayley graphs.

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

\begin{theorem}[\cite{K59}]	\label{th:Kesten}
	Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$,  $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

\begin{theorem}
	\label{th:exp_growth}
	Let $\Gamma$ be a countable finitely presented group.
	If $\Delta_1$ has a spectral gap on 	$\ell_{0,c}^2(E)$ 	then either $\Gamma$ has exponential growth, 	 or $\Gamma$ is virtually infinite cyclic. \end{theorem}

Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:

\begin{theorem}[\cite{K59}]	\label{th:Kesten}
	Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$,  $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

Non-amenability of a group, that is, the existence of a spectral gap for $-\Delta_0$, easily implies exponential growth. The reverse is not true, in general. Thus, it is natural to ask, for example, whether Baumslag--Solitar groups $\langle\mathscr a, \mathscr b\mid \mathscr b^{-1}\mathscr a\mathscr b=\mathscr a^n\rangle$, $n\in\mathbb N$, have a spectral gap for $1$-Laplacian. These groups are \emph{non-elementary amenable} but have exponential growth. Such groups are not covered by Theorem \ref{th:exp_growth}, and it is still unclear for the author whether $1$-Laplacian has a spectral gap on them.

If we assume the contrary to Theorem \ref{th:exp_growth}, then, first, $\Gamma$ cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic, see, e.g., \cite{Me08}. Second, $\Gamma$ cannot have infinitely many ends because then $\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end.

For $\mathscr L>0$, denote by $\mathbb T_{\mathscr{L}}$ a circle of length $\mathscr L$. On $\mathbb T_{\mathscr L}$, one may measure distances along this loop. To prove Theorem \ref{th:exp_growth}, we need the following

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

In fact, we are able to make $\mathscr L$ arbitrarily large with fixed $x$.

Now, let us briefly recall the proof of Kesten Theorem \ref{th:Kesten} to compare it to our argument. Non-amenability of $\Gamma$ means that
\begin{equation}
\label{eq:isoperimetr}
\|du\|_{\ell^1(E)} \gtrsim \|u\|_{\ell^1(\Gamma)}
\end{equation}
for $u=\mathds 1_E$, $E$ ranges all finite subsets in $\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\colon \Gamma\to{\mathbb R}$.
The spectral gap condition $0\notin \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$ means that
$$
\langle u, u\rangle_{\ell^2(\Gamma)} \lesssim \langle du, du\rangle_{\ell^2(E)}, ~~~ u\colon\Gamma\to{\mathbb R} \mbox{ is finitely supported}.
$$
To obtain this from (\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.

In the first step of the latter argument, we assemble a function $u\colon \Gamma\to{\mathbb R}$, say, non-negative one, from its super-level sets $\mathds 1_{\{u\ge \mathscr t\}}$, $\mathscr t$ ranges $[0,+\infty)$; we also assemble $du$ from $d\mathds 1_{\{u\ge \mathscr t\}}$. (Both decompositions are $\ell^1$-convex.) Thus, in Kesten Theorem, we deal with "sets of codimensions $0$ and $1$". At least, we will have such genuine codimensions in the case of a manifold instead of a group, the corresponding result linking spectra and isoperimetry is known as Cheeger--Yau inequality, see \cite{Cheeger}, \cite{Yau}.

Unlike this, in our argument we work with dimension $1$ sets --- loops, in particular, as in Lemma \ref{lemma:embed}. Also, in Section \ref{sec:spectra_homology} we bound $1$-cycles with $2$-dimensional surfaces.

Notice also that an analogue of Cheeger--Yau inequality for $1$-forms was obtained in \cite{BC22} in the case of manifolds. Coexact $1$-Laplacian spectrum is indeed related to appropriate isoperimetric ratio, namely, to $\sup_{\gamma}\inf_h|h|/\length\gamma$ with $\gamma$ ranging homologicaly trivial loops at a manifold $\mathscr M$ and $h$ be a $2$-dimensional chain in $\mathscr M$ bounding $\gamma$; here, $|h|$ is area of $h$. Some Poincar\'e-type estimates for operator $d$ on coclosed $1$-forms are  possible if isoperimetric ratios as above are bounded from the below. But, in \cite{BC22}, authors impose the condition of finite diameter of $\mathscr M$  which is not our case; also, \cite{BC22} does not deal with effects of negative curvature.

What concerns spaces with negative curvature, let us mention recent works \cite{AAGLZ24}, \cite{R23} devoted to $3$-dimensional hyperbolic manifolds. It turns out that, first, $1$-coexact spectral gap is related to exponential growth of torsion $1$-homology of the manifolds; second, there are relations  between the spectral gap and isoperimetric ratios. The latter estimates from \cite{R23} are also volume-dependent, as in \cite{BC22}.

\medskip

This paper is organized as follows. In Section \ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \ref{th:exp_growth}. In Section~\ref{sec:loop}, we prove Lemma \ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \ref{sec:spectra_homology}, we conclude the proof of Theorem~\ref{th:exp_growth}. This is done by approximating the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$ provided that $0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\gamma$ from Lemma \ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\Cay(\Gamma)$ and also  make use of homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$.  Finally, in Section \ref{sec:examples} we check the most natural examples of Cayley graphs.

\medskip

\noindent {\bf Some notation.} For a set $A$ we denote by $\card A$ the number of its elements.
If $v_1, v_2$ are vertices of some oriented graph then we denote by $\edge(v_1, v_2)$ the oriented edge in the graph under consideration provided that the edge exists. If $e$ is an edge in a oriented graph or $\gamma$ is an oriented path in a metric space then we write $\beg e$, $\beg\gamma$ for their beginnings and $\ennd e$, $\ennd\gamma$ for their endpoints, respectively. The notation $\length\gamma$ is obvious.

We write $\mathcal B_X(x, \rho)$ for the open ball in a metric space $X$ centered in a point $x\in X$ and having radius $\rho\ge 0$.

Assuming the contrary to Theorem~\ref{th:exp_growth}, one first reduces the possibilities for the number of ends: $\Gamma$ "cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic"; it also "cannot have infinitely many ends because then $\Gamma$ has exponential growth". Hence, "by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end."

Under the hypothesis that $\Gamma$ has subexponential growth and one end, one proves Lemma~\ref{lemma:embed} ("on loop embedding"), producing for given $C>20$, $x\ge1$ an injective loop $\gamma\colon\mathbb T_{\mathscr L}\to\Cay(\Gamma)$ with $\mathscr L>2DCx$ and the metric control (\ref{eq:x_bilip}): if $\dist_{\Cay(\Gamma)}(\gamma(t_1),\gamma(t_2))\le x$ then $\dist_{\mathbb T_{\mathscr L}}(t_1,t_2)\le Cx$ (and "we are able to make $\mathscr L$ arbitrarily large with fixed $x$"). The lemma is proved "by dropping lots of geodesic perpendiculars in a branching way."

The proof of Theorem~\ref{th:exp_growth} is then completed in Section~\ref{sec:spectra_homology}: assuming a spectral gap, i.e. "$0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$", one "approximat[es] the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$" and applies this to the "$1$-cochain given by curve $\gamma$ from Lemma~\ref{lemma:embed}$." One also "put[s] some metric control on the approximating process" in $\Cay(\Gamma)$ and uses the "homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$", together with the fact that "we bound $1$-cycles with $2$-dimensional surfaces."

Assuming the contrary to the main theorem, one first reduces the possibilities for the number of ends: $\Gamma$ "cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic"; it also "cannot have infinitely many ends because then $\Gamma$ has exponential growth". Hence, "by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end."

Under the hypothesis that $\Gamma$ has subexponential growth and one end, one proves the following lemma.

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

This produces for given $C>20$, $x\ge1$ an injective loop $\gamma\colon\mathbb T_{\mathscr L}\to\Cay(\Gamma)$ with $\mathscr L>2DCx$ and the metric control given by the displayed equation above (and "we are able to make $\mathscr L$ arbitrarily large with fixed $x$"). The lemma is proved "by dropping lots of geodesic perpendiculars in a branching way."

In completing the proof of the main theorem, assuming a spectral gap, i.e. "$0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$", one "approximat[es] the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$" and applies this to the "$1$-cochain given by curve $\gamma$ from the lemma above." One also "put[s] some metric control on the approximating process" in $\Cay(\Gamma)$ and uses the "homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$", together with the fact that "we bound $1$-cycles with $2$-dimensional surfaces."

\label{th:exp_growth}
	Let $\Gamma$ be a countable finitely presented group.
	If $\Delta_1$ has a spectral gap on 	$\ell_{0,c}^2(E)$ 	then either $\Gamma$ has exponential growth,	 or $\Gamma$ is virtually infinite cyclic.,

Question:
Let \(\Gamma\) be a countable finitely presented group. Choose a finite generating set \(S\), let \(\operatorname{Cay}(\Gamma,S)\) be its Cayley graph with edge set \(E\), and form the simply connected 2-complex \(\operatorname{Cay}^{(2)}(\Gamma)\) by gluing oriented polygons corresponding to all translates of the defining relations. Let \(\Delta_1\) be the nonnegative Hodge--Laplace operator on edge functions determined by this 2-complex, and let
\[
\ell^2_{0,c}(E):=\overline{\{\text{finitely supported closed 1-cochains on }\operatorname{Cay}(\Gamma,S)\}}^{\,\ell^2(E)}.
\]
Assume that \(\Delta_1\) has a spectral gap on \(\ell^2_{0,c}(E)\), meaning that for some \(\varepsilon>0\),
\[
\operatorname{spec}\bigl(\Delta_1\big|_{\ell^2_{0,c}(E)}\bigr)\cap[0,\varepsilon)=\varnothing.
\]
Which conclusion about \(\Gamma\) holds?

Either \(\Gamma\) has exponential growth, or \(\Gamma\) is virtually infinite cyclic (that is, it has a finite-index subgroup isomorphic to \(\mathbb Z\)).

Choices:
B. Either \(\Gamma\) is nonamenable, or \(\Gamma\) is virtually infinite cyclic.
C. \(\Gamma\) is either of exponential growth or virtually cyclic.
D. If \(\Delta_1\) has a spectral gap on \(\ell^2_{0,c}(E)\), then \(\Gamma\) has exponential growth unless it is finite.
E. Either \(\Gamma\) has exponential growth, or \(\Gamma\) has exactly one end.

replacing growth conclusion by amenability-type alternative

replace virtually infinite cyclic by weaker virtually cyclic

exceptional virtually-Z case replaced by finite-case exception

one-end reduction under contradiction promoted to final alternative


\begin{tikzpicture}[samples=100, variable=\t]
  \def\a{4}
  \def\b{1.5}
  \draw[domain=0:2*pi]             plot ({\a*cos(\t r)},{\b*sin(\t r)});
  \draw[domain=pi/6:5*pi/6]        plot ({\a*cos(\t r)},{\b*sin(\t r) - 0.7});
  \draw[domain=-0.3+5*pi/4:0.3+7*pi/4]
                                   plot ({\a*cos(\t r)},{\b*sin(\t r) + 0.8});
  \pgfmathsetmacro{\R}{0.8/(2*pi)}
  \draw[domain=0:2*pi, smooth, gray] plot ({-\R*(1-cos(\t r))}, {-\R*(\t - sin(\t r)) - 0.7});
  \draw[domain=0:2*pi, smooth, gray, dotted]
    plot ({\R*(1 - cos(\t r))},
        {-\R*(\t - sin(\t r)) - 0.7});

ll[blue] (0,-0.7) circle(0.05);
ll[blue] (0,-1.5) circle(0.05);
ll[red] (-0.25,-1.2) circle(0.05);
\end{tikzpicture}
\end{subfigure}
\caption{A highly eccentric ellipse (left) and a thin 2-torus (right). Points of large curvature are marked in red, and the narrowest bottlenecks are indicated in blue.}
\label{ellipse_torus}
\end{figure}

To formalize the above discussion, let $\iota : M \to \mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \emph{scaled oriented Grassmannian}  $(\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\mathbf{Gr}_c^+(D,d)} = c \cdot g^{\mathbf{Gr}^+(D,d)}$. Let $\bar{\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map
$$\bar{\mathbf{g}}^+_c : M \xrightarrow{\bar{\mathbf{g}}^+} (\mathbf{Gr}^+(D,d), g^{\mathbf{Gr}^+(D,d)}) \xrightarrow{ \cdot c } (\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)}).$$
We consider the graph embedding $\iota \times \bar{\mathbf{g}}_c^+ : M \to (\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d),\, g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\iota \times \bar{\mathbf{g}}^+_c)^*(g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\mathbf{II}$ for the second fundamental form of $M \subset \mathbb{R}^D$ with the induced Euclidean metric and $\mathbf{II}_c$ for that of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\mathrm{vol}(M)$ for the Euclidean volume of $M \subset \mathbb{R}^D$.
\begin{definition}\label{grassmannian_distance}
For $c>0$, we define a distance $d_c$ on $M$ by
\[
  d_{c}(q_1,q_2) = d_{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}\bigl((q_1, \bar{\mathbf{g}}_c^+(q_1)),(q_2, \bar{\mathbf{g}}_c^+(q_2))\bigr)
  = \sqrt{\|q_1-q_2\|_{\mathbb{R}^D}^2
      + c\, d_{\mathbf{Gr}^+(D,d)}\bigl(\bar{\mathbf{g}}^+(q_1),\bar{\mathbf{g}}^+(q_2)\bigr)^2},
\]
where $\|\cdot\|_{\mathbb{R}^D}$ denotes the Euclidean norm and
$d_{\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian
$\mathbf{Gr}^+(D,d)$.
\end{definition}

We first show that, by embedding $M$ into $\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \in M$ and a nonzero vector $v \in T_pM$, denote by $\kappa_c(v)$ and $\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ at $(p,\bar{\mathbf{g}}_c^+(p))$ in the direction $(v, d\bar{\mathbf{g}}_c^+(v))$ and of $M \subset \mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.
\begin{theorem}\label{principal_curvature}
  Let $M \subset \mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. Fix $p \in M$ and a unit vector $v \in T_p M$. Assume that the normal curvature at $p$ in the direction $v$ is sufficiently large so that
  \begin{equation*}
    \mathrm{vol}_c(M) \leq \bigl(1 + c \,\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2\bigr)^{d/2} \mathrm{vol}(M),
    \qquad
    \kappa(v) > \frac{\|(\nabla_v^{M} \mathbf{II}) (v,\cdot) \|_{\operatorname{HS}}}{\|\mathbf{II} (v,\cdot)\|_{\operatorname{HS}}},
  \end{equation*}
  where $\|\cdot\|_{\operatorname{HS}}$ denotes the Hilbert--Schmidt norm. Then, for every $c>0$,
  \[
    \kappa_c(v)\, \mathrm{vol}_c(M)^{1/d}
    \;\leq\;
    \frac{\sqrt{\kappa(v)^2 + c\|(\nabla_v^{M} \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}}{1 + c\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2}\,
    \mathrm{vol}_c(M)^{1/d}
    \;<\;
    \kappa(v)\, \mathrm{vol}(M)^{1/d},
  \]
and the middle term in the inequalities is strictly decreasing as $c$ grows.
\end{theorem}
In directions with large normal curvatures, even a small displacement along the curve already produces a large change in the tangent space, so the Grassmannian component of $g_c$ contributes an increasingly large portion of the speed. Since the normal curvature is defined as the size of the normal acceleration divided by the square of this speed, reparameterizing the curve to have unit speed with respect to $g_c$ makes the normal curvature decrease as $c$ grows.

Next, to quantify how distances between bottleneck pairs change with respect to $d_c$, we set
\begin{align*}
  L_c(M)
  &= \min\Bigl\{\frac{1}{2}d_c(q_1,q_2) \;\Big|\; (q_1,q_2) \text{ is a bottleneck of } M \subset \mathbb{R}^D\Bigr\} \\
  &= \min\Bigl\{\frac{1}{2}d_c(q_1,q_2) \;\Big|\; \overline{q_1q_2} \perp T_{q_1}M,\ \overline{q_1q_2} \perp T_{q_2}M\Bigr\}.
\end{align*}
Similarly, in the Euclidean case, we define
\[
  L(M)
  = \min\Bigl\{\frac{1}{2}\|q_1-q_2\|_{\mathbb{R}^D} \;\Big|\; (q_1,q_2) \text{ is a bottleneck of } M \subset \mathbb{R}^D\Bigr\}.
\]
Under a mild curvature hypothesis, we show that for a smooth closed orientable hypersurface $M$, the bottlenecks are separated when distances are measured with $d_c$ from Definition~\ref{grassmannian_distance}.
\begin{theorem}\label{normalized_bottleneck}
  Let $M \subset \mathbb{R}^D$ be a smooth closed orientable hypersurface. Suppose that $L(M) \leq \frac{1}{\|\mathbf{II}\|_2}$.
  Then for every $c \in \bigl(0, \tfrac{12L(M)^2}{\pi^2}\bigr]$,
    \[
    \frac{L_c(M)}{\mathrm{vol}_c(M)^{1/(D-1)}}
    \;\geq\;
    \frac{\sqrt{4L(M)^2 + c\pi^2}}{2\,\mathrm{vol}_c(M)^{1/(D-1)}}
    \;>\;
    \frac{L(M)}{\mathrm{vol}(M)^{1/(D-1)}},
  \]
  and the middle term in the inequalities is strictly increasing in $c$ on the interval $\bigl(0, \tfrac{12L(M)^2}{\pi^2}\bigr]$.
\end{theorem}

We obtain an explicit range of radii $r$ for which the ambient Čech complex $\check{C}(M, \mathbb{R}^D\times\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to $M$. In particular, this yields a lower bound on the lengths of the barcodes in the associated persistent homology, from which the homology of $M$ is recovered from the distance $d_c$.
\begin{theorem}\label{length_barcodes}
Let $M \subset \mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. For $c>0$, the ambient Čech complex $\check{C}(M, \mathbb{R}^D\times\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to \(M\) for all
\[
  r <
  \min\!\left(
    \sqrt{\frac{c}{2}}\arctan\sqrt{\frac{2}{c\|\mathbf{II}_c\|_2}},\,
    \frac{\sqrt{c}\,\pi}{2},\,
    L'_c(M)
  \right),
\]
where
\[
  L'_c(M)
  = \min \Bigl\{\frac{1}{2}d_c(q_1,q_2) \;\Bigm|\;
  \overline{(q_1,\bar{\mathbf{g}}_c^+(q_1))(q_2,\bar{\mathbf{g}}_c^+(q_2))}
  \perp T_{(q_i,\bar{\mathbf{g}}_c^+(q_i))}(M,g_c),\ i=1,2 \Bigr\}.
\]
\end{theorem}

Fix a field $\mathbb{k}$. Since the metric $d_c$ depends both on the ambient positions and on the oriented tangent spaces of $M$, the persistence module $H_j(\mathbb{\check{C}}(M, \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d));\mathbb{k})$ is not expected to be stable with respect to the Hausdorff distance on $\mathbb{R}^D$. Instead, we work with a stronger notion of convergence that also takes into account the distance between tangent spaces.
\begin{theorem}\label{stability}
  Let $\{M_i\}_{i \in \mathbb{N}}$ be a sequence of smooth closed orientable $d$-dimensional submanifolds of $\mathbb{R}^D$ converging to a smooth closed orientable submanifold $M_\infty \subset \mathbb{R}^D$ in the Whitney $C^1$ topology (see Definition~\ref{Whitney}). Fix $c>0$. Then for every field $\mathbb{k}$ and every $j \in \mathbb{N} \cup \{0\}$,
  \[
    \lim_{i \to \infty}
    d_I\Bigl(
      H_j(\mathbb{\check{C}}(M_i, \mathbb{R}^D\times\mathbf{Gr}_c^+(D,d));\mathbb{k}),
      H_j(\mathbb{\check{C}}(M_\infty, \mathbb{R}^D\times\mathbf{Gr}_c^+(D,d));\mathbb{k})
    \Bigr) = 0,
  \]
  where $d_I$ denotes the interleaving distance between persistence modules.
\end{theorem}

Based on the above theorems, we describe a method for computing the distance matrix for a finite subset of $M$ with respect to the distance $d_c$ from Definition~\ref{grassmannian_distance}. To illustrate the performance of this method, we display three computational examples: a time-delay embedded attractor, an approximate quasi-halo orbit in the Saturn--Enceladus system, and a classification of three-dimensional image shapes.

\section{Theoretical background}

\subsection{Persistence theory}

A \emph{filtered space (or filtration)} $\mathbb{X}$ is a collection of topological spaces $\{X_i\}_{i \in \mathbb{R}_{\geq 0}}$ equipped with inclusion maps $\iota_i^j : X_i \to X_j$ for all $i \leq j$. Let \( (X, d) \) be a compact metric space and let \( \varepsilon \in \mathbb{R}_{\geq 0} \). The \emph{Vietoris--Rips complex} \( \mathrm{VR}(X;\varepsilon) \) is the abstract simplicial complex whose simplices are finite subsets \( \sigma \subset X \) such that $d(x, y) < \varepsilon$ for all $x, y \in \sigma$. For a non-decreasing sequence of parameters $\varepsilon_0 \leq \varepsilon_1 \leq \varepsilon_2 \leq \dots$, these complexes form a filtered space
\[
\mathrm{VR}(X;\varepsilon_0) \hookrightarrow \mathrm{VR}(X;\varepsilon_1) \hookrightarrow \mathrm{VR}(X;\varepsilon_2) \hookrightarrow \cdots,
\]
called the \emph{Vietoris--Rips filtration}, denoted by $\mathbb{V}R(X)$. The (intrinsic) \emph{Čech complex} $\check{C}(X;r)$ is the nerve of $r$-balls centered at points of $X$, and the corresponding \emph{Čech filtration} $\mathbb{\check{C}}(X)$ is defined analogously. Let $Y$ be a subset of a metric space $(Z,d_Z)$. The \emph{ambient Čech complex} $\check{C}(Y, Z; r)$ is defined as the nerve of the family of $r$-balls $\{B_Z(y, r)\}_{y \in Y}$ taken in the ambient space $Z$. The corresponding filtration is denoted $\mathbb{\check{C}}(Y, Z)$.

For $n \in \mathbb{N} \cup \{0\}$, a field $\mathbb{k}$ and a filtration $\mathbb{X}= \{X_i\}_{i \in \mathbb{R}_{\geq 0}}$, the \emph{$n$-th persistent homology} $ H_n(\mathbb{X}; \mathbb{k}) $ of $\mathbb{X}$ is a family of vector spaces $H_n(X_i; \mathbb{k})$ with induced linear maps $H_n(\iota_i^j) : H_n(X_i; \mathbb{k}) \to H_n(X_j; \mathbb{k})$ for $i\leq j$. If \( H_n(\mathbb{X}; \mathbb{k}) \) is pointwise finite-dimensional and decomposes as a direct sum of interval modules, then it corresponds to a multiset of points \( (b,d) \subset \mathbb{R}_{\geq 0}^2 \), called the \emph{persistence diagram} in degree \( n \).

For a field $\mathbb{k}$, a \emph{persistence module (over $\mathbb{R}_{\geq 0}$)} $\mathbb{V}$ is a family of vector spaces $\{V_{i}\}_{i \in \mathbb{R}_{\geq 0}}$ equipped with linear maps $v_i^j :V_i \to V_j$ for every $0 \leq i \leq j$, satisfying $v_j^k \circ v_i^j = v_i^k$ whenever $i \leq j \leq k$ and $v_i^i$ is the identity map on $V_i$. A \emph{morphism of degree $\varepsilon$} between two persistence modules $\mathbb{V}$ and $\mathbb{W}$ is a family of linear maps $\Phi = \{\phi_i:V_i\to W_{i+\varepsilon}\}_{i\in\mathbb{R}_{\geq 0}}$ such that for all $i\le j$, it holds that
\[
w_{i+\varepsilon}^{j+\varepsilon}\circ \phi_i \;=\; \phi_j\circ v_i^j.
\]
For $\varepsilon\ge 0$, the \emph{shift} $\Sigma^\varepsilon$ of $\mathbb{V}$ by $\varepsilon$ is
\[
(\Sigma^\varepsilon \mathbb{V})_i=V_{i+\varepsilon},\qquad
(\Sigma^\varepsilon)(v_i^j)=v_{i+\varepsilon}^{j+\varepsilon}:V_{i+\varepsilon} \to V_{j+\varepsilon}.
\]
Two persistence modules $\mathbb{V}$ and $\mathbb{W}$ are \emph{$\varepsilon$-interleaved} if there exist morphisms $\Phi=\{\phi_i : V_i \to W_{i+\varepsilon}\}_{i \in \mathbb{R}_{\geq 0}}$ and $\Psi = \{\psi_i : W_i \to V_{i+\varepsilon}\}_{i \in \mathbb{R}_{\geq 0}}$ of degree $\varepsilon$ such that
\[
\psi_{\,i+\varepsilon}\circ \phi_i \;=\; v_{i}^{i+2\varepsilon},
\qquad
\phi_{\,i+\varepsilon}\circ \psi_i \;=\; w_{i}^{i+2\varepsilon},
\]
for every $i \in \mathbb{R}_{\geq 0}$. The \emph{interleaving distance} $d_I$ between $\mathbb{V}$ and $\mathbb{W}$ is
\[
d_I(\mathbb{V},\mathbb{W})\;=\;\inf\bigl\{\varepsilon\ge 0 \,\big|\, \mathbb{V} \text{ and } \mathbb{W} \text{ are } \varepsilon\text{-interleaved}\bigr\}.
\]
For more details, see e.g.\ \cite{oudot2015persistence, chazal2021introduction}.

\subsection{Basic Riemannian geometry}

In this subsection, we refer to \cite{do1992riemannian} and \cite{lee2018introduction}. Let $M$ be a smooth compact $n$-dimensional manifold. For a choice of a smooth section $g \in \Gamma(M; \mathrm{Sym}^2 T^*M)$, a pair $(M,g)$ is called a \emph{Riemannian manifold} if for each $p \in M$, the fiberwise bilinear map $g_p : T_pM \times T_pM \to \mathbb{R}$ is positive definite. The section $g$ is called a \emph{(Riemannian) metric} on $M$.  We call the map
$$\nabla : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M); \quad (X,Y) \mapsto \nabla_X Y$$
an \emph{affine connection} if $\nabla_X Y$ is linear over $C^\infty (M)$ in $X$ and linear over $\mathbb{R}$ in $Y$, satisfying $\nabla_X(fY) = f(\nabla_X Y) + (Xf)Y$ for any $f \in C^\infty(M)$. The \emph{Levi--Civita connection} is the unique affine connection $\nabla$ on $(M,g)$ that satisfies $Z(g(X,Y)) =g(\nabla_Z X, Y) + g(X,\nabla_Z Y)$ and $\nabla_X Y - \nabla_Y X = [X,Y]$ for any $X,Y,Z \in \mathfrak{X}(M).$

For a Riemannian manifold $(M,g)$ with Levi--Civita connection $\nabla$, the \emph{Riemann curvature tensor} $R$ is defined by $R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z- \nabla_{[X,Y]}Z$ for $X,Y,Z \in \mathfrak{X}(M)$. For a tangent $2$-plane $\Sigma_pM \subset T_pM$ with an orthonormal basis $\{u,v\}$, the \emph{sectional curvature} of $\Sigma_p M$ is defined by $K_p(u,v)=g_p(R(u,v)v,u).$

\subsection{Convergence theory}
We refer to \cite{gromov2007metric, hirsch2012differential} in this subsection. Let $(Z,d_Z)$ be a metric space and let $A,B\subset Z$ be nonempty compact subsets. The \emph{Hausdorff distance} between $A$ and $B$ is defined by
\[
d_H^Z(A,B)
=\max\Big\{\,\sup_{a\in A}\inf_{b\in B} d_Z(a,b)\;,\;
\sup_{b\in B}\inf_{a\in A} d_Z(a,b)\,\Big\}.
\]
The following stability theorem establishes the continuity of persistent homology of the ambient Čech complex with respect to the Hausdorff distance.
\begin{theorem}[{\cite[Theorem~5.6]{chazal2014persistence}}]\label{stability_theorem} Let $A,B$ be compact subsets of a metric space $Z$. Then for every $j \in \mathbb{N} \cup \{0\}$ and every field $\mathbb{k}$,
\[
d_I\bigl(H_j(\mathbb{\check{C}}(A, Z); \mathbb{k}), H_j(\mathbb{\check{C}}(B, Z); \mathbb{k})\bigr)
\leq d_H^Z(A,B).
\]
\end{theorem}

Let $(N,g_N)$ be a fixed Riemannian manifold, and let $\{M_i\}_{i\in\mathbb{N}}$ be a family of smooth closed submanifolds of $N$ such that $M_i = F_i(M)$ for every $i \in \mathbb{N}$, where $M$ is a fixed smooth closed manifold and each $F_i : M \to N$ is a smooth embedding.
\begin{definition}[{\cite[Chapter~2]{hirsch2012differential}}]\label{Whitney}
We say that the family of submanifolds $\{M_i\}_{i\in\mathbb{N}}$ \emph{converges to $M_\infty$ in the (Whitney) $C^k$ topology} if there exists an embedding
$$
F_\infty : M \to N, \qquad M_\infty = F_\infty(M),
$$
such that $F_i \to F_\infty$ in the Whitney $C^k$ topology on $C^k(M,N)$. That is, there exist locally finite atlases
$\{(U_\alpha,\varphi_\alpha)\}_{\alpha}$ of $M$ and $\{(V_\alpha,\psi_\alpha)\}_{\alpha}$ of $N$ with $F_\infty(U_\alpha) \subset V_\alpha$ such that, for each $\alpha$ and each $\varepsilon_\alpha > 0$, there exists $n_\alpha \in \mathbb{N}$ with the property that for every $i \geq n_\alpha$,
\[
\max_{|\beta|\le k}
\sup_{x \in \varphi_\alpha(U_\alpha)}
\left\|
D^\beta\big(\psi_\alpha \circ F_i \circ \varphi_\alpha^{-1}\big)(x)
-
D^\beta\big(\psi_\alpha \circ F_\infty \circ \varphi_\alpha^{-1}\big)(x)
\right\|_2
< \varepsilon_\alpha.
\]
\end{definition}

\subsection{Geometry of an embedded submanifold}

Let $M \subset \mathbb{R}^D$ be a smooth closed $d$-dimensional submanifold with $d < D$. Two distinct points $q_1,q_2 \in M$ form a \emph{bottleneck} in $\mathbb{R}^D$ if the line segment $\overline{q_1 q_2}$ is orthogonal to both tangent spaces $T_{q_1}M$ and $T_{q_2}M$. The \emph{width} of such a bottleneck is defined as $\tfrac{1}{2}\|q_1-q_2\|_{\mathbb{R}^D}$. This notion should not be confused with the bottleneck distance between persistence modules
or with bottlenecks in graph theory.

Define the (Euclidean) \emph{reach} of $M \subset \mathbb{R}^D$ by
\[
\textsf{rch}_{\mathbb{R}^D}(M)
= \sup \Bigl\{ r \geq 0 \,\Big|\,
\text{every } p \in \mathbb{R}^D \text{ with } d_{\mathbb{R}^D}(p,M)< r
\text{ admits a unique nearest point projection onto } M \Bigr\}.
\]
\begin{theorem}[{\cite[Theorem~3.4]{aamari2019estimating}, \cite[Theorem~7.8]{breiding2024metric}}]\label{reach}
    Suppose that the reach of a closed submanifold $M \subset \mathbb{R}^D$ is $\tau>0$. Then at least one of the following holds.
    \begin{itemize}
        \item There exist a point $q \in M$ and a unit-speed geodesic $\gamma : (-\varepsilon, \varepsilon) \to M$ for some $\varepsilon>0$ such that $\gamma(0)=q$ and $|\gamma''(0)| = \frac{1}{\tau}$.
        \item There exist distinct points $q_1, q_2 \in M$ forming a bottleneck of $M \subset \mathbb{R}^D$ such that $\|q_1 - q_2\|_{\mathbb{R}^D} = 2\tau$.
    \end{itemize}
    In particular, the reach $\tau$ of $M$ is realized either as the reciprocal of the normal curvature of a geodesic or as the width of a bottleneck pair.
\end{theorem}

Suppose a $d$-dimensional smooth compact manifold $M$ is embedded in a closed Riemannian manifold $(N,g)$. Denote the Levi--Civita connection of $N$ by $\nabla$ and the normal bundle of $M$ by $\nu$. The \emph{second fundamental form} of $M$ is defined to be the tensor $\mathbf{II} : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \Gamma(\nu)$ given by $\mathbf{II}(X,Y) = (\nabla_X Y)^{\perp}$, where the map $(\cdot)^{\perp}$ denotes the orthogonal projection onto $\nu$. The \emph{normal exponential map} $\exp_\nu : \nu \to N$ is defined by
$$ \exp_\nu(q,v) = \exp_q(v),\quad (q,v) \in \nu.$$
For $p \in M$ and a nonzero vector $v \in T_pM$, the \emph{(norm of the) normal curvature} $\kappa(v)$ at $p$ in the direction $v$ is
\[
\kappa(v)
=
\sup_{\substack{w \in \nu_p \\ \|w\|_2=1}}
\frac{g_p\bigl(\mathbf{II}_p(v,v),\, w\bigr)}{\|v\|_2^2}.
\]
The \emph{operator norm} (or \emph{2-norm}) of $\mathbf{II}_p$ is
\[
\|\mathbf{II}_p\|_2
=
\sup_{\substack{v \in T_pM \\ \|v\|_2=1}}
\kappa(v)
=
\sup_{\substack{v \in T_pM \\ \|v\|_2=1}}
\bigl\|\mathbf{II}_p(v,v)\bigr\|_2.
\]
Denote $\|\mathbf{II}\|_2 = \sup_{p \in M} \|\mathbf{II}_p\|_2$. For two linear operators $A_p,B_p : T_pM \to \nu_p$ and an orthonormal basis $\{e_i\}_{i=1}^d$ of $T_pM$, the \emph{Hilbert--Schmidt inner product} between them is
$$\langle A_p, B_p \rangle_{\operatorname{HS}} =\sum_{i=1}^d g_p\bigl(A_p(e_i), B_p(e_i)\bigr).$$
Denote by $\|\cdot\|_{\operatorname{HS}}$ the induced norm.

\begin{definition}[{\cite[Definition~2.5]{prasad2023cut}, \cite[Definitions~12 and 13]{attali2022tight}}]
Let $S$ be a closed subset embedded in a closed Riemannian manifold $N$. Denote the distance function on $N$ by $d_N$ and the length of a curve $\gamma \subset N$ by $\mathrm{len}_N(\gamma)$. We define a geodesic $\gamma : [0,T] \to N$ to be ($S$-)\emph{distance-minimal} if $\mathrm{len}_N(\gamma|_{[0,t]}) = d_N(S,\gamma(t))$ for all $t \in [0,T]$. The \emph{cut locus} of $S$, denoted by $\mathsf{Cu}_N(S)$, is the set of points $p \in N$ for which there exists a distance-minimal geodesic $\gamma$ from $S$ to $p$ such that any extension of $\gamma$ beyond its endpoint $p$ is not distance-minimal. The \emph{(cut locus) reach} (or \emph{normal injectivity radius}) of $S$ in $N$ is
$$\mathsf{rch}_N(S) = \inf\{ d_N(p,q) \mid q \in S,\ p \in \mathsf{Cu}_N(S)\}.$$
\end{definition}
If $S$ is a smooth closed submanifold, then $\mathsf{rch}_N(S)$ is the supremum of all $\varepsilon>0$ such that the restriction of the normal exponential map
\[
\exp_\nu : \{(p,v)\in \nu \mid \|v\|_2<\varepsilon\} \longrightarrow N
\]
is an embedding. We give an analogue of Theorem~\ref{reach} below.
\begin{theorem}[{\cite[Section~2]{singh1988closest}, \cite[Lemma~A.2]{basu2023connection}}]\label{cutlocus}
    Let $M$ be a smooth closed submanifold of a Riemannian manifold $(N,g)$, and suppose $\mathsf{rch}_N(M)=T>0$. Denote the normal bundle of $M$ by $\nu$ and its unit normal bundle by $S(\nu)$. Then one of the following holds (see Figure~\ref{cutlocus_figure}):
    \begin{itemize}
        \item[\textnormal{(focal)}] There exists a pair $(p,v) \in S(\nu)$ such that the differential of the normal exponential map $d(\exp_\nu)_{(p,Tv)}$ is not of full rank, whereas $d(\exp_\nu)_{(p',tv')}$ has full rank for every $(p',v') \in S(\nu)$ and every $t$ with $0 <t<T$. In this case, the point $\exp_\nu(Tv)$ is called a \emph{first focal point} and $T$ a \emph{first focal time}.
        \item[\textnormal{(non-focal)}] There exist at least two distinct unit-speed distance-minimal geodesics $\gamma_1, \gamma_2 : [0,T] \to N$ from (not necessarily distinct) two points in $M$ satisfying $\gamma_1(T) = \gamma_2(T)$ and $ \gamma_i'(0) \perp M$ for $i=1,2$.
    \end{itemize}
\end{theorem}

\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[scale=1.5]

  \begin{scope}[yshift=-0.8cm]
    \draw[thick,black,domain=-1.1:1.1,samples=200] plot (\x,{\x*\x});
ll[red] (0,0.5) circle(0.03);
ll[blue] (0,0) circle(0.03);
    \draw[gray,->] (0,0) -- (0,0.5);

    \pgfmathsetmacro{\aone}{0.12}
    \pgfmathsetmacro{\atwo}{-0.10}
    \pgfmathsetmacro{\xone}{\aone}
    \pgfmathsetmacro{\yone}{\aone*\aone}
    \pgfmathsetmacro{\xtwo}{\atwo}
    \pgfmathsetmacro{\ytwo}{\atwo*\atwo}

ll[blue] (\xone,\yone) circle(0.03);
ll[blue] (\xtwo,\ytwo) circle(0.03);
    \draw[gray,->] (\xone,\yone) -- (0,0.5);
    \draw[gray,->] (\xtwo,\ytwo) -- (0,0.5);
  \end{scope}

  \begin{scope}[xshift=3.2cm]
    \def\az{25}\def\el{35}
    \pgfmathsetmacro{\caz}{cos(\az)} \pgfmathsetmacro{\saz}{sin(\az)}
    \pgfmathsetmacro{\cel}{cos(\el)} \pgfmathsetmacro{\sel}{sin(\el)}

    \draw[lightgray] (0,0) circle (1);

    \def\ang{30}\def\hw{16}

    \pgfmathsetmacro{\cang}{cos(\ang)} \pgfmathsetmacro{\sang}{sin(\ang)}
    \pgfmathsetmacro{\PX}{\cang*\caz - \sang*\saz}
    \pgfmathsetmacro{\PY}{(\cang*\saz + \sang*\caz)*\cel}
    \pgfmathsetmacro{\QX}{-\PX} \pgfmathsetmacro{\QY}{-\PY}
ll[blue] (\PX,\PY) circle (0.03);
ll[red]  (\QX,\QY) circle (0.03);

    \draw[gray,domain=0:360,samples=360,variable=\x]
      plot ({cos(\x)*(\cang*\caz - \sang*\saz)},
            {cos(\x)*(\cang*\saz + \sang*\caz)*\cel - sin(\x)*\sel});

    \draw[gray,->]
      ({cos(70)*(\cang*\caz - \sang*\saz)},
       {cos(70)*(\cang*\saz + \sang*\caz)*\cel - sin(70)*\sel})
      --
      ({cos(100)*(\cang*\caz - \sang*\saz)},
       {cos(100)*(\cang*\saz + \sang*\caz)*\cel - sin(100)*\sel});

    \draw[gray,->]
      ({cos(270)*(\cang*\caz - \sang*\saz)},
       {cos(270)*(\cang*\saz + \sang*\caz)*\cel - sin(270)*\sel})
      --
      ({cos(240)*(\cang*\caz - \sang*\saz)},
       {cos(240)*(\cang*\saz + \sang*\caz)*\cel - sin(240)*\sel});
  \end{scope}

  \begin{scope}[xshift=6.4cm,scale=0.7]
    \draw[thick,black,domain=-1.3:1.3,samples=300]
      plot ({ sqrt(1+\x*\x) },{\x});
    \draw[thick,black,domain=-1.3:1.3,samples=300]
      plot ({-sqrt(1+\x*\x) },{\x});

ll[blue] ( 1,0) circle(0.03);
ll[blue] (-1,0) circle(0.03);
ll[red] (0,0) circle(0.03);

    \draw[gray,->] ( 1,0) -- (0,0);
    \draw[gray,->] (-1,0) -- (0,0);
  \end{scope}
\end{tikzpicture}

\emph{Persistent homology} computes barcodes of filtered chain complexes constructed from a metric space $X$ \cite{zomorodian2004computing, oudot2015persistence}. For $r>0$, the \emph{\v{C}ech complex} \(\check{C}(X;r)\) is the nerve of the open cover of $r$-balls $\{B_X(x,r)\}_{x\in X}$, and the \emph{Vietoris--Rips complex} $\mathrm{VR}(X;r)$ is the abstract simplicial complex whose simplices are the finite subsets $\sigma\subset X$ with $\operatorname{diam}(\sigma)<r$. If $M$ is a Riemannian manifold, then the Čech complex $\check{C}(M;r)$ is homotopy equivalent to $M$ for $r<r_{\operatorname{conv}}(M)$ by the nerve lemma, while the systole and the filling radius bound the range of $r$ for which the Vietoris--Rips complex $\mathrm{VR}(M;r)$ has the same first and top-dimensional homology groups as $M$ \cite{virk20201, lim2024vietoris, balitskiy2025geometric}.

The \emph{reach} (or \emph{normal injectivity radius}) $\mathsf{rch}_{\mathbb{R}^D}(M)$ of a smooth submanifold $M \subset \mathbb{R}^D$ is the supremum of $\tau>0$ such that every point $x \in \mathbb{R}^D$ with $d_{\mathbb{R}^D}(x,M)<\tau$ has a unique nearest point in $M$ \cite{federer1959curvature, aamari2019estimating}. For smooth embedded submanifolds, this agrees with the supremum of radii for which the normal exponential map is injective. The reach of $M$ gives bounds on the scales $r$ where the Vietoris--Rips or Čech complex built from sufficiently dense points of $M$ in the ambient Euclidean metric \cite{niyogi2008finding, kim2019homotopy}, and the \emph{metric thickening} of $M$ \cite{adams2019metric, adams2022metric, adams2024persistent}, the 1-Wasserstein analogue of the Vietoris--Rips complex, are homotopy equivalent to $M$.

Aamari et al. \cite{aamari2019estimating} showed that the reach of a submanifold $M \subset \mathbb{R}^D$ can be expressed as the minimum of the reciprocal of the \emph{maximal normal curvature} of $M$ and the minimal width of its \emph{bottlenecks}, pairs of points in $M$ whose connecting line segment is orthogonal to both tangent spaces. For example, for a highly eccentric ellipse in $\mathbb{R}^2$ or a thin $2$-torus with small systole in $\mathbb{R}^3$ as in Figure~\ref{ellipse_torus}, the normal exponential map ceases to be injective at a radius comparable to the reach, since the points of large curvature (marked in red) induce focal points and the bottlenecks (marked in blue) lie close to each other in the Euclidean metric.

In both examples, the indicated bottlenecks have opposite normal directions. This suggests that, for an orientable submanifold $M \subset \mathbb{R}^D$, embedding $M$ into an augmented ambient space, such as the \emph{oriented Grassmannian bundle} $\mathbb{R}^D \times \mathbf{Gr}^+(D,d)$, can enlarge the normal injectivity radius of $M$. In the product metric on $\mathbb{R}^D \times \mathbf{Gr}^+(D,d)$, the contribution from the Grassmannian factor enlarges the lengths of curves near points of large normal curvature, so that unit-speed geodesics have smaller normal curvature, and bottleneck pairs with opposite normal directions are pushed farther apart because their oriented tangent spaces are far from each other in $\mathbf{Gr}^+(D,d)$.

\begin{figure}
\centering
\begin{subfigure}[t]{0.47\textwidth}
\centering
\begin{tikzpicture}
  \draw[thick] (0,0) ellipse [x radius=3.0cm, y radius=0.5cm];
ll[blue] (0,0.5) circle(0.05);
ll[blue] (0,-0.5) circle(0.05);
ll[red] (3.0,0) circle(0.05);
\end{tikzpicture}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.47\textwidth}
\centering
\begin{tikzpicture}[samples=100, variable=\t]
  \def\a{4}
  \def\b{1.5}
  \draw[domain=0:2*pi]             plot ({\a*cos(\t r)},{\b*sin(\t r)});
  \draw[domain=pi/6:5*pi/6]        plot ({\a*cos(\t r)},{\b*sin(\t r) - 0.7});
  \draw[domain=-0.3+5*pi/4:0.3+7*pi/4]
                                   plot ({\a*cos(\t r)},{\b*sin(\t r) + 0.8});
  \pgfmathsetmacro{\R}{0.8/(2*pi)}
  \draw[domain=0:2*pi, smooth, gray] plot ({-\R*(1-cos(\t r))}, {-\R*(\t - sin(\t r)) - 0.7});
  \draw[domain=0:2*pi, smooth, gray, dotted]
    plot ({\R*(1 - cos(\t r))},
        {-\R*(\t - sin(\t r)) - 0.7});

ll[blue] (0,-0.7) circle(0.05);
ll[blue] (0,-1.5) circle(0.05);
ll[red] (-0.25,-1.2) circle(0.05);
\end{tikzpicture}
\end{subfigure}
\caption{A highly eccentric ellipse (left) and a thin 2-torus (right). Points of large curvature are marked in red, and the narrowest bottlenecks are indicated in blue.}
\label{ellipse_torus}
\end{figure}

To formalize the above discussion, let $\iota : M \to \mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \emph{scaled oriented Grassmannian}  $(\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\mathbf{Gr}_c^+(D,d)} = c \cdot g^{\mathbf{Gr}^+(D,d)}$. Let $\bar{\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map
$$\bar{\mathbf{g}}^+_c : M \xrightarrow{\bar{\mathbf{g}}^+} (\mathbf{Gr}^+(D,d), g^{\mathbf{Gr}^+(D,d)}) \xrightarrow{ \cdot c } (\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)}).$$
We consider the graph embedding $\iota \times \bar{\mathbf{g}}_c^+ : M \to (\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d),\, g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\iota \times \bar{\mathbf{g}}^+_c)^*(g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\mathbf{II}$ for the second fundamental form of $M \subset \mathbb{R}^D$ with the induced Euclidean metric and $\mathbf{II}_c$ for that of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\mathrm{vol}(M)$ for the Euclidean volume of $M \subset \mathbb{R}^D$.
\begin{definition}\label{grassmannian_distance}
For $c>0$, we define a distance $d_c$ on $M$ by
\[
  d_{c}(q_1,q_2) = d_{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}\bigl((q_1, \bar{\mathbf{g}}_c^+(q_1)),(q_2, \bar{\mathbf{g}}_c^+(q_2))\bigr)
  = \sqrt{\|q_1-q_2\|_{\mathbb{R}^D}^2
      + c\, d_{\mathbf{Gr}^+(D,d)}\bigl(\bar{\mathbf{g}}^+(q_1),\bar{\mathbf{g}}^+(q_2)\bigr)^2},
\]
where $\|\cdot\|_{\mathbb{R}^D}$ denotes the Euclidean norm and
$d_{\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian
$\mathbf{Gr}^+(D,d)$.
\end{definition}

We first show that, by embedding $M$ into $\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \in M$ and a nonzero vector $v \in T_pM$, denote by $\kappa_c(v)$ and $\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ at $(p,\bar{\mathbf{g}}_c^+(p))$ in the direction $(v, d\bar{\mathbf{g}}_c^+(v))$ and of $M \subset \mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.

\emph{Persistent homology} computes barcodes of filtered chain complexes constructed from a metric space $X$ \cite{zomorodian2004computing, oudot2015persistence}. For $r>0$, the \emph{\v{C}ech complex} \(\check{C}(X;r)\) is the nerve of the open cover of $r$-balls $\{B_X(x,r)\}_{x\in X}$, and the \emph{Vietoris--Rips complex} $\mathrm{VR}(X;r)$ is the abstract simplicial complex whose simplices are the finite subsets $\sigma\subset X$ with $\operatorname{diam}(\sigma)<r$. If $M$ is a Riemannian manifold, then the Čech complex $\check{C}(M;r)$ is homotopy equivalent to $M$ for $r<r_{\operatorname{conv}}(M)$ by the nerve lemma, while the systole and the filling radius bound the range of $r$ for which the Vietoris--Rips complex $\mathrm{VR}(M;r)$ has the same first and top-dimensional homology groups as $M$ \cite{virk20201, lim2024vietoris, balitskiy2025geometric}.

The \emph{reach} (or \emph{normal injectivity radius}) $\mathsf{rch}_{\mathbb{R}^D}(M)$ of a smooth submanifold $M \subset \mathbb{R}^D$ is the supremum of $\tau>0$ such that every point $x \in \mathbb{R}^D$ with $d_{\mathbb{R}^D}(x,M)<\tau$ has a unique nearest point in $M$ \cite{federer1959curvature, aamari2019estimating}. For smooth embedded submanifolds, this agrees with the supremum of radii for which the normal exponential map is injective. The reach of $M$ gives bounds on the scales $r$ where the Vietoris--Rips or Čech complex built from sufficiently dense points of $M$ in the ambient Euclidean metric \cite{niyogi2008finding, kim2019homotopy}, and the \emph{metric thickening} of $M$ \cite{adams2019metric, adams2022metric, adams2024persistent}, the 1-Wasserstein analogue of the Vietoris--Rips complex, are homotopy equivalent to $M$.

Aamari et al. \cite{aamari2019estimating} showed that the reach of a submanifold $M \subset \mathbb{R}^D$ can be expressed as the minimum of the reciprocal of the \emph{maximal normal curvature} of $M$ and the minimal width of its \emph{bottlenecks}, pairs of points in $M$ whose connecting line segment is orthogonal to both tangent spaces. For example, for a highly eccentric ellipse in $\mathbb{R}^2$ or a thin $2$-torus with small systole in $\mathbb{R}^3$ as in Figure~\ref{ellipse_torus}, the normal exponential map ceases to be injective at a radius comparable to the reach, since the points of large curvature (marked in red) induce focal points and the bottlenecks (marked in blue) lie close to each other in the Euclidean metric.

In both examples, the indicated bottlenecks have opposite normal directions. This suggests that, for an orientable submanifold $M \subset \mathbb{R}^D$, embedding $M$ into an augmented ambient space, such as the \emph{oriented Grassmannian bundle} $\mathbb{R}^D \times \mathbf{Gr}^+(D,d)$, can enlarge the normal injectivity radius of $M$. In the product metric on $\mathbb{R}^D \times \mathbf{Gr}^+(D,d)$, the contribution from the Grassmannian factor enlarges the lengths of curves near points of large normal curvature, so that unit-speed geodesics have smaller normal curvature, and bottleneck pairs with opposite normal directions are pushed farther apart because their oriented tangent spaces are far from each other in $\mathbf{Gr}^+(D,d)$.

To formalize the above discussion, let $\iota : M \to \mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \emph{scaled oriented Grassmannian} $(\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\mathbf{Gr}_c^+(D,d)} = c \cdot g^{\mathbf{Gr}^+(D,d)}$. Let $\bar{\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map
$$\bar{\mathbf{g}}^+_c : M \xrightarrow{\bar{\mathbf{g}}^+} (\mathbf{Gr}^+(D,d), g^{\mathbf{Gr}^+(D,d)}) \xrightarrow{ \cdot c } (\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)}).$$
We consider the graph embedding $\iota \times \bar{\mathbf{g}}_c^+ : M \to (\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d),\, g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\iota \times \bar{\mathbf{g}}^+_c)^*(g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\mathbf{II}$ for the second fundamental form of $M \subset \mathbb{R}^D$ with the induced Euclidean metric and $\mathbf{II}_c$ for that of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\mathrm{vol}(M)$ for the Euclidean volume of $M \subset \mathbb{R}^D$.
\begin{definition}\label{grassmannian_distance}
For $c>0$, we define a distance $d_c$ on $M$ by
\[
 d_{c}(q_1,q_2) = d_{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}\bigl((q_1, \bar{\mathbf{g}}_c^+(q_1)),(q_2, \bar{\mathbf{g}}_c^+(q_2))\bigr)
 = \sqrt{\|q_1-q_2\|_{\mathbb{R}^D}^2
 + c\, d_{\mathbf{Gr}^+(D,d)}\bigl(\bar{\mathbf{g}}^+(q_1),\bar{\mathbf{g}}^+(q_2)\bigr)^2},
\]
where $\|\cdot\|_{\mathbb{R}^D}$ denotes the Euclidean norm and
$d_{\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian
$\mathbf{Gr}^+(D,d)$.
\end{definition}

We first show that, by embedding $M$ into $\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \in M$ and a nonzero vector $v \in T_pM$, denote by $\kappa_c(v)$ and $\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ at $(p,\bar{\mathbf{g}}_c^+(p))$ in the direction $(v, d\bar{\mathbf{g}}_c^+(v))$ and of $M \subset \mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.

\begin{proof}[Proof of Theorem~\ref{principal_curvature}]
Denote by $\nabla^c$ and $\nu_c$ the Levi--Civita connection on $\mathbb{R}^D \times \mathbf{Gr}^+_c(D,d)$ and the normal bundle of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$, respectively. Suppose that there exists a unit vector $(w - cS(Q),Q)$ of $\nu_c$ at $p$ such that
\[
\|w\|_{2}^2 + c^2\|S(Q)\|_{2}^2 + c\|Q\|_{\operatorname{HS}}^2 = 1,
\]
as in Lemma~\ref{normal_bundle}, where $w \perp T_pM$. By Lemma~\ref{grass_connection},
\[
\nabla^c_{(v,d\bar{\mathbf{g}}_c^+(v))}(v,d\bar{\mathbf{g}}_c^+(v))
= \bigl(\nabla_{v}^{\mathbb{R}^D} v,\,
\nabla^{\mathbf{Gr}_c^+(D,d)}_{d\bar{\mathbf{g}}_c^+(v)} d\bar{\mathbf{g}}_c^+(v)\bigr)
= \bigl(\nabla_{v}^M v + \mathbf{II}(v,v),\, (\nabla_{v}^M \mathbf{II})(v,\cdot) + \mathbf{II}(\nabla_{v}^M v,\cdot)\bigr).
\]
Since
\[
d(\iota \times \bar{\mathbf{g}}_c^+)_p(\nabla_{v}^M v)
= \bigl(\nabla_{v}^M v,\, \mathbf{II}(\nabla_{v}^M v,\cdot)\bigr),
\]
the projections of $\nabla^c_{(v,d\bar{\mathbf{g}}_c^+(v))}(v,d\bar{\mathbf{g}}_c^+(v))$ and $\bigl(\mathbf{II}(v,v),(\nabla_{v}^M \mathbf{II})(v,\cdot)\bigr)$ onto $\nu_c$ coincide.
On the other hand,
\[
g^{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}\bigl((w,0),(\mathbf{II}(v,v),(\nabla_{v}^M \mathbf{II})(v,\cdot))\bigr)
= \bigl\langle w,\mathbf{II}(v,v)\bigr\rangle_{\mathbb{R}^D},
\]
and since $S(Q) \perp T_pM$,
\[
g^{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}\bigl((-cS(Q),Q),(\mathbf{II}(v,v),(\nabla_{v}^M \mathbf{II})(v,\cdot))\bigr)
= c \bigl\langle Q,(\nabla_{v}^M \mathbf{II})(v,\cdot)\bigr\rangle_{\operatorname{HS}}.
\]
Therefore,
\begin{flalign*}
 |g^{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}((w-cS(Q),Q), (\mathbf{II}({v},{v}) ,(\nabla_{v}^M \mathbf{II})({v},\cdot)))| &= |\langle w, \mathbf{II}(v,v) \rangle_{\mathbb{R}^D} + c\langle Q, (\nabla_{v}^M \mathbf{II})({v},\cdot) \rangle_{\operatorname{HS}}|
 \\ &\leq (\kappa(v) \|w\|_2 + c\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}\|Q\|_{\operatorname{HS}})
 \\ &\leq \sqrt{\kappa(v)^2 + c\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}\sqrt{\|w\|_2^2 + c\|Q\|_{\operatorname{HS}}^2}
 \\ &\leq \sqrt{\kappa(v)^2 + c \|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2},
\end{flalign*}
where we used $\|w\|_2^2 + c^2\|S(Q)\|_2^2 + c\|Q\|_{\operatorname{HS}}^2 = 1$ in the last inequality. Hence,
$$\kappa_c(v)
\leq \frac{\sqrt{\kappa(v)^2 + c\|(\nabla_v^M\mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}}{\|(v,d\bar{\mathbf{g}}_c^+(v))\|_2^2}
= \frac{\sqrt{\kappa(v)^2 + c\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}}{1 + c\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2}.$$
By the volume assumption in Theorem~\ref{principal_curvature}, we obtain
\begin{flalign*}
\kappa_c(v)\,(\mathrm{vol}_c(M))^{1/d}
&\leq \frac{\sqrt{\kappa(v)^2 + c\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}}{\sqrt{1 + c\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2}}\,
(\mathrm{vol}(M))^{1/d}.
\end{flalign*}
We compute
\begin{flalign*}
\frac{d}{dc}\left(\frac{\kappa(v)^2 + c\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}{1 + c\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2}\right)
&= \frac{\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2 - \kappa(v)^2\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2}
{(1 + c\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2)^2} < 0,
\end{flalign*}
by the assumption. This completes the proof.
\end{proof}

\begin{remark}
In codimension one, if $\{e_i\}_{i=1}^{D-1}$ is an orthonormal eigenbasis of $T_qM$ with principal curvatures
$\lambda_1(q),\dots,\lambda_{D-1}(q)$, then in this basis, the metric $g_c$ satisfies
\[
 [(g_c)_q] = I + c H_q,\qquad
 H_q = \operatorname{diag}(\lambda_1(q)^2,\dots,\lambda_{D-1}(q)^2),
\]
so that $\operatorname{tr}H_q = \sum_{i=1}^{D-1} \lambda_i(q)^2$. Therefore, for small $c>0$, we obtain
\[
 d\mathrm{vol}_c
 = \sqrt{\det(I + cH_q)}\,d\mathrm{vol}
 = \Bigl(1 + \frac{c}{2}\,\sum_{i=1}^{D-1} \lambda_i(q)^2 + O(c^2)\Bigr)\,d\mathrm{vol}.
\]
In particular, regions where many principal curvatures are simultaneously large
contribute most to the first-order increase of $\mathrm{vol}_c(M)$. On the other hand, for any $p\in M$ and any unit vector $v \in T_pM$, the volume condition in Theorem~\ref{principal_curvature} states
\[
 \mathrm{vol}_c(M) \;\leq\; \bigl(1 + c \,\|\mathbf{II}_p(v,\cdot)\|_{\operatorname{HS}}^2\bigr)^{(D-1)/2} \mathrm{vol}(M) = \Bigl(1 + \frac{c\,(D-1)}{2}\|\mathbf{II}_p(v,\cdot)\|_{\operatorname{HS}}^2 + O(c^2)\Bigr)\,\mathrm{vol}(M).
\]
Thus, for small $c$, one may interpret this condition as requiring that at points $p$ and directions $v$ the normal curvature is significantly larger than the average curvature of $M$. In higher codimension, the analogous condition may require such a largeness assumption for all sections of the normal bundle.
\end{remark}

The condition
$\kappa(v) > \frac{\|(\nabla_v^M \mathbf{II})(v,\cdot) \|_{\operatorname{HS}}}{\|\mathbf{II} (v,\cdot)\|_{\operatorname{HS}}}$ in Theorem~\ref{principal_curvature}
indicates that the normal curvature of $M$ in the direction $v$ dominates the logarithmic variation of the second fundamental form in the same direction.
\begin{theorem}\label{log_II_derivative_bound}
 Let $M \subset \mathbb{R}^D$ be a smooth $d$-dimensional submanifold. For a unit-speed geodesic $\gamma \subset M$ such that $\gamma'(t)=v(t)$, suppose that the tensor $\mathbf{II}_{\gamma(t)}(v(t),\cdot)$ does not vanish along $\gamma$. Then
\[
\left|
\frac{d}{dt}\log \|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}
\right|
\le
\frac{
\bigl\|(\nabla_v^M \mathbf{II})(v,\cdot)\bigr\|_{\operatorname{HS}}
}{
\bigl\|\mathbf{II}(v,\cdot)\bigr\|_{\operatorname{HS}}
}.
\]
\end{theorem}
The proof is given in Appendix~\ref{proof_log_II}.

For each $x \in M$, fix an ordered orthonormal basis for $T_xM$ compatible with the given
orientation on $M$, and let $B_x$ be the matrix whose columns are this basis. Let
$\widehat{T_xM}$ be the estimated tangent space at $x$, and choose an ordered orthonormal basis for
$\widehat{T_xM}$ so that it induces the same orientation as $T_xM$. Denote the corresponding basis matrix by
$\widehat{B_x}$. To establish that such orientations can be chosen consistently across tangent spaces, we prove the following theorem.
\begin{theorem}\label{orientation}
 Let $M \subset \mathbb{R}^D$ be a smooth closed connected orientable $d$-dimensional submanifold with $\mathsf{rch}_{\mathbb{R}^D}(M) = \tau >0$. If two points $p,q \in M$ satisfy $\|p-q\|_{\mathbb{R}^D} < \frac{\tau}{2}$, then
 $$ \det (B_p^\top B_q ) > 0.$$
\end{theorem}
We postpone the proof to Appendix~\ref{orientation_proof}. If two points $x_1$ and $x_2$ in $\mathbf{X}$ are adjacent and the basis matrix $\widehat{B_{x_1}}$ is oriented, then we orient $\widehat{B_{x_2}}$ by flipping the sign of its last row whenever $\det (\widehat{B_{x_1}}^\top \widehat{B_{x_2}})<0$. This yields a consistent orientation of the estimated tangent spaces over $\mathbf{X}$. We choose $c > 0$ so that the diameters of $M$ and $\mathbf{Gr}_c^+(D,d)$ coincide.
Approximating the diameter of $M$ by $\widehat{\mathrm{diam}}(M) = \mathrm{diam}(\mathbf{Y})$ and using $\mathrm{diam}(\mathbf{Gr}^+(D,d)) = \max\bigl(\pi, \frac{\pi}{2}\,\sqrt{\min(d,D-d)}\bigr)$ from \cite[Theorem 12.6]{kozlov2000geometry}, we set
\begin{equation*}\label{parameter}
c = \frac{\mathrm{diam}(\mathbf{Y})^2}{\max(\pi, \frac{\pi}{2}\sqrt{\min(d,D-d)})^2}.
\end{equation*}
We then compute the distance matrix \(\mathbf{D}\) with entries
\begin{equation*}\label{grass_distance}
\bigl(\mathbf{D}\bigr)_{ij} = \sqrt{\|y_i - y_j\|_{\mathbb{R}^D}^2 + c \, d_{\mathbf{Gr}^+(D,d)}(\widehat{T_{y_i}M},\widehat{T_{y_j}M})^2}
\end{equation*}
for $1\leq i,j \leq |\mathbf{Y}|$ and perform Vietoris--Rips persistent homology with respect to \(\mathbf{D}\).
We compute persistent homology using the Ripser library~\cite{bauer2021ripser}, with coefficients in $\mathbb{Z}/2$.

Question:
Let $M\subset \mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold, let $p\in M$, and let $v\in T_pM$ be a unit tangent vector. For $c>0$, let $\bar{\mathbf g}^+:M\to \mathbf{Gr}^+(D,d)$ be the Gauss map to the oriented Grassmannian of oriented $d$-planes in $\mathbb{R}^D$, equip $\mathbf{Gr}^+(D,d)$ with the rescaled metric $c\,g^{\mathbf{Gr}^+(D,d)}$, and define the pullback metric
$$
g_c=(\iota\times \bar{\mathbf g}_c^+)^*\bigl(g^{\mathbb{R}^D}\oplus g^{\mathbf{Gr}_c^+(D,d)}\bigr)
$$
on $M$, where $\iota:M\hookrightarrow \mathbb{R}^D$ is the embedding. Let $\mathrm{vol}(M)$ be the Euclidean volume of $M$, let $\mathrm{vol}_c(M)$ be the Riemannian volume of $(M,g_c)$, let $\mathbf{II}$ be the second fundamental form of $M\subset \mathbb{R}^D$, and write $\|\cdot\|_{\mathrm{HS}}$ for the Hilbert--Schmidt norm. Let $\kappa(v)$ denote the normal curvature of $M$ at $p$ in the direction $v$, and let $\kappa_c(v)$ denote the corresponding normal curvature for the graph embedding $(M,g_c)\subset \mathbb{R}^D\times \mathbf{Gr}_c^+(D,d)$. Assume
$$
\mathrm{vol}_c(M)\le \bigl(1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2\bigr)^{d/2}\,\mathrm{vol}(M),
\qquad
\kappa(v)>\frac{\|(\nabla_v^M\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}}{\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}}.
$$
Under these assumptions, which quantitative estimate holds?

For every $c>0$,
$$
\kappa_c(v)\,\mathrm{vol}_c(M)^{1/d}
\;\le\;
\frac{\sqrt{\kappa(v)^2+c\,\|(\nabla_v^{M}\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}^2}}{1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2}\,\mathrm{vol}_c(M)^{1/d}
\;<\;
\kappa(v)\,\mathrm{vol}(M)^{1/d},
$$
and the middle expression is strictly decreasing as $c$ grows.

Choices:
B.
For every $c>0$,
$$
\kappa_c(v)\,\mathrm{vol}_c(M)^{1/d}
\;\le\;
\frac{\sqrt{\kappa(v)^2+c\,\|(\nabla_v^{M}\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}^2}}{\sqrt{1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2}}\,\mathrm{vol}_c(M)^{1/d}
\;<\;
\kappa(v)\,\mathrm{vol}(M)^{1/d},
$$
and the middle expression is strictly decreasing as $c$ grows.
C.
For every $c>0$,
$$
\kappa_c(v)\,\mathrm{vol}_c(M)^{1/d}
\;<\;
\kappa(v)\,\mathrm{vol}(M)^{1/d}.
$$
D.
There exists $c_0>0$ such that for every $c\ge c_0$,
$$
\kappa_c(v)\,\mathrm{vol}_c(M)^{1/d}
\;\le\;
\frac{\sqrt{\kappa(v)^2+c\,\|(\nabla_v^{M}\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}^2}}{1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2}\,\mathrm{vol}_c(M)^{1/d}
\;\le\;
\kappa(v)\,\mathrm{vol}(M)^{1/d},
$$
and the middle expression is nonincreasing as $c$ grows.
E.
For every $c>0$,
$$
\kappa_c(v)\,\mathrm{vol}(M)^{1/d}
\;\le\;
\frac{\sqrt{\kappa(v)^2+c\,\|(\nabla_v^{M}\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}^2}}{1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2}\,\mathrm{vol}(M)^{1/d}
\;<\;
\kappa(v)\,\mathrm{vol}_c(M)^{1/d},
$$
and the middle expression is strictly decreasing as $c$ grows.

speed-squared denominator from reparameterization

explicit intermediate upper bound and monotonicity statement

for every c and strict decrease replaced by eventual/nonstrict version

volume factor attached to the wrong side of the comparison


For a positive integer $n$, let $[n]:=\{1,2,\ldots,n\}$. A {\em set composition} $\phi$ of $[n]$,
denoted by $\phi\vDash[n]$, is a list of mutually disjoint nonempty subsets
$\phi_1/\phi_2/\ldots/\phi_{\ell}$ of $[n]$ whose union is $[n]$.
Each $\phi_i$ is called a {\em block} of $\phi$ and the number of blocks of $\phi$ is denoted
by $\ell(\phi)$.
A set composition $\phi$ is said to be {\em odd} if each block of $\phi$ consists of odd number
of elements. For instance, $\phi=7/148/9/6/235$ is an odd set composition of $[9]$ with $\ell(\phi)=5$.
Let
$$
O(n,k):=|\{\phi\vDash[n]: \phi\text{ odd}, \ell(\phi)=k\}|.
$$
Note that $O(n,k)=0$ if $n$ and $k$ have different parity. By the compositional formula for
the exponential generating functions (see~\cite[Proposition~5.1.3]{St}), we have
\begin{align*}
&\quad\sum_{n,k} O(n,k)t^k\frac{x^n}{n!}=\frac{t\sinh(x)}{1-t\sinh(x)}\\
&=tx+2t^2\frac{x^2}{2!}+(t+6t^3)\frac{x^3}{3!}+(8t^2+24t^4)\frac{x^3}{3!}+
(t+60t^3+120t^5)\frac{x^5}{5!}+\cdots.
\end{align*}

The {\em Catalan numbers}
$$
\left(C_n=\frac{1}{n+1}{2n\choose n}\right)_{n\geq0}=(1,1,2,5,14,42,132,\ldots)
$$
and the {\em tangent numbers} $\{T_{2n+1}\}_{n\geq0}$ defined by the Taylor expansion
$$
\tan(x)=\sum_{n\geq0}T_{2n+1}\frac{x^{2n+1}}{(2n+1)!}=
x+2\frac{x^3}{3!}+16\frac{x^5}{5!}+272\frac{x^7}{7!}+7936\frac{x^9}{9!}+\cdots
$$
are well-known combinatorial sequences in enumerative combinatorics.

The following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,
was found by Aliniaeifard and Li~\cite{AL} in their study of non-commutative peak algebra
and later proved by Zhao, Lin and Zang~\cite{ZLZ} using generating functions.

The following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,
was found by Aliniaeifard and Li~\cite{AL} in their study of non-commutative peak algebra
and later proved by Zhao, Lin and Zang~\cite{ZLZ} using generating functions.

For instance, when $n=2$ identity~\eqref{eq:Cat-Tan} reads
$$
1\cdot2^4C_0-60\cdot2^2C_1+120\cdot2^0C_2=16.
$$
Although Catalan numbers and tangent numbers have been extensively studied in combinatorics
(see~\cite[Exercise~6.19]{St} and~\cite{St10}), there seems to be no combinatorial identity relating
them other than~\eqref{eq:Cat-Tan}. As observed by Aliniaeifard and Li~\cite{AL},
combining~\cite[Corollary~11.5]{AL} and
Theorem~\ref{thm:Cat-Tan} results in the following intriguing identity for tangent numbers
\begin{equation}\label{eq:Genocchi}
\sum_{k=0}^{n-1}(-1)^{k}{2n\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.
\end{equation}
A generating function proof of Theorem~\ref{thm:Cat-Tan} and a combinatorial proof of~\eqref{eq:Genocchi},
together with its surprising arithmetic applications, were given in~\cite{ZLZ}.
However, a combinatorial proof of Theorem~\ref{thm:Cat-Tan} remains elusive.

The main objective of this paper is to provide a combinatorial involution proof of
Theorem~\ref{thm:Cat-Tan}.
In the course, we find a new combinatorial identity similar to~\eqref{eq:Genocchi}:
\begin{equation}\label{eq:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.
\end{equation}
This identity is also reminiscent of the one proved by Andrews and Gessel~\cite{AG}:
\begin{equation}\label{eq:TanAG}
T_{2n+1}+\sum_{k=1}^{n}(-1)^{k}{2n+1\choose 2k}2^{2k-1}T_{2n-2k+1}=(-1)^n2^{2n}.
\end{equation}

Let $\LB_{2n+1}$ be the set of all labeled binary trees on $[2n+1]$ and let $\IB_{2n+1}$ be the set of
all complete increasing binary trees on $[2n+1]$. It is clear that $\IB_{2n+1}\subseteq\LB_{2n+1}$.
For a labeled binary tree $T\in\LB_{2n+1}$, let $\h(T)$ be half the number of edges of $T$.
The following lemma shows that the left-hand side of~\eqref{eq:Cat-Tan} is
a signed counting of labeled binary trees on $[2n+1]$.
\begin{lemma}\label{lem:sign}
For $n\geq0$, we have
\begin{equation}\label{eq:sign}
\sum_{T\in\LB_{2n+1}}(-1)^{\h(T)}=\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k.
\end{equation}
\end{lemma}
\begin{proof}
Each labeled binary tree in $\LB_{2n+1}$ with $2k$ edges can be constructed by
\begin{itemize}
\item choosing a complete binary tree $T$ with $2k+1$ nodes,
\item choosing a set composition $\phi$ of $[2n+1]$ with $2k+1$ blocks,
\item forming an odd unimodal permutation using letters in $\phi_i$ for the label of the $i$-th node
(under the in-order traversal) of $T$ for each $1\leq i\leq 2k+1$.
\end{itemize}
There are $C_k$ choices for $T$, $O(2n+1,2k+1)$ choices for $\phi$, and $2^{2n+1-(2k+1)}$ ways to form
the labels of $T$ from $\phi$, whence~\eqref{eq:sign} follows.
\end{proof}

A permutation $\pi=\pi_1\pi_2\cdots\pi_n$ of $[n]$ is {\em down-up} (or {\em alternating}) if
$$
\pi_1>\pi_2<\pi_3>\pi_4<\cdots.
$$
If the above inequalities are reversed, then $\pi$ is said to be {\em up-down}.
Let $\A_n$ be the set of all down-up permutations of $[n]$.
A result attributed to Andr\'e~\cite{An} asserts that
\begin{equation}\label{eul:egf}
1+\sum_{n\geq1}|\A_n|\frac{x^n}{n!}=\sec(x)+\tan(x).
\end{equation}
Thus, $T_{2n+1}=|\A_{2n+1}|$.
It is a classical result (see~\cite{St10}) that $T_{2n+1}(q)$ has the combinatorial interpretation
\begin{equation}\label{int:qtan}
T_{2n+1}(q)=\sum_{\pi\in\A_{2n+1}}q^{\inv(\pi)},
\end{equation}
where $\inv(\pi):=|\{(i,j)\in[n]^2: i<j,\pi_i>\pi_j\}|$ is the {\em inversion number} of a permutation $\pi$.
The following is our first $q$-analog of~\eqref{eq:Tan2}.
\begin{theorem}\label{thm:q-analog1}
For $n\geq0$, we have
\begin{equation}\label{q1:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\brack 2k}(-q;q)_{2n-2k}\widetilde T_{2k+1}(q)=(-1)^nT_{2n+1}(q),
\end{equation}
where $\widetilde T_{1}(q)=1$ and
\begin{equation}
\widetilde T_{2k+1}(q)=\sum_{i=0}^{k-1} {2k\brack 2i+1} T_{2i + 1}(q) T_{2k - 2i - 1}(q) \quad (k \geq 1).
\end{equation}
\end{theorem}

For a labeled tree $T\in\LB_{2n+1}$ with $2k+1$ nodes, consider the word concatenation
$w(T)=\alpha_1\alpha_2\cdots\alpha_{2k+1}$, where $\alpha_i$ is the labeling of the unimodal
permutation of the $i$-th node (under the in-order traversal) of $T$. For instance,
if $T$ is the first tree in Fig.~\ref{ex:lbt}, then $w(T)=467381952$.
It is clear that $w(T)$ is a permutation of $[2n+1]$. The {\em inversion number of $T$},
denoted by $\inv(T)$, is thus defined as
$$
\inv(T):=\inv(w(T)).
$$
The following interpretation of signed $q$-tangent numbers is a consequence of
Lemma~\ref{lem:invo}.
\begin{lemma}\label{lem:newint}
For $n\geq0$, we have
\begin{equation}\label{new:tan}
\sum_{T\in\LB_{2n+1}}(-1)^{\h(T)}q^{\inv(T)}=(-1)^nT_{2n+1}(q).
\end{equation}
\end{lemma}
\begin{proof}It is plain to check that our involution $\kappa$ has the following feature:
$$
\inv(T)=\inv(\kappa(T)) \quad\text{for any $T\in\LB_{2n+1}$},
$$
that is, it preserves the inversion numbers of labeled trees. The result then follows from Lemma~\ref{lem:invo}, interpretation~\eqref{int:qtan} and the known fact that $T\mapsto w(T)$ establishes a one-to-one correspondence between $\IB_{2n+1}$ and $\A_{2n+1}$ preserving inversion numbers.
\end{proof}
 We are ready for the proof of Theorem~\ref{thm:q-analog1}.
\begin{proof}[{\bf Proof of Theorem~\ref{thm:q-analog1}}]
Recall the well-known combinatorial interpretation for the $q$-binomial coefficients:
\begin{equation}\label{eq:qmul}
{n\brack k}=\sum_{({\mathcal A}, {\mathcal B})}q^{\inv({\mathcal A}, {\mathcal B})},
\end{equation}
summed over all set compositions $({\mathcal A}, {\mathcal B})$ of $[n]$ such that $|{\mathcal A}|=k$, and
$$\inv({\mathcal A}, {\mathcal B}):=|\{(a,b)\in{\mathcal A}\times{\mathcal B}: a>b\}|.$$
A labeled binary tree $T\in\LB_{2n+1}$ can be decomposed as $(T_g,r,T_d)$, where $r$ is the root and $T_g$ and $T_d$ (possibly empty) are left branch and right branch of $r$, respectively. It follows from this decomposition, Lemma~\ref{lem:unimodal}, the interpretation~\eqref{new:tan} of $q$-tangent numbers and the interpretation~\eqref{eq:qmul} of $q$-binomial coefficients that
$$
(-1)^nT_{2n+1}(q)=(-q;q)_{2n}+\sum_{k=1}^{n}(-1)^{k}{2n+1\brack 2k}(-q;q)_{2n-2k}\sum_{i=0}^{k-1}{2k\brack 2i+1}T_{2i+1}(q)T_{2k-2i-1}(q),
$$
where $(-q;q)_{2n}$ counts single-node labeled binary trees on $[2n+1]$ by inversion numbers.
This completes the proof of the theorem.
\end{proof}

In view of~\eqref{eul:egf}, the number $|\A_{2n}|:=S_{2n}$ is known as a {\em secant number}.
Consider the {\em$q$-secant number }
$$
S_{2n}(q):=\sum_{\pi\in\A'_{2n}}q^{\inv(\pi)},
$$
where $\A'_n$ denotes the set of all up-down permutations of $[n]$. For convenience, set $S_0(q)=1$. It was known (see~\cite{FH10}) that
$$
\sum_{n\geq0}S_{2n}(q)\frac{x^{2n}}{(q;q)_{2n}}=\frac{1}{\cos_q(x)}.
 $$
 The following is our second $q$-analog of~\eqref{eq:Tan2}.
\begin{theorem}\label{thm:q-analog2}
For $n\geq0$, we have
\begin{equation}\label{q2:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\brack 2k}(-q;q)_{2n-2k} T_{2k+1}(q)=(-1)^n\widehat T_{2n+1}(q),
\end{equation}
where
\begin{equation}
(-1)^n\widehat T_{2n+1}(q)=\sum_{k=0}^{n} (-1)^{k} {2n+1\brack 2k}q^{2k} S_{2k}(q).
\end{equation}
Equivalently,
$$
\sum_{k=0}^{n}(-1)^{k}{2n+1\brack 2k}\bigl((-q;q)_{2n-2k} T_{2k+1}(q)-q^{2k} S_{2k}(q)\bigr)=0.
$$
\end{theorem}

\begin{theorem}\label{thm:q-sec:tan}
For $n\geq0$, we have
\begin{equation}\label{eq:q-sec:tan}
\sum_{k=0}^n(-1)^k{2n+1\brack 2k}S_{2k}(q)=(-1)^nT_{2n+1}(q).
\end{equation}
\end{theorem}

\begin{equation}\label{eq:Cat-Tan}
\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}.
\end{equation}

\begin{theorem}\label{thm:Cat-Tan}
For $n\geq0$, we have
\begin{equation}\label{eq:Cat-Tan}
\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}.
\end{equation}
\end{theorem}

For instance, when $n=2$ identity~\eqref{eq:Cat-Tan} reads
$$
1\cdot2^4C_0-60\cdot2^2C_1+120\cdot2^0C_2=16.
$$
Although Catalan numbers and tangent numbers have been extensively studied in combinatorics
(see~\cite[Exercise~6.19]{St} and~\cite{St10}), there seems to be no combinatorial identity relating
them other than~\eqref{eq:Cat-Tan}. As observed by Aliniaeifard and Li~\cite{AL},
combining~\cite[Corollary~11.5]{AL} and
Theorem~\ref{thm:Cat-Tan} results in the following intriguing identity for tangent numbers
\begin{equation}\label{eq:Genocchi}
\sum_{k=0}^{n-1}(-1)^{k}{2n\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.
\end{equation}
A generating function proof of Theorem~\ref{thm:Cat-Tan} and a combinatorial proof of~\eqref{eq:Genocchi},
together with its surprising arithmetic applications, were given in~\cite{ZLZ}.
However, a combinatorial proof of Theorem~\ref{thm:Cat-Tan} remains elusive.

The main objective of this paper is to provide a combinatorial involution proof of
Theorem~\ref{thm:Cat-Tan}.
In the course, we find a new combinatorial identity similar to~\eqref{eq:Genocchi}:
\begin{equation}\label{eq:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.
\end{equation}
This identity is also reminiscent of the one proved by Andrews and Gessel~\cite{AG}:
\begin{equation}\label{eq:TanAG}
T_{2n+1}+\sum_{k=1}^{n}(-1)^{k}{2n+1\choose 2k}2^{2k-1}T_{2n-2k+1}=(-1)^n2^{2n}.
\end{equation}

The rest of this paper is organized as follows. In Section~\ref{sec:2}, we introduce the combinatorial
structure of complete binary trees labeled by odd unimodal permutations and then construct
a sign-reversing involution on them in Section~\ref{sec:3} to prove Theorem~\ref{thm:Cat-Tan}
combinatorially. In Section~\ref{sec:4}, we prove two different $q$-analogs of~\eqref{eq:Tan2}
using our involution on labeled binary trees and some other involutions on permutation pairs.

To prove Theorem~\ref{thm:Cat-Tan} combinatorially, the paper (i) introduces in Section~\ref{sec:2} “the combinatorial structure of complete binary trees labeled by odd unimodal permutations,” and then (ii) “construct[s] a sign-reversing involution on them in Section~\ref{sec:3} to prove Theorem~\ref{thm:Cat-Tan} combinatorially.”

To prove the main theorem combinatorially, the paper (i) introduces in the next section “the combinatorial structure of complete binary trees labeled by odd unimodal permutations,” and then (ii) “construct[s] a sign-reversing involution on them in a later section to prove the main theorem combinatorially.”

Question:
For a positive integer \(m\), let \([m]=\{1,2,\ldots,m\}\). A set composition \(\phi\vDash [m]\) is an ordered list of mutually disjoint nonempty subsets whose union is \([m]\); it is called odd if every block has odd cardinality. Let \(O(m,r)\) denote the number of odd set compositions of \([m]\) with exactly \(r\) blocks. Also let the Catalan numbers be \(C_k=\frac{1}{k+1}\binom{2k}{k}\), and let the tangent numbers \(T_{2j+1}\) be defined by
\[
\tan x=\sum_{j\ge 0} T_{2j+1}\frac{x^{2j+1}}{(2j+1)!}.
\]
For every integer \(n\ge 0\), which explicit identity holds?

\[
\sum_{k=0}^{n}(-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_k = (-1)^n T_{2n+1}.
\]

Choices:
B.
\[
\sum_{k=0}^{n}(-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_k = T_{2n+1}.
\]
C.
\[
\sum_{k=0}^{n}(-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_k = \pm T_{2n+1}.
\]
D.
\[
\sum_{k=0}^{n}(-1)^k\,O(2n+1,2k+1)\,2^{2n+1-2k}C_k = (-1)^n T_{2n+1}.
\]
E.
\[
\sum_{k=0}^{n}(-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_{2k+1} = (-1)^n T_{2n+1}.
\]

global sign from sign-reversing involution

exact parity-dependent sign factor (-1)^n

factor 2^{2n-2k} from odd unimodal labelings

Catalan index tied to complete binary trees with 2k+1 nodes