Given positive integers $k$ and $\ell$, we say a graph $G$ is
\emph{Ramsey} for $(K_k,K_\ell)$ if every colouring of the edges of
$G$ with red and blue contains a red copy of $K_k$ or a blue copy of
$K_\ell$ and we denote this property by $G \rightarrow (K_k,K_\ell)$.
In a seminal work~\cite{Ramsey}, Ramsey proved that for all positive
integers $k$ and $\ell$, there exists a positive integer $n$ such that
$K_n \rightarrow (K_k, K_{\ell})$. In the special case $k=\ell$, we
simply write $G \rightarrow K_k$ and we define the \emph{Ramsey
 number} $R(k)$ as the minimum $n$ such that $K_n \rightarrow K_k$.

Estimating $R(k)$ has proven to be notoriously difficult and a central
problem in Ramsey theory is to determine the Ramsey number $R(k)$.
Classical results due to Erd\H{o}s~\cite{Erd} and
Erd\H{o}s–-Szekeres~\cite{Erd.Szek} established the bounds
$2^{k/2} \leq R(k) \leq 2^{2k}$. Despite several refinements, these
exponents remained essentially unchanged for decades. Only in recent
years, significant breakthroughs have been achieved (see,
e.g.,~\cite{Conlon, ConlonFoxSudakov2015, Sah2023}). In a striking
advance, Campos, Griffiths, Morris, and Sahasrabudhe~\cite{CGMS}
proved that there exists an $\varepsilon > 0$ such that
$R(k)\leqslant (4-\varepsilon)^k$ for sufficiently large $k$. This
result provides an exponential improvement over the classical
Erd\H{o}s–-Szekeres upper bound. More generally, one can think of the
minimum $n$ such that $K_n\rightarrow (K_k,K_\ell)$, for which there
was another major breakthrough recently by Mattheus and
Verstraete~\cite{MV}, who showed that
$n=\Omega\big(t^3/(\log^4 t)\big)$ vertices are enough to force red
copies of $K_4$ or blue copies of $K_t$ in red-blue colourings of the
edges of $K_n$.

Although much effort has been put into estimating Ramsey numbers, a
parallel and rich direction of research investigates the structure of
graphs that are Ramsey for given pairs of graphs. In this context, we
study Ramsey phenomena of the form $G \rightarrow (K_{s},K_{t})$.

A classical result of Ne\v{s}et\v{r}il and R\"odl~\cite{NR} shows
that for every $k\geq 2$ there are graphs with no copies of $K_k$
that are Ramsey for $K_{k-1}$.

\begin{abstract}
 Given positive integers $k$ and $\ell$ we write
 $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of
 $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we
 denote by $R(k)$ the minimum $n$ such that
 $K_n\rightarrow (K_k,K_k)$. By using probabilistic methods and
 hypergraph containers we prove that for every integer $k \geq 3$,
 there exists a graph $G$ such that $G \nrightarrow (K_k,K_k)$ and
 $G \rightarrow (K_{R(k)-1},K_{k-1})$. This result can be viewed as
 a variation of a classical theorem of Ne\v{s}et\v{r}il and R\"odl
 [The Ramsey property for graphs with forbidden complete subgraphs,
 {\em Journal of Combinatorial Theory, Series B}, \textbf{20} (1976),
 243–249], who proved that for every integer $k\geq 2$ there exists a
 graph $G$ with no copies of $K_k$ such that
 $G\rightarrow(K_{k-1}, K_{k-1})$.
\end{abstract}

\section{Introduction}
Given positive integers $k$ and $\ell$, we say a graph $G$ is
\emph{Ramsey} for $(K_k,K_\ell)$ if every colouring of the edges of
$G$ with red and blue contains a red copy of $K_k$ or a blue copy of
$K_\ell$ and we denote this property by $G \rightarrow (K_k,K_\ell)$.
In a seminal work~\cite{Ramsey}, Ramsey proved that for all positive
integers $k$ and $\ell$, there exists a positive integer $n$ such that
$K_n \rightarrow (K_k, K_{\ell})$. In the special case $k=\ell$, we
simply write $G \rightarrow K_k$ and we define the \emph{Ramsey
 number} $R(k)$ as the minimum $n$ such that $K_n \rightarrow K_k$.

Although much effort has been put into estimating Ramsey numbers, a
parallel and rich direction of research investigates the structure of
graphs that are Ramsey for given pairs of graphs. In this context, we
study Ramsey phenomena of the form $G \rightarrow (K_{s},K_{t})$.

We remark that Theorem~\ref{thm:main} also relates to the theory of
Ramsey equivalence. Szab\'o, Zumstein, and Z\"urcher \cite{Szabo}
introduced the notion of \emph{Ramsey-equivalent graphs}: two graphs
$H_1$ and $H_2$ are {Ramsey-equivalent} if for every graph $G$, we
have $G \rightarrow H_1$ if and only if $G \rightarrow H_2$
(see~\cite{Bloom, Fox}) for results on Ramsey equivalence). More
generally, two pairs of graphs $(F_1, H_1)$ and $(F_2, H_2)$ are
{Ramsey-equivalent} if for every graph $G$ we have
$G \rightarrow (F_1, H_1)$ if and only if $G \rightarrow (F_2, H_2)$.
In other words, the two pairs share exactly the same family of Ramsey
graphs. In this direction, our result implies that the pairs $(K_k,K_k)$
and $(K_{s},K_{k-1})$ for any $s\leq R(k)-1$ are not
Ramsey-equivalent.

\begin{theorem}
 \label{thm:part1}
 For all integers $s\geq 2$ and $k\geq 3$, there exists \(C>0\) such that the
 following holds with high probability for $\cH = \hnp$ when
 $\pp \geq C n^{2-s-1/m_2(K_{k-1})}$. For every subhypergraph
 $\ho \subseteq \cH$ with at least $(1 - o(1))e(\cH)$ hyperedges, we
 have
 \[
 G[\ho] \rightarrow (K_{s}, K_{k-1}).
 \]
\end{theorem}

From Lemma~\ref{lem:quasi}, we have
$\PP{S_i \sqsubset \cH} \leq \pt^{|S_i|}$
for $\pt = \pp \binom{n-2}{s-2}$. Let $m = Dn^{2- 1/m_2(K_{k-1})}$ and
note that from the choice of $C$ we have
$m \leq (D/C)\pp n^{s} \leq \pt n^ 2$. Since $|S_i| \leq m$ for every
$i \in [t]$ and there are at most ${n^2\choose \ell}$ sources $S_i$
with exactly $\ell$ edges, we have
 $$
 \sum_{i=1}^{t} \mathbb{P}[S_i \sqsubset \cH]
 \leq \sum_{i=1}^{t} \pt^{|S_i|}
 \leq \sum_{\ell=1}^{m} \binom{n^2}{\ell} \pt^\ell
 \leq \sum_{\ell=1}^{m} \left(\frac{e\pt n^2}{\ell} \right)^{\ell}.
 $$
 Since $(e\pt n^2/\ell)^{\ell}$ is increasing for
 $\ell \leq \pt n^2$, we may replace $m$ with its upper bound
 $(D/C)\pp n^{s}$ in the above estimation. This together with
 $\pt n^2 \leq \pp n^s$ gives
\begin{equation}
 \sum_{i=1}^{t} \mathbb{P}[S_i \sqsubset \cH]
 \leq m \left(\frac{e\pt n^2}{m} \right)^{m}
 \leq n^2 {\left(\frac{eC}{D}\right)}^{(D/C)\pp n^{s}}
 \leq \exp\left(\delta\pp n^{s}/16\right)\label{eq:bound-si},
\end{equation}
where the last inequality follows from the fact that $C$ is
sufficiently large. Finally, using \eqref{eq:bound-xi} and
\eqref{eq:bound-si}, the bound on \eqref{eq:bound-main} becomes
 \begin{align*}
 \PP{\exists i \in [t]: X_i \leq \delta e(\cH) \text{ and } S_i \sqsubset \cH}
 & \leq \exp\left\{-\frac{\delta \pp n^{s}}{16} \right\} = o(1).
 \end{align*}
 Therefore, with high probability, every $\ho\subseteq \cH$ with
 $e(\ho) \geq (1-\delta)e(\cH)$ is such that
 $G[\ho]\rightarrow (K_{s},K_{k-1})$, which finishes the proof.
\end{proof}

In this short section we combine Theorems~\ref{thm:part1}
and~\ref{thm:part2} to prove our main result.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
 Let $k \geq 3$ be an integer and $s = R(k)-1$. Consider $\pp$ such that $n^{2-s-1/m_2(K_{k-1})} \ll \pp \ll n^{2-s - 1/m_2(K_k)}$ and let $\cH = \hnp$.
 By Theorem~\ref{thm:part1}, with high probability, every subhypergraph $\ho \subseteq \cH$ with \(e(\ho) = (1-o(1))e(\cH)\) satisfies
 \begin{align}
 G[\ho]\rightarrow(K_{s},K_{k-1}).
 \label{eq:H0a}
 \end{align}
 On the other hand, by Theorem~\ref{thm:part2}, with high probability
 there exists a subhypergraph $\ho \subseteq \cH$ with
 \(e(\ho) = (1-o(1))e(\cH)\) such that
 \begin{align}
 G[\ho]\nrightarrow K_k.
 \label{eq:H0b}
 \end{align}
 Since both events can occur with high probability, there exists a
 hypergraph $\ho$ such that both~\eqref{eq:H0a}
 and~\eqref{eq:H0b} hold. Therefore, $G[\ho]$ is the desired graph.

It is possible to adapt our proof to obtain the following
generalization of Theorem~\ref{thm:main} by considering a linear
$k$-conformal subhypergraph of $\cH_{s}(n,p)$, by choosing
$n^{2-s-1/m_2(K_{\ell-1})} \ll p \ll n^{2-s-1/m_2(K_{\ell})}$.
\begin{theorem}\label{thm:generalization}
 For any integers $k \ge \ell \ge 3$, there exists a graph $G$ such
 that $G \nrightarrow (K_k,K_{\ell})$ and
 $G \rightarrow (K_{s},K_{\ell-1})$ for $s\leq R(k,\ell)-1$.
\end{theorem}
We propose the following conjecture as a variation of the previous
theorem for three colours.
\begin{conjecture}\label{conj:first}
 For any integers $k \ge \ell \ge 2$, there exists a graph $G$ such
 that $G \nrightarrow (K_k,K_{\ell})$ and
 $G \rightarrow (K_{k-1},K_{k-1},K_{\ell})$.
\end{conjecture}
Note that the case $\ell = 2$ of the above conjecture is precisely the
result of Ne\v{s}et\v{r}il and R\"odl (Theorem~\ref{thm_NR}). We
conclude proposing the following conjecture that relates to
Conjecture~\ref{conj:first} in the same way that
Theorem~\ref{thm:generalization} relates to Theorem~\ref{thm_NR}.
\begin{conjecture}
 For any integers $k \ge \ell \ge 2$, there exists a graph $G$ such
 that $G \nrightarrow (K_k,K_k,K_{\ell})$, but
 $G \rightarrow (K_{k+1},K_{k-1},K_{\ell})$.
\end{conjecture}

\begin{theorem}\label{thm:main}
	For every integer $k \ge 3$, there exists a graph $G$ such that $G
	\nrightarrow K_{k}$ and $G \rightarrow (K_{s},K_{k-1})$, for
  $s=R(k)-1$.
\end{theorem}

\begin{theorem}[Ne{\v{s}}et{\v{r}}il \& R{\"o}dl, 1976]\label{thm_NR}
	For every $k\geq 2$ there is a graph $G$ such that
	$K_k \nsubseteq G$ and $G\rightarrow K_{k-1}$.
\end{theorem}

Our main result, Theorem~\ref{thm:main} below, can be seen as a
variation of Theorem~\ref{thm_NR}. We prove that for any $k\geq 3$
there exists a graph $G$ that is not Ramsey for $K_k$ but it is Ramsey
for the pair $(K_{s},K_{k-1})$, for $s=R(k)-1$, i.e., we replace the
condition $K_k \nsubseteq G$ in Theorem~\ref{thm_NR} with the weaker
condition $G\not\rightarrow K_k$, which allows $G$ to contain copies of
$K_k$, but still there is a colouring of $E(G)$ avoiding monochromatic
copies of $K_k$; and we strengthen the conclusion
$G\rightarrow K_{k-1}$ by showing that
$G \rightarrow (K_{s},K_{k-1})$, for $s=R(k)-1$ (note that $s$ cannot
be any larger).
\begin{theorem}\label{thm:main}
	For every integer $k \ge 3$, there exists a graph $G$ such that $G
	\not\rightarrow K_{k}$ and $G \rightarrow (K_{s},K_{k-1})$, for
  $s=R(k)-1$.
\end{theorem}

We remark that Theorem~\ref{thm:main} also relates to the theory of
Ramsey equivalence. Szab\'o, Zumstein, and Z\"urcher \cite{Szabo}
introduced the notion of \emph{Ramsey-equivalent graphs}: two graphs
$H_1$ and $H_2$ are {Ramsey-equivalent} if for every graph $G$, we
have $G \rightarrow H_1$ if and only if $G \rightarrow H_2$
(see~\cite{Bloom, Fox}) for results on Ramsey equivalence). More
generally, two pairs of graphs $(F_1, H_1)$ and $(F_2, H_2)$ are
{Ramsey-equivalent} if for every graph $G$ we have
$G \rightarrow (F_1, H_1)$ if and only if $G \rightarrow (F_2, H_2)$.
In other words, the two pairs share exactly the same family of Ramsey
graphs. In this direction, our result implies that the pairs $(K_k,K_k)$
and $(K_{s},K_{k-1})$ for any $s\leq R(k)-1$ are not
Ramsey-equivalent.

The proof of Theorem~\ref{thm:main} combines probabilistic methods
with the hypergraph container
framework~\cite{balogh2015independent,SaxtonThomason2015} and is
inspired by ideas from~\cite{bollobas2001ramsey}. The rest of the
paper is organized as follows. In Section~\ref{sec:mono}, we show that
with high probability\footnote{Meaning with probability going to $1$
  as $n$ tends to infinity.} the graph $G$ obtained in a natural way from
every ``dense'' subhypergraph of a suitable $n$-vertex random
$s$-uniform hypergraph satisfies $G\rightarrow(K_{s},K_{k-1})$ for
$s = R(k)-1$. In Section~\ref{sec:hyper}, we show that with high
probability a suitable random hypergraph $\mathcal{H}$ contains a dense
subhypergraph $\cH_0$ that will allow us to obtain a graph $G$ such
that $G\not\rightarrow K_k$. These results are then combined in
Section~\ref{sec:proof}. Finally, in Section~\ref{sec:conc}, we
outline some directions for future research.

The post-theorem introduction gives the following proof outline for Theorem~\ref{thm:main}: it “combines probabilistic methods with the hypergraph container framework~\cite{balogh2015independent,SaxtonThomason2015} and is inspired by ideas from~\cite{bollobas2001ramsey}.” The paper is then organized into steps: (1) In Section~\ref{sec:mono}, “with high probability … the graph $G$ obtained in a natural way from every ‘dense’ subhypergraph of a suitable $n$-vertex random $s$-uniform hypergraph satisfies $G\rightarrow(K_s,K_{k-1})$ for $s=R(k)-1$.” (2) In Section~\ref{sec:hyper}, “with high probability a suitable random hypergraph $\mathcal{H}$ contains a dense subhypergraph $\cH_0$ that will allow us to obtain a graph $G$ such that $G\not\rightarrow K_k$.” (3) “These results are then combined in Section~\ref{sec:proof}” to produce a graph $G$ satisfying both $G\not\rightarrow K_k$ and $G\rightarrow(K_s,K_{k-1})$ (with $s=R(k)-1$).

The post-theorem introduction gives the following proof outline for the main theorem: it “combines probabilistic methods with the hypergraph container framework Balogh--Morris--Samotij, *Independent sets in hypergraphs* (2015) and Saxton--Thomason, *Hypergraph containers* (2015) and is inspired by ideas from Bollobás--Erdős, *Ramsey graphs* (2001).” The paper is then organized into steps: (1) Next, “with high probability … the graph $G$ obtained in a natural way from every ‘dense’ subhypergraph of a suitable $n$-vertex random $s$-uniform hypergraph satisfies $G\rightarrow(K_s,K_{k-1})$ for $s=R(k)-1$.” (2) Then, “with high probability a suitable random hypergraph $\mathcal{H}$ contains a dense subhypergraph $\cH_0$ that will allow us to obtain a graph $G$ such that $G\not\rightarrow K_k$.” (3) “These results are then combined later” to produce a graph $G$ satisfying both $G\not\rightarrow K_k$ and $G\rightarrow(K_s,K_{k-1})$ (with $s=R(k)-1$).

Question: For a graph $H$, write $G \rightarrow H$ to mean that every 2-colouring of the edges of $G$ with red and blue contains a monochromatic copy of $H$. For an integer $k \ge 2$, which statement holds regarding complete graphs?

For every integer $k \ge 2$, there exists a graph $G$ that contains no copy of $K_k$ (that is, $K_k \nsubseteq G$) and nevertheless satisfies $G \rightarrow K_{k-1}$; equivalently, every red-blue colouring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.

Choices:
B. For every integer $k \ge 2$, there exists a graph $G$ that contains no copy of $K_k$ and nevertheless satisfies $G \rightarrow K_k$; equivalently, every red-blue colouring of the edges of $G$ contains a monochromatic copy of $K_k$.
C. For every integer $k \ge 2$, there exists a graph $G$ such that $G \rightarrow K_{k-1}$; equivalently, every red-blue colouring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.
D. For every integer $k \ge 3$, there exists a graph $G$ that contains no copy of $K_k$ and nevertheless satisfies $G \rightarrow K_{k}$; equivalently, every red-blue colouring of the edges of $G$ contains a monochromatic copy of $K_{k}$.
E. For every integer $k \ge 2$, there exists a graph $G$ such that $K_{k-1} \nsubseteq G$ and nevertheless satisfies $G \rightarrow K_{k-1}$; equivalently, every red-blue colouring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.

shift from forcing $K_{k-1}$ to forcing $K_k$ while retaining $K_k$-free host

dropped the forbidden-subgraph condition $K_k \nsubseteq G$

changed the conclusion to $G \rightarrow K_k$ and also restricted the range to $k\ge 3$

replaced the forbidden clique $K_k$ by the smaller clique $K_{k-1}$, contradicting the Ramsey conclusion


\begin{equation}\label{mH1}
	\mH(R):= \left\{ h\in W^{1,2}(B_R) \,\bigg\vert\, \exists \zeta \in W^{1,2}(\mathbb{S}^{N-1}): \ h(x)=\frac{1}{|x|}\zeta\big( \frac{x}{|x|} \big) \ \text{for $x\in B_R$ a.e.} \right\}.
\end{equation}

\begin{theorem}\label{TH-00}
Let $u$ be a suitable weak solutions. Suppose that
the following two conditions hold:
\begin{equation}
\liminf_{r\to 0}M(r):=\liminf_{r\to 0}\int_{B_r}\!\! \left( \frac{|u|^2}{r^3} + \frac{|\nabla u|^2}{r} \right) <\infty, \quad \liminf_{r\to 0}\frac1{r^2}\int_{B_r}\!\!(|u|^2+2P)u\cdot\frac x{|x|}>0.
\end{equation}
Then  for any $\{r_k\}_{k=1}^\infty, r_k\downarrow 0$ there is a subsequence $r_{k_m}$ such that
the scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to
a homogenous vector field of degree negative one, and hence
$x=0$ is a regular point.
\end{theorem}

\begin{proposition}\label{prop:cubicEst}
	For fixed constants $\delta_{1}, \delta_{2}>0$, there exists $\ep>0$ such that if $(u,p)$ is a Leray-Hopf solution to the Navier-Stokes equations satisfying
	\begin{equation}\label{h-A}
		\frac{1}{R^3} \int_{B_R} \big|u- \mP_R[u] \big|^2 + \frac{1}{R} \int_{ B_R} \big|\nabla u - \nabla \mP_R[u] \big|^2 \le \ep M[u](R)
	\end{equation}
	for some $R>0$, then
	\begin{equation}
		\bigg|\frac{1}{R^2}\int_{B_{R/2}} \big( |u|^2 + 2 p \big) \, u \cdot \frac{x}{|x|} \bigg|  \le \delta_{1} + \delta_{2} \big(M[u](R)\big)^{\frac{3}{2}}.
	\end{equation}
\end{proposition}

In this paper we study the local behavior of the weak solutions of the stationary incompressible
Navier-Stokes equations in five space dimensions
\begin{equation}\label{eq:problem}
\left.\begin{array}{rrr}
u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3, 4, 5,\\
\div u =0,
\end{array}
\right\}
\quad\mbox{in}\ \Omega\subset \mathbb{R}^5
\end{equation}
where $\Omega\subset \mathbb{R}^5$ is a domain .

The existence of weak solutions under various assumptions on the
boundary data and $\Omega$ has been established in \cite{Galdi}, \cite{FR-Pisa}, \cite{Struwe-per}.
Moreover, in  \cite{FR-Pisa}, \cite{FR-arma}, \cite{Struwe-per} the authors constructed
smooth solutions of \eqref{eq:problem-3d}.

The problem \eqref{eq:problem} has a number of similarities with the
dynamic Navier-Stokes system in three space dimensions
\begin{equation}\label{eq:problem-3d}
\left.\begin{array}{rrr}
u_t^i+u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3,\\
\div u =0,
\end{array}
\right\}
\mbox{in}\ \Omega\times(0, T), \Omega\subset \mathbb{R}^3.
\end{equation}
For instance, in both cases $u\in L^{\frac{10}3}_{loc}, P\in L^{\frac 53}_{loc}, \nabla P\in L^{\frac54}_{loc}$, see  \cite{Seregin-book}.
Due to this a number of mathematicians studied the stationary
Navier-Stokes equations in higher dimensions in order to develop
stronger analytical methods which may be applicable to the
dynamic case \eqref{eq:problem-3d}, see \cite{Galdi}.

In this context, of particular interest is the problem of estimating the
dimension of the singular points of suitable weak solutions $u$, i.e. the points where
$u$ is not bounded. Scheffer \cite{Scheffer}
proved such results  for \eqref{eq:problem-3d} and later
Caffarelli, Kohn and Nirenberg \cite{CKN} improved upon it
showing that  the Hausdorff dimension of the singular set in space-time is atmost one.
We note that the latter result can be established by a different method
by looking at the
small perturbations of the Stokes system \cite{Lin}.
For \eqref{eq:problem} the partial regularity is proved in \cite{Struwe}.

At the possible singular point $(x_0, t_0)$ the scale invariance  $u(x, t)\mapsto ru(x_0+rx, t_0+r^2), r>0$ suggests that at the scale $r$, $u$ behaves like $1/r$ near $(x_0, t_0)$.
A natural question that follows from this observation is whether
one can classify the scale invariant solutions.
This has been the main approach towards understanding the structure of possible singularities.
\v{S}ver\'{a}k's classification for the self-similar solutions \cite{Sverak} for Navier-Stokes equations \eqref{eq:problem} shows that
a solution of the form $h(x)=\frac{\zeta(\frac x{|x|})}{|x|}$, with some smooth vectorfield $\zeta$, must be identically zero.

Another questions following from this result is whether the solutions sufficiently close to the self-similar one
are in fact zero. It is easy to see that the self-similar vectorfields $h=\frac{\zeta(\frac x{|x|})}{|x|}$ form a Hilbert subspace $\mathcal{H}(R)$ of
the Sobolev space $\mathcal W^{1, 2}(B_R)$ in $\mathbb{R}^5$ with an appropriately scaled invariant norms,
\begin{equation}\label{mH1}
	\mathcal{H}(R):= \left\{ h\in W^{1,2}(B_R) \,\bigg\vert\, \exists \zeta \in W^{1,2}(\mathbb{S}^{N-1}): \ h(x)=\frac{1}{|x|}\zeta\big( \frac{x}{|x|} \big) \ \text{for $x\in B_R$ a.e.} \right\}.
\end{equation}
Thus the Hilbert projection theorem yields that $\mathcal W^{1,2}(B_R) = \mathcal{H}(R) \oplus \mathcal{H}(R)^{\perp}$ and we can define the the corresponding projection operator as $\mathcal{P}_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mathcal{H}(R)$. Using this, we can measure the error $u-\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.

Our work is motivated by the following question.

{\textit{  If a suitable weak solution to the stationary Navier-Stokes system develops a singularity,
 can it asymptotically  become self-similar?}}

Our  main result in this direction can be stated as follows:

In this paper we study the local behavior of the weak solutions of the stationary incompressible
Navier-Stokes equations in five space dimensions
\begin{equation}\label{eq:problem}
\left.\begin{array}{rrr}
u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3, 4, 5,\\
\div u =0,
\end{array}
\right\}
\quad\mbox{in}\ \Omega\subset \mathbb{R}^5
\end{equation}
where $\Omega\subset \mathbb{R}^5$ is a domain .

The problem \eqref{eq:problem} has a number of similarities with the
dynamic Navier-Stokes system in three space dimensions
\begin{equation}\label{eq:problem-3d}
\left.\begin{array}{rrr}
u_t^i+u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3,\\
\div u =0,
\end{array}
\right\}
\mbox{in}\ \Omega\times(0, T), \Omega\subset \mathbb{R}^3.
\end{equation}
For instance, in both cases $u\in L^{\frac{10}3}_{loc}, P\in L^{\frac 53}_{loc}, \nabla P\in L^{\frac54}_{loc}$, see \cite{Seregin-book}.
Due to this a number of mathematicians studied the stationary
Navier-Stokes equations in higher dimensions in order to develop
stronger analytical methods which may be applicable to the
dynamic case \eqref{eq:problem-3d}, see \cite{Galdi}.

In this context, of particular interest is the problem of estimating the
dimension of the singular points of suitable weak solutions $u$, i.e. the points where
$u$ is not bounded. Scheffer \cite{Scheffer}
proved such results for \eqref{eq:problem-3d} and later
Caffarelli, Kohn and Nirenberg \cite{CKN} improved upon it
showing that the Hausdorff dimension of the singular set in space-time is atmost one.
We note that the latter result can be established by a different method
by looking at the
small perturbations of the Stokes system \cite{Lin}.
For \eqref{eq:problem} the partial regularity is proved in \cite{Struwe}.

Another questions following from this result is whether the solutions sufficiently close to the self-similar one
are in fact zero. It is easy to see that the self-similar vectorfields $h=\frac{\zeta(\frac x{|x|})}{|x|}$ form a Hilbert subspace $\mathcal{H}(R)$ of
the Sobolev space $\mathcal W^{1, 2}(B_R)$ in $\mathbb{R}^5$ with an appropriately scaled invariant norms,
\begin{equation}\label{mH1}
 \mathcal{H}(R):= \left\{ h\in W^{1,2}(B_R) \,\bigg\vert\, \exists \zeta \in W^{1,2}(\mathbb{S}^{N-1}): \ h(x)=\frac{1}{|x|}\zeta\big( \frac{x}{|x|} \big) \ \text{for $x\in B_R$ a.e.} \right\}.
\end{equation}
Thus the Hilbert projection theorem yields that $\mathcal W^{1,2}(B_R) = \mathcal{H}(R) \oplus \mathcal{H}(R)^{\perp}$ and we can define the the corresponding projection operator as $\mathcal{P}_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mathcal{H}(R)$. Using this, we can measure the error $u-\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.

{\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity,
 can it asymptotically become self-similar?}}

Our main result in this direction can be stated as follows:

\begin{equation}\label{eq:problem}
\left.\begin{array}{rrr}
u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3, 4, 5,\\
\div u =0,
\end{array}
\right\}
\quad\mbox{in}\ \Omega\subset \R^5
\end{equation}

\begin{equation}\label{eq:problem-3d}
\left.\begin{array}{rrr}
u_t^i+u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3,\\
\div u =0,
\end{array}
\right\}
\mbox{in}\ \Omega\times(0, T), \Omega\subset \R^3.
\end{equation}

{\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity,
 can it asymptotically become self-similar?}}

If $\liminf_{r\to 0}M(r)<\infty$
then the singularity may occur only if the function
\begin{equation}
\text{\Large$\wp$}(r) =\frac{1}{r^2} \int_{B_r} \Big\{{|u|^2} + 2P \Big\} (u\cdot \frac{x}{|x|}) \, dx .
\end{equation}
takes nonpositive values as $r\to 0$. Moreover,
if $u$ is of the form $\frac{\zeta(\frac x{|x|})}{|x|}$ then one can check that
$\text{\Large$\wp$}(r) =0$. This observation motivates the formulation of
a condition in our next result that allows to control $\text{\Large$\wp$}(r)$.
\begin{theorem}\label{TH}
Let $u$ be a suitable weak solution of \eqref{eq:problem}, $B_1\subset \Omega$, and
\[
m:=\liminf_{R\to 0}M(R)<\infty \qquad \text{where} \quad M(R):=\int_{B_R} \Big( \frac{|u|^2}{R^3} + \frac{|\nabla u|^2}{R} \Big).
\] Let $\mP_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mH(R)$ be the projection operator for the space \eqref{mH1}. There exists $\ep(m)>0$
such that
if
\begin{equation}\label{eq:13-alt}
\frac{1}{R^3} \int_{B_R} \left| u - \mP_R[u] \right|^2 + \frac{1}{R} \int_{B_R} \left| \nabla u - \nabla \mP_R[u] \right|^2 \le \ep(m) M(R)
\end{equation}
holds
for all $R\in (0, R_0)$ then $u$ is regular at $x=0$.
\end{theorem}

\begin{proposition}\label{prop:dAdr}
 Suppose $N=5$ and $(u,P)$ is a suitable weak solution of \eqref{st-NS-N}. For $r>0$, define
 \begin{equation}
 \left\{
 \begin{aligned}
 &D(r)\vcentcolon= \int_{B_r} \Big\{ \frac{15}{4r^3}|u|^2 + \frac{1}{4r}|\nabla u |^2 + \frac{3}{4r^3}\left| \nabla (|x|u) \right|^2 + \frac{3(r^2-|x|^2)}{4r^3} |\nabla u|^2\Big\}\, dx,\\
 &A(r) \vcentcolon= \frac{1}{r^3}\int_{B_r}(x\cdot \nabla)\frac{|u|^2}{2}\, dx + \frac{9}{4r^3}\int_{ B_r}|u|^2 \, dx - \frac{1}{r^{2}} \int_{B_r} \big\{ \frac{|u|^2}{2} + P \big\} u\cdot \frac{x}{|x|} \, dx .
 \end{aligned}
 \right.
 \end{equation}
 Then the following differential equation holds for $r>0$,
 \begin{equation*}
 \frac{d A}{d r} \ge \frac{1}{r} D(r) + \frac{2}{r^3} \int_{B_r} \Big\{\frac{|u|^2}{2} + P \Big\} (u\cdot \frac{x}{|x|}) \, dx .
 \end{equation*}
\end{proposition}
\begin{proof}
 Let us consider the function
 \begin{equation}
 \phi(x)=
 \left\{
 \begin{array}{lll}
 1 \quad & \mbox{if} \ |x|<r,\\
 \frac{r-|x|}\epsilon \quad &\mbox{if} \ r\le |x|\le r+\epsilon,\\
 0 \quad & \mbox{if}\ |x|>r+\epsilon.\\
 \end{array}
 \right.
 \end{equation}
 We then mollify this function and take $\psi=\phi*\rho_\delta$,
 where $\rho$ is the mollification kernel.
 Note that $\psi\in \mathcal C_c^\infty(\R^N)$.
 We use $\psi$ as a test function in the local
 energy inequality to obtain
 \begin{equation}
 \int |\nabla u|^2 \psi
 \le
 \int \left(-u \nabla u +(|u|^2+2P)u\right)\cdot \nabla \psi.
 \end{equation}

\begin{lemma}\label{lemma:reg}
Suppose $m:=\liminf_{R\to 0}M(R)<\infty$.
 If there exists a sufficiently small $\ep>0$, depending on $m$, such that $(u,P)$ is a suitable weak solution to the Navier-Stokes equations satisfying
 \begin{equation}\label{H-Assump}
 \ep \frac{1}{R^{3}}\int_{B_R} |p| + \frac{1}{R^3} \int_{B_R} \left| u - \mP_R[u] \right|^2 + \frac{1}{R} \int_{B_R} \left| \nabla u - \nabla \mP_R[u] \right|^2 \le \ep M[u](R),
 \end{equation}
 for all $R\in(0,1]$ then $u$ is regular at $x=0$.
\end{lemma}
\begin{proof}
If $m\le 8C_E$ then we can apply Proposition \ref{prop:fM}, and
hence the result follows.
 Now suppose $m\in(8C_E, \infty)$. In light of Lemma \ref{lemma:MBound}, there exists $\ep_1>0$ such that if $(u,P)$ satisfies \eqref{H-Assump} with $\ep\in (0,\ep_1)$ then $M(R)$ is uniformly bounded in $R\in(0,1]$ and we set
 \begin{equation}\label{Mast}
 M_{\ast} \vcentcolon= \sup\limits_{0<R\le 1} M(R) \le 2^5 C_0(m),
 \end{equation}
 where $C_0(m)$ is given by \eqref{eq:Cmm}.
 At this point we can apply Proposition \ref{prop:fM}
 with $C_0=2^5C_0(m)$ by choosing $\ep_0$ sufficiently small, however we can avoid this by using the monotonicity formula.

\begin{lemma}\label{lem:Lin}
If $u$ is a suitable weak solution and
\[
\int_{B_1} |u|^3+|p|^{\frac32}<\ep^*
\]
for some sufficiently small $\ep^*$,
then
\begin{equation}\label{eq:Lin1}
\frac1{\theta^5}\int_{B_\theta}\frac{|u-[u]_\theta|^3}{\theta^{\alpha_0}}+\frac{|p-[p]_\theta|^\frac32}{\theta^{\alpha_0}}\le \frac12 \int_{B_1} |u|^3+|p|^{\frac32},
\end{equation}
for some positive $\theta$ and $\alpha_0\in (0, \frac12)$.
\end{lemma}
\begin{proof}
If \eqref{eq:Lin1} fails, then there would be a sequence of solutions
$(u_i, p_i)$ such that $\int_{B_1}|u_i|^3+|p_i|^{\frac32}:=\epsilon_i\to 0$ but
\eqref{eq:Lin1} is not valid. Introduce
\[
\bar u_i=\frac{u_i}{\epsilon_i}, \quad \bar p_i=\frac{p_i}{\epsilon_i},
\]
then
\begin{equation}\label{eq:Lin4}
\epsilon_i \bar u_i\cdot \nabla \bar u_i+\nabla \bar p_i=\Delta \bar u_i,
\quad
\frac1{\theta^5}\int_{B_\theta}\frac{|\bar u-[\bar u]_\theta|^3}{\theta^{\alpha_0}}
+
\frac{|\bar p-[\bar p]_\theta|^\frac32}{\theta^{\alpha_0}}> \frac12,
\quad \int_{B_1} |\bar u_i|^3+|\bar p_i|^{\frac32}\le 2.
\end{equation}
From the local energy inequality $u\in \mathcal W^{1, 2}_{loc}(B_1).$
Moreover, the following equation is satisfied in distributional sense
\begin{equation}
\Delta \bar p_i=-\epsilon_i \frac{\partial^2 (\bar u^k\bar u^l)}{\partial x_l\partial x_k},
\quad \mbox {in}\ B_1.
\end{equation}
From the Poisson representation theorem we can write $\bar p_i=h_i+g_i$,
where $h_i$ is harmonic in $B_1$, and
\begin{equation}
\left\{
\begin{array}{lll}
\Delta g_i=-\epsilon_i \frac{\partial^2 (\bar u^k\bar u^l)}{\partial x_l\partial x_k}
\quad \mbox {in}\ B_{\frac23},\\
g_i=0 \quad \mbox {on}\ \partial B_{\frac23}.
\end{array}
\right.
\end{equation}
From the Calder\'on-Zygmund estimates
$g_i$ is uniformly bounded in ${L^{5/3}(B_{2/3})}$.
Consequently, $h_i\in L^{3/2}(B_{2/3})$ uniformly, hence
from the local estimates for the harmonic functions
\begin{align}
\int_{B_\theta}|\bar p_i-[\bar p_i]_\theta|^{\frac32}\le
\int_{B_\theta}| h_i-[h_i]_\theta|^{\frac32}
\int_{B_\theta}|g_i-[g_i]_\theta|^{\frac32}\\
\le C_0 \theta^5\theta^{3/2}+C_0\epsilon_i\int_{B_{2/3}}|\bar u_i|^3.
\end{align}
For a suitable subsequence $\bar u_i\to \bar u$ in $\mathcal W^{1, 2}(B_{2/3})$
and $\bar p_i\to \bar p$ strongly in ${L^{3/2}(B_{2/3})}$.
Consequently, for sufficiently large $i$, we have
\begin{equation}\label{eq:Lin2}
\int_{B_\theta}|\bar p_i-[\bar p_i]_\theta|^{\frac32} \le C_0 \theta^5\theta^{3/2}
\end{equation}
Since the limit $\bar u$ solves the Stokes system, then it follows that
$\bar u$ is H\"older continuous with, say, exponent $2\alpha_0$, and therefore
$\int_{B_\theta}|\bar u-[\bar u]_\theta|^{\frac32}\le \frac14 \theta^5\theta^{\alpha_0}$.
From the strong convergence $\bar u_i\to \bar u$ in $L^3(B_{2/3})$, we infer that
\begin{equation}\label{eq:Lin3}
\int_{B_\theta}|\bar u-[\bar u]_\theta|^{\frac32}\le \frac13 \theta^5\theta^{\alpha_0}.
\end{equation}
Combining \eqref{eq:Lin2} and \eqref{eq:Lin2} we get a contradiction with the
second inequality in \eqref{eq:Lin4}.
\end{proof}

The proof of Theorem \ref{TH-00} uses the monotonicity
formula introduced in Proposition \ref{prop:dAdr}, and
a scaling argument. See Lemma \ref{lem:positiveinf} for the proof.
Note that there are no smallness assumptions in the
statement of Theorem \ref{TH-00}.

If $\liminf_{r\to 0}M(r)<\infty$
then the singularity may occur only if the function
\begin{equation}
\text{\Large$\wp$}(r)  =\frac{1}{r^2} \int_{B_r}  \Big\{{|u|^2} + 2P \Big\} (u\cdot \frac{x}{|x|})  \, dx .
\end{equation}
takes nonpositive values as $r\to 0$. Moreover,
if $u$ is of the form $\frac{\zeta(\frac x{|x|})}{|x|}$ then one can check that
$\text{\Large$\wp$}(r) =0$. This observation motivates the formulation of
a condition in our next result that allows to control $\text{\Large$\wp$}(r)$.
\begin{theorem}\label{TH}
Let $u$ be a suitable weak solution of \eqref{eq:problem}, $B_1\subset \Omega$, and
\[
m:=\liminf_{R\to 0}M(R)<\infty \qquad \text{where} \quad M(R):=\int_{B_R} \Big( \frac{|u|^2}{R^3} + \frac{|\nabla u|^2}{R} \Big).
\] Let $\mP_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mathcal{H}(R)$ be the projection operator for the space \eqref{mH1}. There exists $\varepsilon(m)>0$
such that
if
\begin{equation}\label{eq:13-alt}
\frac{1}{R^3} \int_{B_R} \left| u - \mP_R[u] \right|^2 + \frac{1}{R} \int_{B_R} \left| \nabla u - \nabla \mP_R[u] \right|^2 \le  \varepsilon(m) M(R)
\end{equation}
holds
for all $R\in (0, R_0)$ then $u$ is regular at $x=0$.
\end{theorem}

It is known that if $u\in \mathcal{H}$ then $u=0$ \cite{Sverak}. In this context, Theorem \ref{TH}
states that if $\|u-\mP_R[u]\|_{\mathcal{W}^{1,2}(R)}$ is small compared to $\|u\|_{\mathcal{W}^{1,2}(R)}$ then
$\text{\Large$\wp$}(r)$ is smaller than $M^{\frac32}(r)$, which
after application of Proposition \ref{prop:dAdr} implies that
$u=0$.

As opposed to the main result in \cite{CKN}, we do not assume that $u$ is small in some scale invariant seminorm, reminiscent to
the ``local" Reynolds number $\frac1{r}\int_{B_r}|\nabla u|^2$. This  leads us to the classification of the
self-similar solution of the incompressible Euler equations in $\mathbb{R}^5$.
In fact, we prove that for such solutions
the Bernoulli pressure is zero. This is the first  key point in our proof of the main technical result, Proposition \ref{prop:cubicEst}.

The second key point is the construction of a monotonicity formula for the suitable weak solutions, which follows from
the weak energy inequality.

We compare Theorem \ref{TH} with the well-known
regularity criteria for suitable weak solutions of
\eqref{eq:problem-3d}, which in its most general form, can be stated as follows:
let $Q(R)=B_R\times(-R^2, 0)$ and define the local Reynolds numbers
\[
E(R)=\frac1R\int_{Q(R)}|\nabla u|^2, \quad C(R)=\frac1{R^2}\int_{Q(R)}|u|^3.
\]

Then the following statement holds: for every $M>0$ there is
$\varepsilon(M)>0$ such that
$\limsup_{R\to 0} C(R)<M, \liminf_{R\to 0}E(R)<\epsilon(M)$ imply that the origin is a regular point.

This result can be found in
Seregin's paper \cite{Seregin}, Theorem 1.4. Note that
 $\limsup_{R\to 0} C(R)<M$ implies that there is a constant $C_0(M)$
 depending on $M$ such that $\limsup_{R\to 0} E(R)<C_0(M)$ \cite{Lewis}.

As opposed to this result, we do not impose the finiteness of upper limit
of the Reynolds number. Instead, we assume that the lower limit is finite, i.e.
\[
\liminf_{R\to 0} M(R)<\infty.
\]

This is the replacement of the condition $\limsup_{R\to 0} C(R)<M$ in \cite{Seregin}.

As for the other condition, $\liminf_{R\to 0}E(R)<\epsilon(M)$, it is replaced by closeness assumption: more precisely, we assume that there is a vectorfield $h$
homogeneous of degree negative one such that
$u-h$ has a suitable small norm compared to $M(R)$.
See Section \ref{sec:energy}
for precise definitions.
Hence our conditions are weaker.

The paper is organized as follows:
In Section \ref{sec:mon} we introduce one of our main technical tools, the
monotonicity formula and prove Theorem \ref{TH-00}. In the next section
we classify the self-similar solutions of the incompressible Euler equations in
$\mathbb R^5$, and prove that for such solutions the Bernoulli
pressure  is zero.
In Section \ref{sec:energy} we prove some estimates for the pressure.
Section \ref{sec:technical} contains one of our main estimates of the
cubic term that appears in the local energy inequality.
In order to control the growth of $M(R)$ we prove an iteration result in Section \ref{sec:iteration}, and apply it to obtain a local bound in Section
\ref{sec:sup-est}.
The proof of Theorem \ref{TH} is given in Section.
\ref{sec:proofofmain}.
We also added an appendix at the end of the paper that contains
some estimates and computations used in the proof of Theorem \ref{TH}.

The post-theorem introduction says: “The proof of Theorem \ref{TH-00} uses the monotonicity formula introduced in Proposition \ref{prop:dAdr}, and a scaling argument. See Lemma \ref{lem:positiveinf} for the proof.”

The post-theorem introduction says: “To prove the main theorem, the proof uses the monotonicity formula introduced in the following proposition.

\begin{proposition}\label{prop:dAdr}
	Suppose $N=5$ and $(u,P)$ is a suitable weak solution of \eqref{st-NS-N}. For $r>0$, define
	\begin{equation}
		\left\{
		\begin{aligned}
			&D(r)\vcentcolon=   \int_{B_r} \Big\{ \frac{15}{4r^3}|u|^2 + \frac{1}{4r}|\nabla u |^2 + \frac{3}{4r^3}\left| \nabla (|x|u) \right|^2 + \frac{3(r^2-|x|^2)}{4r^3} |\nabla u|^2\Big\}\, dx,\\
			&A(r) \vcentcolon= \frac{1}{r^3}\int_{B_r}(x\cdot \nabla)\frac{|u|^2}{2}\, dx + \frac{9}{4r^3}\int_{ B_r}|u|^2 \, dx  - \frac{1}{r^{2}} \int_{B_r} \big\{ \frac{|u|^2}{2} + P \big\} u\cdot \frac{x}{|x|}  \, dx .
		\end{aligned}
		\right.
	\end{equation}
	Then the following differential equation holds for $r>0$,
	\begin{equation*}
		\frac{d A}{d r} \ge \frac{1}{r} D(r) + \frac{2}{r^3} \int_{B_r}  \Big\{\frac{|u|^2}{2} + P \Big\} (u\cdot \frac{x}{|x|})  \, dx .
	\end{equation*}
\end{proposition}

It also uses a scaling argument. For the proof, see the following lemma.

\begin{lemma}\label{lem:positiveinf}
Let
\begin{equation}
Q(r):= \int_{B_r} \Big\{ \frac{15}{4r^3}|u|^2
+
\frac{1}{4r}|\nabla u |^2
+
\frac{3(r^2-|x|^2)}{4r^3} |\nabla u|^2\Big\}\, dx,
\end{equation}
and
\begin{equation}
\text{\Large$\wp$}(r)  =\frac{1}{r^2} \int_{B_r}  \Big\{{|u|^2} + 2P \Big\} (u\cdot \frac{x}{|x|})  \, dx .
\end{equation}
If  $\liminf_{r\to 0} M(r)<\infty$ and
\[
\liminf_{r\to0^+}\left[ Q(r)
+\text{\Large$\wp$}(r)
\right]> 0
\]
then $x=0$ is a regular point.
\end{lemma}”,

\label{TH-00}
Let $u$ be a suitable weak solutions. Suppose that
the following two conditions hold:
\begin{equation}
\liminf_{r\to 0}M(r):=\liminf_{r\to 0}\int_{B_r}\!\! \left( \frac{|u|^2}{r^3} + \frac{|\nabla u|^2}{r} \right) <\infty, \quad \liminf_{r\to 0}\frac1{r^2}\int_{B_r}\!\!(|u|^2+2P)u\cdot\frac x{|x|}>0.
\end{equation}
To prove the main theorem, we show that for any $\{r_k\}_{k=1}^\infty, r_k\downarrow 0$ there is a subsequence $r_{k_m}$ such that
the scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to
a homogenous vector field of degree negative one, and hence
$x=0$ is a regular point.,

Question:
Let (u,P) be a suitable weak solution of the stationary incompressible NavierStokes system in a neighborhood of x=0 in \mathbb{R}^5,
\[
u^j\partial_j u^i+\partial_i P=\Delta u^i,\qquad \operatorname{div}u=0,\qquad i=1,\dots,5.
\]
For r>0, define
\[
M(r)=\int_{B_r}\left(\frac{|u|^2}{r^3}+\frac{|\nabla u|^2}{r}\right),
\]
and assume
\[
\liminf_{r\to 0} M(r)<\infty,
\qquad
\liminf_{r\to 0}\frac1{r^2}\int_{B_r} (|u|^2+2P)\,u\cdot\frac{x}{|x|}>0.
\]
If the rescaled velocities are defined by u_r(x)=r\,u(rx), which conclusion holds?

For every sequence \(r_k\downarrow 0\), there exists a subsequence \(r_{k_m}\) such that the scaled solutions \(u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)\) converge to a vector field that is homogeneous of degree \(-1\); consequently, \(x=0\) is a regular point of \(u\) (equivalently, \(u\) is bounded near \(0\)).

Choices:
B. For every sequence \(r_k\downarrow 0\), the full family of scaled solutions \(u_{r_k}(x)=r_k u(r_kx)\) converges, without passing to a subsequence, to a vector field that is homogeneous of degree \(-1\); consequently, \(x=0\) is a regular point of \(u\).
C. The point \(x=0\) is a regular point of \(u\) (equivalently, \(u\) is bounded near \(0\)).
D. There exists \(r_0>0\) such that for every \(0<r<r_0\), the scaled velocity \(u_r(x)=r\,u(rx)\) is itself homogeneous of degree \(-1\); consequently, \(u\) is asymptotically self-similar near \(0\) and \(x=0\) is a regular point.
E. For every sequence \(r_k\downarrow 0\), there exists a subsequence \(r_{k_m}\) such that the scaled solutions \(u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)\) converge to a vector field that is homogeneous of degree \(-1\); consequently, \(x=0\) is a singular point of \(u\).


\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, and let $\iota:M\to\mathbb{R}^D$ be the inclusion. For $c>0$, equip the oriented Grassmannian $\mathbf{Gr}^+(D,d)$ with the rescaled metric $g^{\mathbf{Gr}_c^+(D,d)}=c\,g^{\mathbf{Gr}^+(D,d)}$, let $\bar{\mathbf g}_c^+:M\to \mathbf{Gr}_c^+(D,d)$ be the rescaled Gauss map, and define the metric
$$g_c=(\iota\times \bar{\mathbf g}_c^+)^*(g^{\mathbb{R}^D}\oplus g^{\mathbf{Gr}_c^+(D,d)}).$$
Write $\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\mathrm{vol}(M)$ for the Euclidean volume of $M\subset\mathbb{R}^D$. Let $\mathbf{II}$ be the second fundamental form of $M\subset\mathbb{R}^D$, let $\nabla_v^M\mathbf{II}$ be its covariant derivative in the direction $v$, and let $\|\cdot\|_{\mathrm{HS}}$ denote the Hilbert--Schmidt norm. Fix a point $p\in M$ and a unit vector $v\in T_pM$. Let $\kappa(v)$ be the normal curvature of $M\subset\mathbb{R}^D$ at $p$ in the direction $v$, and let $\kappa_c(v)$ be the corresponding normal curvature for the graph embedding $(M,g_c)\subset \mathbb{R}^D\times \mathbf{Gr}_c^+(D,d)$.

Assume that
$$\mathrm{vol}_c(M) \le \bigl(1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2\bigr)^{d/2}\,\mathrm{vol}(M),$$
and
$$\kappa(v)>\frac{\|(\nabla_v^M\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}}{\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}}.$$
Which conclusion about these quantities 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 a strictly decreasing function of $c$.

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

speed contribution enters quadratically in the denominator

explicit intermediate bound and monotonicity statement removed

uniform quantifier for all c and strict monotonicity weakened to eventual/nonincreasing

direction of the comparison between \kappa_c(v) and the intermediate reparameterized bound