# Volume of Tubes and Concentration of Measure in Riemannian Geometry

S.L. Cacciatori1,2 and P. Ursino3 1 Department of Science and High Technology,
Università dell’Insubria, Via Valleggio 11, IT-22100 Como, Italy  2 INFN
sezione di Milano, via Celoria 16, IT-20133 Milano, Italy  3 Department of
Mathematics and Informatics, Università degli Studi di Catania, Viale Andrea
Doria 6, 95125 Catania, Italy

###### Abstract.

We investigate the notion of concentration locus introduced in [CacUrs22], in
the case of Riemann manifolds sequences and its relationship with the volume
of tubes. After providing a general formula for the volume of a tube around a
Riemannian submanifold of a Riemannian manifold, we specialize it to the case
of totally geodesic submanifolds of compact symmetric spaces. In the case of
codimension one, we prove explicitly concentration. Then, we investigate for
possible characterizations of concentration loci in terms of Wasserstein and
Box distances.

## Introduction

A concentration locus is roughly speaking a sequence of sub-manifolds
$(M_{n},\sigma_{n},g_{n})$ (where $g_{n}$ is the geodetic metric and $\mu_{n}$
the volume measure) which approximates the concentration behaviour of the
manifolds $(N_{n},\mu_{n},g_{n})$ where they are embedded. In a sense, the
concentration character of the “big” sequence is fully determined by the
“thin” one. This phenomenon is particularly significant, from the point of
view of applications, whenever it is possible to single out, inside a sequence
of manifolds of unknown concentration behaviour, a sequence of much simpler
sub-manifolds which is a concentration locus and therefore, provided the
concentration behaviour of the sub-manifolds is known, it determines the
concentration behaviour of the big one.

Both sequences can be regarded as sequences of metric-measurable spaces (mm-
spaces). In the space $\mathfrak{M}$ of all mm-spaces Gromov, in his
celebrated green book [Gro99], defines the notion of observable distance
($d_{conc}$ in [Shi]) which fully generalizes the classical phenomenon of
concentration of measure to a point and Levy family (see for example [GM]).
Practically, we say that a sequence of mm-spaces concentrates to an mm-space,
whenever the former $d_{conc}$ converges to the latter.

Our primary goal consists in finding a way to detect if a sequence of sub-
manifolds is or is not a concentration locus by investigating the sequences of
volumes of tubes built around the sub-manifolds, with decreasing rays.

The problem of calculating the volume of a tube in a Riemann manifold is very
interesting in itself and it is treated in Section 1. We start from the well-
known article of Weyl [We] and the beautiful book of Gray [Gray]. We succeed,
using the approach of Gray, in finding a general formula for the volume of
tubes which involves the codimension, the ray of the tube, the curvature of
the ambient manifold and the killing curvatures of the sub-manifold.
Unfortunately, it is widely useless in computing concentration locus unless
you have an estimate of the curvature derivatives of every order. The case of
symmetric spaces is much more practicable, instead. Indeed, we find a concrete
formula and we are able to extend the system of coordinates used for
calculating the volume of the tubes to the entire ambient manifold and
therefore calculate the asymptotic conditions to detect a concentration locus.
We believe that our results can be easily extended to the case of homogeneous
spaces.

Observable distance is a very difficult tool to deal with, fortunately, there
are more practical distances that are related to $d_{conc}$. For example,
Wasserstein distance $d_{W}$, which is related to optimal transport, and
$d_{box}$, which makes $\mathfrak{M}$ a complete metric space. The following
holds $d_{W}\Rightarrow d_{box}\Rightarrow d_{conc}$ even if $d_{W}\nLeftarrow
d_{box}\nLeftarrow d_{conc}$ (there are $d_{conc}$ cauchy sequence which does
not converge [ShiKa]). On the other side, it is very hard to find
differentiable optimal transport. Indeed, McCann et al. showed that the Monge-
Kantorovich problem is solvable for smooth Riemannian manifolds [McCannFeld].
Regarding differentiable solutions, Figalli, Rifford, and Villani [FigRiffVil]
solved it in the positive for the spheres [FigRiffVil], while McCann and
Young-Heon for $\mathbb{CP}^{n}$ [McCannYoung].

However, our Wasserstein problem is a special one, since we are looking for a
transport to the push forward measure induced by the transport itself. In this
case, the problem assumes differential geometry features and a viscosity
solution of the following Hamilton-Jacobi equation

$\left\\{\begin{array}[]{@{}l@{}}|\nabla u|^{2}=1\ \mbox{in}\ M\setminus C\\\
u=0\ \mbox{in}\ C\end{array}\right.$

provides an answer to our problem. In particular, the distance function, which
solves the previous equation, singles out a projection function which is
exactly the function that minimizes the distance from the target sub-manifold.
This solution is unique up to a measure null set (focal points) and up to this
set it is differentiable.

By a well-known result, which is a consequence of Hopf-Rinow Theorem (see
Proposition 5 Cheeger [CheeEbin]), the trajectories (the geodesics) of this
projection, through which the mass is transported, are orthogonal to the
boarder of $C$. In particular, $\gamma$, the geodesic or trajectory which has
length exactly the distance, is uniquely determined and, it is the super-
differential of the distance function.

All these results can be seen in [CanSin] in $R^{n}$ context or in [ManteMen]
or in [Fath] in the Riemann geometry context.

We succeed in Section 2 in finding a characterization of concentration locus
in terms of $d_{box}$ and projection distance, and this allows us to prove
Corollary 2, which guarantees that whenever a concentration locus converges
$d_{conc}$ to an mm-space $(H,\mu,g)$, the sequence of ambient manifolds
concentrates to $(H,\mu,g)$.

## Preliminaries.

Given a set $X$, we denote by $l^{\infty}(X)$ the unital Banach algebra of all
bounded real-valued functions on $X$ equipped with the supremum norm. Let $X$
be a topological space. If the topology of $X$ is generated by a metric $d$,
then we call $d$ a compatible metric on $X$.
We will denote by $C(X)$ the set of all continuous real-valued functions on
$X$, and we set $CB(X)=l^{\infty}(X)\cap C(X)$. Let us denote by
$\mathcal{B}(X)$ the Borel $\sigma$-algebra of $X$ and by $P(X)$ the set of
all Borel probability measures on $X$. The weak topology on $P(X)$ is defined
to be the initial topology on $P(X)$ generated by the maps of the form
$P(X)\rightarrow\mathbb{R},\mu\rightarrow\int fd\mu$ where $f\in CB(X)$.
The support of a measure $\mu\in P(X)$ is defined as

$\displaystyle spt\mu=\\{x\in X\mid\forall U\subseteq X\mbox{ open: }x\in
U\Rightarrow\mu(U)>0\\}.$

Given $\mu\in P(X)$ and a Borel subset $B\subseteq X$ with $\mu(B)=1$, we let
$\mu|_{B}:=\mu|_{\mathcal{B}(B)\subset P(B)}$. The push-forward of a measure
$\mu\in P(X)$ along a Borel map $f:X\rightarrow Y$ into another topological
space $Y$ is defined to be

$\displaystyle\sharp
f\mu:\mathcal{B}(Y)\rightarrow[0,1],B\mapsto\mu(f^{-1}(B)).$

Furthermore, let us note that each $\mu\in P(X)$ gives rise to a pseudo-metric
$me_{\mu}$ on the set of all Borel measurable real-valued functions $X$
defined by

$\displaystyle me_{\mu}(f,g):=\inf\\{\epsilon>0\mid\mu(\\{x\in X\mid\mid
f(x)-g(x)\mid>\epsilon\\})\leq\epsilon\\},$

for any two Borel functions $f,g:X\rightarrow\mathbb{R}$. Let $(X,d)$ be a
pseudo-metric space. Given a subset $A\subseteq X$, we abbreviate
$d|_{A}:=d|_{A\times A}$ and define ${\rm diam}(A,d):=\sup\\{d(x,y)\mid x,y\in
A\\}$. For $x\in A\subseteq X$ and $\epsilon>0$ we set

$\displaystyle B_{d}(x,\epsilon):=\\{y\in X\mid d(x,y)<\epsilon\\}\ \
B_{d}(A,\epsilon):=\\{y\in X\mid\exists a\in A;d(x,y)<\epsilon\\}.$

Then the Hausdorff distance between any two subsets $A,B\subseteq X$ is given
by

$\displaystyle d_{H}(A,B):=\inf\\{\epsilon>0\mid B\subseteq
B_{d}(A,\epsilon),\ A\subseteq B_{d}(B,\epsilon)\\}.$

For $l,r\geq 0$, we denote by $Lip_{l}(X,d)$ the set of all $l$-Lipschitz
real-valued functions on $(X,d)$, and we define

$\displaystyle Lip_{l}^{\infty}(X,d):=Lip_{l}(X,d)\cap l^{\infty}(X),\
Lip_{l}^{r}(X,d):=\\{f\in Lip_{l}(X,d)\mid\|f\|_{\infty}\leq r\\}.$

Moreover, we set $Lip(X,d):=\bigcup\\{Lip_{l}(X,d)\mid l\geq 0\\}$ and
$Lip(X,d)^{\infty}:=Lip(X,d)\cap l^{\infty}(X)$.
Whenever $(X,d)$ is a separable metric space, the Wasserstein distance
$W_{1}(\mu,\nu)$111Different names appearing in the literature include Monge-
Kontorovich distance, bounded Lipschitz distance, mass transportation
distance, and Fortet-Mourier distance [VilBook] is a compatible metric for a
weak topology on $P(X)$ defined by

$\displaystyle W_{1}(\mu,\nu):=\sup_{f\in Lip_{1}^{1}(X,d)}\left|\int
fd\mu-\int fd\nu\right|\ \ (\mu.\nu\in P(X)).$

######  Definition 1.

(Gromov-Milman [Gro99])
A space with a metric $g$ and a measure $\mu$, or an mm-space, is a triple
$(X,\mu,g)$, consisting of a set $X$, a metric $g$ on $X$ and a probability
Borel measure such that $(X,g)$ is a separable complete metric space.

Moreover, an mm-space $(X,\mu,d)$ is called compact if $(X,d)$ is compact, and
fully supported if $spt\mu=X$. Henceforth, we will denote by $\lambda$ the
Lebesgue measure on $[0,1)$.
A parametrization of an mm-space $(X,\mu,d)$ is a Borel measurable map
$\phi:[0,1)\rightarrow X$ such that $\sharp\phi\lambda=\mu$. It is well known
that any mm-space admits a parametrization (see, e.g.[Shi]). In the set of
isomorphism classes of mm-spaces, $\mathfrak{M}$, we can define the box
distance, $d_{box}$, that we are going to define ([Shi]). For two pseudo-
metrics $\rho_{1}$ and $\rho_{1}$ on the unit interval $I$, we define
$d_{box}(\rho_{1},\rho_{2})$ to be the infimum of $\epsilon>0$ satisfying that
there exists a Borel subset $I_{0}\subseteq I$ such that

  1. (1)

$|\rho_{1}(s,t),\rho_{2}(s,t)|\leq\epsilon$ for any $s,t\in I_{0}$,

  2. (2)

$\mathfrak{L}^{1}(I_{0})\geq 1-\epsilon$ where $\mathfrak{L}^{1}$ denotes the
one-dimensional Lebesgue measure.

######  Definition 2.

let $X$ be a topological space with a Borel probability measure $\mu_{X}$. A
map $\varphi:I\rightarrow X$ is called a parameter of $X$ if $\varphi$ is a
Borel measurable map such that

$\sharp\varphi\mathfrak{L}^{1}=\mu_{X}$

We define box distance $d_{box}$ between two isomorphism classes of mm-spaces
$X,Y$ to be the infimum of $d_{box}(\sharp\varphi d_{X},\sharp\psi d_{Y})$
where $\varphi:I\rightarrow X$ and $\psi:I\rightarrow Y$ run over all
parameters of $X$ and $Y$, respectively, and where $\sharp\varphi
d_{X}(s,t):=d_{X}(\varphi(s),\varphi(t))$.

######  Definition 3.

(Gromov-Milman [Gro99][Shi])
In the set of isomorphism classes of mm-spaces we can define the following
distance:

$\displaystyle
d_{conc}(X,Y):=\inf\\{(me_{\lambda})_{H}(Lip_{1}(X)\circ\phi,Lip_{1}(Y)\circ\psi\mid\phi\mbox{
param. of }X,\psi\mbox{ param. of }Y\\}.$

Two mm-spaces $X$ and $Y$ are isomorphic if there exists an isomorphism
between mm-spaces $(X,\mu,d),(Y,\nu,d^{\prime})$ i.e an isometry

$\displaystyle f:(spt\mu,d\mid X)\rightarrow(spt\nu,d^{\prime}\mid Y)$

such that $\sharp f(\mu\mid{spt\mu})=\nu\mid{spt\nu}$. A sequence of mm-spaces
$(X_{n},\mu_{n},g_{n})$ is said to concentrate to an mm-space $(X,\mu,g)$ if

$\displaystyle\lim_{n}d_{conc}(X_{n},X)=0.$

In this case, we denote $(X,\mu,g)$ as a concentration set for the sequence of
mm-spaces $(X_{n},\mu_{n},g_{n})$.
Finally, let us recall the definition of Concentration Locus as defined in
[CacUrs22]

######  Definition 4.

Let $\\{X_{n},\mu_{n}\\}_{n\in\mathbb{N}}$ be a family of metric spaces with
metrics $d_{n}$, and Borel’s measures $\mu_{n}$ w.r.t. which nonempty open set
have non-vanishing measure. Assume the measures to be normalized,
$\mu_{n}(X_{n})=1$. Let $\\{S_{n}\\}_{n\in\mathbb{N}}$ be a family of proper
closed subsets, $S_{n}\subset X_{n}$. Fix a sequence
$\\{\varepsilon_{n}\\}_{n\in\mathbb{N}}$ such that $\varepsilon_{n}>0$,
$\lim_{n\to\infty}\varepsilon_{n}=0$, and let
$\\{U^{\varepsilon_{n}}_{n}\\}_{n\in\mathbb{N}}$ be the sequence of tubular
neighbourhoods of $S_{n}$ of radius $\varepsilon_{n}$. We say that the family
$\\{S_{n}\\}$ is a Concentration Locus if

$\displaystyle\lim_{n\to\infty}\mu_{n}(X_{n}-U^{\varepsilon_{n}}_{n})=0.$
(0.1)

Moreover, if such a sequence $\varepsilon_{n}$ converges to 0 at rate $k$ (so
that $\lim_{n\to\infty}n^{k}\varepsilon_{n}=c$ for some constant $c$), we say
that the family $\\{S_{n}\\}$ is a Concentration Locus at least at rate $k$.

##  1\. Volume of tubes in compact manifolds.

Let $M$ be a compact Riemannian submanifold $M\subseteq N$ of codimension $q$
of a manifold $N$. We call $M_{\epsilon}$ the tube generated by all geodetic
segments of length $\epsilon$, outgoing perpendicularly from $M$. It is a
well-known result established by Weyl that the volume of the tube can be
expressed in terms of the Lipschitz-Killing curvatures $K_{2j}$ of $M$ and the
ambient space, and the codimension $q$. More precisely:

######  Theorem 1 (Weyl, [We]).

Let $M$ be a compact Riemannian submanifold of $\mathbb{R}^{N}$ of codimension
$q=N-n$. Let $M_{\varepsilon}$ a tubular neighbourhood of $M$ of radius
$\varepsilon$. Then, for all $r>0$ sufficiently small, it holds

$\displaystyle{\rm
Vol}_{\mathbb{R}^{N}}(M_{\varepsilon})=\frac{\pi^{\frac{q}{2}}\varepsilon^{q}}{\Gamma(\frac{q}{2}+1)}\left(K_{0}(M)+\sum_{j=1}^{\lfloor
n/2\rfloor}\frac{K_{2j}(M)\varepsilon^{2j}}{(q+2)(q+4)\cdots(q+2j)}\right),$
(1.1)

where

$\displaystyle K_{2j}(M)=\int_{M}k_{2j}(\Omega),$ (1.2)

are the integrated Lipschitz-Killing curvatures, and $\Omega$ is the curvature
2-form of $M$.

Remember that if $e^{a}$ is the dual basis to an orthonormal frame $V_{a}$,
$a=1,\ldots,n$, then

$\displaystyle k_{2j}(\Omega)=\frac{1}{2^{j}j!(n-2j)!}\sum_{\sigma\in
S_{n}}\epsilon_{\sigma}\Omega_{\sigma(1)\sigma(2)}\wedge\cdots\wedge\Omega_{\sigma(2j-1)\sigma(2j)}e^{\sigma(2j+1)}\wedge\cdots\wedge
e^{\sigma(n)},$ (1.3)

where $S_{n}$ is the set of permutations of $n$ elements and
$\epsilon_{\sigma}$ the sign of the permutation. In particular,

$\displaystyle k_{0}(\Omega)=d{\rm Vol}_{M},$ (1.4)

is the volume form on $M$,

$\displaystyle k_{2}(\Omega)=$ $\displaystyle\frac{1}{2}R\ d{\rm Vol}_{M},$
(1.5) $\displaystyle k_{n}(\Omega)=$ $\displaystyle{\rm Pf}(\Omega),$ (1.6)

where $R$ is the scalar curvature of $M$ and Pf the Pfaffian. Finally, notice
that

$\displaystyle\frac{\pi^{\frac{q}{2}}\varepsilon^{q}}{\Gamma(\frac{q}{2}+1)}={\rm
Vol}_{\mathbb{R}^{q}}(D_{\varepsilon})$ (1.7)

is the volume of the $q$ dimensional disc of radius $\varepsilon$ in
$\mathbb{R}^{q}$. So, if we define the mean Lipschitz-Killing curvatures
$\kappa_{2j}$ as

$\displaystyle\kappa_{2j}=\frac{K_{2j}(M)}{{\rm Vol}_{M}(M)},$ (1.8)

then we can rewrite Weyl’s formula as

$\displaystyle{\rm Vol}_{\mathbb{R}^{N}}(M_{\varepsilon})={\rm Vol}_{M}(M){\rm
Vol}_{\mathbb{R}^{q}}(D_{\varepsilon})\left(1+\sum_{j=1}^{\lfloor
n/2\rfloor}\frac{\kappa_{2j}(M)\varepsilon^{2j}}{(q+2)(q+4)\cdots(q+2j)}\right).$
(1.9)

We are interested in understanding the volumes of tubular neighborhoods of
submanifolds of compact manifolds (and, in general, on manifolds with positive
curvature). However, it is interesting to do some general considerations on
this formula before discussing the more general case.

Since there is no curvature in the directions of $\mathbb{R}^{N}$ orthogonal
to $M$, the deformations of the volume are only due to the bending of $M$. If
$M$ is flat, then the volume of the tube is just the product of the volumes of
the submanifold and the disc. The terms in the parenthesis then give the
contributions of the deformations of the tube to the volume, when we bend the
tube along a curved $M$. For example, if we bend the tube neighborhood of a
segment to the one of a circle, the tube will be compressed along the most
internal circle and stretched along the most external one. However, the volume
doesn’t change (if we don’t change the length of the segment), indeed the
scalar curvature of the circle is $R_{S^{1}}=0$.
Now, since $\kappa_{2j}$ are mean curvatures, we can get some hints about the
dependence on curvatures by considering $M$ to be a manifold of constant
sectional curvature $1/r^{2}$. In this case, one has

$\displaystyle\varepsilon^{2j}\kappa_{2j}(\Omega)=\frac{n!}{2^{j}j!(n-2j)!}\left(\frac{\varepsilon}{r}\right)^{2j}.$
(1.10)

We want to see under which conditions the curvature terms become relevant in
the formula (1.9) when $n$ grows. In general, this also could imply that
generically also $q$ grows. Keeping $j$ fixed, we see that Stirling’s formula
implies

$\displaystyle\varepsilon^{2j}\kappa_{2j}(\Omega)\approx\frac{1}{2^{j}j!}\left(\frac{n\varepsilon}{r}\right)^{2j}.$
(1.11)

If $\varepsilon/r$ is small but constant, the curvature terms become dominant
for large $n$, at least if $q$ is constant. Despite we are considering spheres
embedded in $\mathbb{R}^{n+q}$, let us for a moment imagine assuming $q=1$ and
that $\mathbb{R}^{n+q}$ is replaced by $S_{r}^{n+q}$. In this case, it is well
known that the measure of the whole sphere concentrates in a tube of radius
$\varepsilon\sim rn^{-\frac{1}{2}}$ around the equator $M$. In this situation,
we see that

$\displaystyle\varepsilon^{2j}\kappa_{2j}(\Omega)\approx\frac{1}{2^{j}j!}n^{j},$
(1.12)

so the curvature terms are dominant w.r.t. the 1. If also the codimension
$q\equiv q_{n}$ increases unboundedly, then, including the denominators, we
see that the contributions are of the order

$\displaystyle\frac{1}{2^{j}j!}(n/q_{n})^{j},$ (1.13)

so that the dominance of the curvatures persists if $q_{n}$ grows slower than
$n^{1-a}$ for any fixed arbitrarily small but positive $a$. This is also
another well-known condition for concentration. Why should we consider these
considerations acceptable if we replace the flat ambient space with spheres?
The reason is that the spheres have curvature $1/r^{2}$ much smaller than the
inverse square radius of the tube, $\sim n/r^{2}$. This suggests that in
general, we may obtain the same results if we have an estimation of the bound
of the curvature of the ambient manifold $N$ and a control on the error we
make in using the flat formulas as a function of the radius of the tube.

Since we are interested in compact manifolds, we recall that Weyl deduced the
exact formula for the case when the embedding space is a sphere. From this
result we can deduce:

######  Proposition 1.

Let $M\subset S_{R}^{n+q}$ a $n$ dimensional smooth compact submanifold of
codimension $q$ of a sphere of radius $R$, and $M_{\varepsilon}$ the tube of
radius $\varepsilon$ around $M$. Then, we can write

$\displaystyle{\rm Vol}_{S_{R}^{n+q}}(M_{\varepsilon})={\rm
Vol}^{flat}(M_{\varepsilon})\left(1+o(\varepsilon/R)\right),$ (1.14)

where ${\rm Vol}^{flat}_{N}(M_{\varepsilon})\equiv{\rm
Vol}_{\mathbb{R}^{n+q}}(M_{\varepsilon})$ is given by Weyl’s formula for the
embedding in the flat $\mathbb{R}^{n+q}$.

Remark: Before giving the proof, let us notice that this proposition has an
immediate consequence: if $\varepsilon\ll R$ then we can use the result of the
previous section to deduce the properties of concentration of the measure
around $M$, up to a relative error controlled by $\varepsilon/R$.

###### Proof.

Let us set $N=n+q$. In [We], Weyl deduced the following formula for the volume
of a tube of “radius $a$” in a sphere of radius $R$:

$\displaystyle{\rm
Vol}_{S^{N}}(M_{\varepsilon})=\frac{2\pi^{\frac{q}{2}}}{\Gamma(\frac{q}{2})}\sum_{j=0}^{\lfloor
n/2\rfloor}\frac{K_{2j}(M)R^{2j+q}}{q(q+2)\cdots(q+2j-2)}\int_{0}^{\frac{\varepsilon}{R}}(\sin\rho)^{q+2j-1}(\cos\rho)^{n-2j}d\rho.$
(1.15)

Here, the radius $a$ is not the geodesic radius along the sphere but the
Euclidean radius in the tangent space. It is related to the geodesic radius
$\varepsilon$ through the relation $a=R\tan\frac{\varepsilon}{R}$. If we use
the change of variable $x=\sin^{2}\rho/\sin^{2}(\varepsilon/R)$ we get

$\displaystyle\int_{0}^{\frac{\varepsilon}{R}}(\sin\rho)^{q+2j-1}(\cos\rho)^{n-2j}d\rho=$
$\displaystyle\frac{1}{2}(\sin(\varepsilon/R))^{q+2j}\int_{0}^{1}dx\
\frac{x^{\frac{q}{2}+j-1}}{\left(1-\sin^{2}(\varepsilon/R)x\right)^{j-\frac{n-1}{2}}}$
(1.16) $\displaystyle=$
$\displaystyle\frac{(\sin(\varepsilon/R))^{q+2j}}{q+2j}{}_{2}F_{1}\left(j+\frac{q}{2},j-\frac{n-1}{2};j+\frac{q}{2};\sin^{2}\frac{\varepsilon}{R}\right).$
(1.17)

Thus,

$\displaystyle{\rm
Vol}_{S^{N}}(M_{\varepsilon})=\frac{2\pi^{\frac{q}{2}}}{\Gamma(\frac{q}{2})}\sum_{j=0}^{\lfloor
n/2\rfloor}\frac{K_{2j}(M)R^{2j+q}(\sin(\varepsilon/R))^{q+2j}}{q(q+2)\cdots(q+2j)}{}_{2}F_{1}\left(j+\frac{q}{2},j-\frac{n-1}{2};j+\frac{q}{2};\sin^{2}\frac{\varepsilon}{R}\right).$
(1.18)

Using that $\sin x=x+o(x)$ and that ${}_{2}F_{1}(a,b;c;x^{2})=1+o(x)$ for
$x\to 0$, and that the volume of the discs on a sphere are the same as in the
tangent space up to correction of order $\varepsilon/R$, we get the proof of
our assert. ∎

This corroborates our previous discussion, showing that we can use formula
(1.9) when the radius of the tube is small compared with the radius of
curvature of the sphere. Notice that the factor containing the mean curvatures
seems to suggest that one should look for submanifolds having large curvatures
in order to look for concentration phenomena. However, it is well-known that
in the case of spheres, the concentration is on equators, which are totally
geodesic submanifolds with the property that the extrinsic curvature vanishes.
Therefore, in this case, the Lipschitz-Killing curvatures are all zero,
contradicting our intuition. Before explaining why this happens, let us see
how things work in the more general case.

###  1.1. The general case

Let $M$ be a Riemannian submanifold of codimension $q$ in a compact Riemannian
manifold $N$ of dimension $q+n$. Let $M_{\varepsilon}$ a tube of radius
$\varepsilon$ around $M$. We can coordinatize the tube as follows. Fix a point
$p$ on $M$ with local coordinates $\bar{x}$. Consider the normal bundle
$\mathcal{N}M$ of $M$ in $\Sigma$ and let $\hat{n}_{j}(\bar{x})$,
$j=1,\ldots,q$, an orthonormal basis of $\mathcal{N}_{p}M$. We can introduce
polar coordinates $\theta_{a}$, $a=1,\ldots,q-1$, and director cosines
$\omega^{j}(\bar{\theta})$ to parametrise the arbitrary direction orthogonal
to $M$ as
$n(\bar{\theta};\bar{x})=\sum_{j}\omega^{j}(\bar{\theta})\hat{n}_{j}(\bar{x})$.
Let us consider the geodesic $\gamma_{\bar{x},\bar{\theta}}(t)$ in $N$
starting at $t=0$ from $p$ in the direction $n(\bar{\theta};\bar{x})$, where
$t$ is the geodesic length parameter. The tube of radius $\varepsilon$ is
defined by all such geodesics for $t\leq\varepsilon$. For $\varepsilon$ small
enough it is well defined. We use the coordinates $(\bar{x},\bar{\theta},t)$
in the tube. In order to compute the volume of the tube in the measure of $N$,
we need to compute the measure in the given local coordinates. We do it
assuming that the coordinate $\underline{x}$ cover the whole $M$ up to a
subset of vanishing measure. This is not a restriction since this hypothesis
can be replaced by the introduction of a partition of unity. In
$p\equiv\bar{x}$, let us choose an orthonormal frame in $T_{p}M$, say $e_{a}$,
$a=1,\ldots,n$, to be used as a frame of Fermi along the normal geodesic
$\gamma_{\bar{x},\bar{\theta}}(t)$. The Jacobian we are interested in is

$\displaystyle
J=(\partial_{\bar{x}}\gamma_{\bar{x},\bar{\theta}};\partial_{\bar{\theta}}\gamma_{\bar{x},\bar{\theta}};\partial_{t}\gamma_{\bar{x},\bar{\theta}}).$
(1.19)

It is clear that by construction
$\partial_{\bar{\theta}}\gamma_{\bar{x},\bar{\theta}};\partial_{t}\gamma_{\bar{x},\bar{\theta}}$
just provides the measure of the volume form of the disc generated by the
normal geodesics from $p$, say $dVol_{D_{\bar{x}}}(\bar{\theta},t)$. Notice
that for generic $M$ it is expected to depend on $\bar{x}$. The remaining
contribution can be computed by employing the Jacobi equation w.r.t. the Fermi
frame. It is (the tilde just means we are restricting to the directions of the
Fermi frame)

$\displaystyle\ddot{\tilde{J}}_{ab}+\sum_{c}R(\dot{\gamma},e_{a},\dot{\gamma},e_{c})\tilde{J}_{cb}=0,$
(1.20)

where the dot indicates derivative w.r.t. to $t$. Here
$\gamma\equiv\gamma_{\bar{x},\bar{\theta}}$ and $e_{a}$ are to be intended as
Fermi transported along the geodesic. Because of this, we have

$\displaystyle\frac{d}{dt}R(\dot{\gamma},e_{a},\dot{\gamma},e_{c})=(\nabla_{\dot{\gamma}}R)(\dot{\gamma},e_{a},\dot{\gamma},e_{c}).$
(1.21)

Notice that for $N$ compact the matrix
$R(\dot{\gamma},e_{a},\dot{\gamma},e_{c})$ is symmetric and positive definite.
In general, the solution of equation (1.20) is completely determined by the
Cauchy data $\tilde{J}_{cb}|_{t=0}$, $\dot{\tilde{J}}_{cb}|_{t=0}$. For
example, deriving (1.20) in $t=0$, we get

$\displaystyle\dddot{\tilde{J}}_{ab}+\sum_{c}R(\dot{\gamma},e_{a},\dot{\gamma},e_{c})\dot{\tilde{J}}_{cb}+\sum_{c}(\nabla_{\dot{\gamma}}R)(\dot{\gamma},e_{a},\dot{\gamma},e_{c})\tilde{J}_{cb}=0,$
(1.22)

and iterating this operation, one gets
$\left.\frac{d^{n}{\tilde{J}}_{ab}}{dt^{n}}\right|_{t=0}$ as a function of
$\tilde{J}_{ab}$, $\dot{\tilde{J}}_{ab}$ and
$R(\dot{\gamma},e_{a},\dot{\gamma},e_{c})$ and all its covariant derivatives
up to order $n-2$ along the direction $\dot{\gamma}$, in the point $p$. More
in general, we can write formally the solution in the form

$\displaystyle\tilde{J}(\bar{x},\bar{\theta},t)=J_{0}(\bar{x},\bar{\theta})+\sum_{j=1}^{\infty}(A_{j}J_{0}+B_{j}\dot{J}_{0})(\bar{x},\bar{\theta})\frac{t^{j}}{j!},$
(1.23)

where $J_{0}$, $\dot{J}_{0}$, $A_{j}$, $B_{j}$ are $n\times n$ matrix-valued
functions of $(\bar{x},\bar{\theta},t)$ defined by

$\displaystyle J_{0}=$ $\displaystyle\tilde{J}(\bar{x},\bar{\theta},0),$
(1.24) $\displaystyle\dot{J}_{0}=$
$\displaystyle\frac{d\tilde{J}}{dt}(\bar{x},\bar{\theta},0),$ (1.25)
$\displaystyle(A_{1})_{ab}=$ $\displaystyle 0,\qquad(B_{1})_{ab}=\delta_{ab},$
(1.26) $\displaystyle A_{j+1}=$
$\displaystyle\nabla_{\dot{\gamma}}(A_{j})_{ab}-\sum_{c}(B_{j})_{ac}R(\dot{\gamma},e_{c},\dot{\gamma},e_{b}),$
(1.27) $\displaystyle B_{j+1}=$
$\displaystyle\nabla_{\dot{\gamma}}(B_{j})_{ab}+(A_{j})_{ab}.$ (1.28)

Let us assume the analyticity condition that for any given
$(\bar{x},\bar{\theta})$ the series (1.23) has a strictly positive convergence
radius. Since $N$ is compact and $M$ is closed, then also $M$ is compact and
there is a minimum positive radius $\tau$, such that (1.23) converges
uniformly in any region $t\leq\varepsilon<\tau$. We can fix such an
$\varepsilon$ to define the tube. Moreover, notice that
$\tilde{J}(\bar{x},\bar{\theta},0)$ determines the change of variables along
$M$, and its determinant does not depend on $\bar{\theta}$, while

$\displaystyle\dot{\tilde{J}}(\bar{x},\bar{\theta},0)_{ab}=\sum_{c=1}^{n}\sum_{s=1}^{q}\omega^{s}(\bar{\theta})K^{s}_{ac}(\bar{x}){\tilde{J}}(\bar{x},\bar{\theta},0)_{cb},$
(1.29)

where $K^{j}_{ab}$ is the second fundamental form of the embedding of $M$
along the direction $n^{j}$ and $\omega^{j}$ are the director cosines defined
above. Therefore, we have proven the following proposition:

######  Proposition 2.

Let $M$ be a Riemannian submanifold of codimension $q$ in a compact Riemannian
manifold $N$ of dimension $q+n$. Let $M_{\varepsilon}$ be a tube of radius
$\varepsilon$ around $M$, coordinatized as above. Then, the volume element
$dV$ in the tube is

$\displaystyle
dV=dVol_{D_{\bar{x}}}(\bar{\theta},t)dVol_{M}(\bar{x})\det\left(I_{n}+\sum_{j=1}^{\infty}\left(A_{j}(\bar{x},\bar{\theta})+B_{j}(\bar{x},\bar{\theta})\sum_{s=1}^{q}\omega^{s}(\bar{\theta})K^{s}(\bar{x})\right)\frac{t^{j}}{j!},\right),$
(1.30)

where $K^{s}$ is the symmetric matrix with components $K^{s}_{ab}$, that is
the second fundamental form along the normal direction $s$, $I_{n}$ is the
$n\times n$ identity matrix.

This very general formula is clearly of poor practical usage since for
applications one needs to have control of the Riemann tensor and all its
covariant derivatives. In any case, we can see that if we look for the
concentration of the measure around $M$, the main ingredients entering into
the game are the extrinsic curvatures $K^{s}(\bar{x})$ and the volume of $M$.
Of course, large values of the curvatures may amplify the last factor.
However, if the sign of the curvatures is constant because we are looking for
convex regions, then, large values of the curvatures may correspond to a small
value of the volume of $M$. For example, it is well known that the measure of
the spheres concentrates on equators which are totally geodesic subvarieties,
thus having zero extrinsic curvatures. This maximizes the volume of $M$.
This suggests that the best candidates for the concentration of the measure
are totally geodesic subvarieties. The contribution of the curvatures should
then be to maximize the dependence on $t$ in $t=0$ through the coefficients
$A_{j}$. However, it is quite hard to say more in the general case, both
because it is not guaranteed the existence of totally geodesic subvarieties
and because it is quite hard to have a uniform control on the coefficients
$A_{j}$ and $B_{j}$. For these reasons, we now move to specific examples.

###  1.2. Compact Symmetric spaces

We want to apply our general formula to the case of compact symmetric spaces
$\Sigma=G/H$, where $G$ is a compact Lie group and $H$ is a symmetrically
embedded subgroup. The reason is that they are simple enough to allow for a
very explicit calculation of the coefficients $A_{j}$ and $B_{j}$, and, at the
same time, they contain several totally geodesic submanifolds. In a sense,
they are the simplest generalizations of $S^{N}=SO(N+1)/SO(N)$. Here we
consider the case where $G$. is a simple group, but our construction can be
extended to semisimple groups in an obvious way. We assume that the dimension
of $\Sigma$ is $N=n+q$ while $M$ is an $n$ dimensional submanifold. $\Sigma$
is endowed by a metric $g_{ij}$ that is invariant under both the left and the
right translations generated by $G$. Since $G$ is compact, the metric is
induced by the Killing form of $Lie(G)$, up to a (negative) constant. The
corresponding Riemann tensor is covariantly constant. In particular, $\Sigma$
is an Einstein manifold with Ricci tensor $R_{ij}=\frac{S}{N}g_{ij}$, where
the scalar curvature $S$ is a constant. We have the following orthogonal
decomposition

$\displaystyle Lie(G)=Lie(H)\oplus Lie(H)^{\perp}.$ (1.31)

The elements of $Lie(H)$ act as infinitesimal isometries leaving fixed the
points of $\Sigma$, while the elements of $Lie(H)^{\perp}$ generate
translations. At each point $p$ of $M$ we can take of vectors
$\\{\vec{n}_{1},\ldots,\vec{n}_{q}\\}$ forming a basis of the normal space of
$T_{p}M$ in $T_{p}\Sigma$. We can take $\vec{n}_{j}$ as elements of
$Lie(H)^{\perp}$. The disc of radius $a$ in $T_{p}M^{\perp}$ defined by

$\displaystyle
D_{a}(p)=\\{x_{1}\vec{n}_{1}+\cdots+x_{q}\vec{n}_{q}|x_{1}^{2}+\cdots+x_{q}^{2}\leq
a^{2}\\}$ (1.32)

is mapped to a geodesic disc by the exponential map. The orthogonal geodesics
from $p$ are thus of the form

$\displaystyle\gamma(t)=e^{t\sum_{j}v^{j}\vec{n}_{j}}\cdot p$ (1.33)

where $\cdot$ indicates the action of the elements of $G$ on $\Sigma$,
$\sum_{j}(v^{j})^{2}=1$, and the exponential is in the sense of groups. Then,
$\varepsilon$ is the geodesic distance of $\gamma(a)$ from $p$.
The main point now is that, since the Riemann tensor is covariantly constant,
we have

$\displaystyle(\nabla_{\dot{\gamma}}R)(\dot{\gamma},e_{a},\dot{\gamma},e_{c})=0.$
(1.34)

Therefore, the matrix

$R(\dot{\gamma},e_{a},\dot{\gamma},e_{c})=R(n(\bar{x};\bar{\theta}),e_{a},n(\bar{x};\bar{\theta}),e_{c})_{\bar{x}}=:A(\bar{x},\bar{\theta})_{ac}$

is constant in $t$ and we need just to evaluate it in the point $p$. Moreover,
$\Sigma$ is compact so that the matrix $A$ is symmetric and positive definite.
Hence, it exists an orthogonal matrix $\Omega(\bar{x},\bar{\theta})$ such that

$\displaystyle
A(\bar{x},\bar{\theta})=\Omega(\bar{x},\bar{\theta})D^{2}(\bar{x},\bar{\theta})\Omega(\bar{x},\bar{\theta})^{T},$
(1.35)

where $D^{2}$ is a diagonal matrix with positive eigenvalues
$d_{j}^{2}(\bar{x},\bar{\theta})$. If, for any given real function $f$, we
define $f(Dt)$ as the diagonal matrix having $f(d_{j}t)$ as diagonal elements,
and $f(\sqrt{A}t)=\Omega f(Dt)\Omega^{T}$, then we get for the volume element
$dV$ in the tube is given by the following

######  Proposition 3.

Let $M$ be a Riemannian submanifold of codimension $q$ in a compact Riemannian
symmetric manifold $\Sigma$ of dimension $q+n$. Let $M_{\varepsilon}$ be a
tube of radius $\varepsilon$ around $M$, coordinatized as above. Then, the
volume element $dV$ in the tube is

$\displaystyle
dV=dVol_{D_{\bar{x}}}(\bar{\theta},t)dVol_{M}(\bar{x})\det\left(\cos(\sqrt{A(\bar{x},\bar{\theta})}t)+\frac{\sin(\sqrt{A(\bar{x},\bar{\theta})}t)}{\sqrt{A(\bar{x},\bar{\theta})}}\sum_{s=1}^{q}\omega^{s}(\bar{\theta})K^{s}(\bar{x})\right).$
(1.36)

This is what we get by a direct application of (1.30) with a constant matrix
$R$. Notice that in general, the volume element of the disc depends on its
center $\bar{x}$, as well as the matrix $A(\bar{x},\bar{\theta})$.
We now restrict further ourselves to the case when $M$ is a totally geodesic
submanifold of $\Sigma$. In this case $K^{s}(\bar{x})=0$. Since the sub-
manifold $\Sigma$ is totally geodetic, any two points in $M$ are connected by
a geodetic of $\Sigma$ which is also a geodesic for $M$. Moreover, $\Sigma$ is
symmetric, so it has covariantly constant Riemann tensor (and metric,
obviously). Then $dVol_{D_{\bar{x}}}(\bar{\theta},t)$ and
$A(\bar{x},\bar{\theta})$ are independent on $\bar{x}$ and the formula further
reduces to

$\displaystyle
dV=dVol_{D}(\bar{\theta},t)dVol_{M}(\bar{x})\prod_{a=1}^{n}\cos(d_{a}(\bar{\theta})t).$
(1.37)

Notice that for a compact symmetric space, analyticity is guaranteed, and $t$
can thus be extended so that the above parametrization covers the whole
manifold up to a measure null set (the set of focal points). Therefore, we can
prove the following proposition. We can now notice that the range is such that
the cosine factors remain non-negative. In this case, we can notice that for
$x\in[0,\pi]$ one has

$\displaystyle\cos x\leq e^{-\frac{x^{2}}{2}},$ (1.38)

from which we get that everywhere

$\displaystyle 0\leq\prod_{a=1}^{n}\cos(d_{a}(\bar{\theta})t)\leq
e^{-\frac{t^{2}}{2}\sum_{a=1}^{n}d^{2}_{a}(\bar{\theta})}$ (1.39)

######  Proposition 4.

Let $(M_{n},\Sigma_{n})$ be a family, labeled by $n$, of $n$-dimensional
totally geodesic submanifolds $M_{n}$ of symmetric spaces $\Sigma_{n}$ of
dimension $n+1$ and constant diameter. Let $M^{\varepsilon_{n}}_{n}$ be the
tube of geodesic radius $\varepsilon_{n}$ centered in $M_{n}$. Finally, let
$m_{n}$ the Riemannian measure over $\Sigma_{n}$ normalized so that
$\mu_{n}(\Sigma_{n})=1$. If $\lim_{n\to\infty}\sqrt{n}\varepsilon_{n}=\infty$
then

$\displaystyle\lim_{n\to\infty}\mu(\Sigma-M^{\varepsilon_{n}}_{n})=0.$ (1.40)

###### Proof.

Suppose we consider the measure (1.37) normalised to 1. We also assume that
$\Sigma_{n}$ has diameter $L$. We need to consider

$\displaystyle\lim_{n\to\infty}\int_{\Sigma_{n}}dVol_{D}(\bar{\theta},t)dVol_{M}(\bar{x})f_{n}$
(1.41)

where

$\displaystyle
f_{n}=\prod_{a=1}^{n}\cos(d_{a}(\bar{\theta})t)\chi_{(\Sigma-M^{\varepsilon_{n}}_{n})},$
(1.42)

and $\chi_{E}$ is the characteristic function of the set $E$. Since the
codimension of $\Sigma$ is 1, we have that

$\displaystyle\sum_{a=1}^{n}d^{2}_{a}(\bar{\theta})=\sum_{a=1}^{n}R(n(\bar{x};\bar{\theta}),e_{a},n(\bar{x};\bar{\theta}),e_{c})_{\bar{x}}=Ric(n(\bar{x};\bar{\theta}),n(\bar{x};\bar{\theta})),$

where $Ric$ is the Ricci tensor of $M$. It is known that for a symmetric
manifold of constant diameter and dimension $s$, the Ricci tensor has the form
$R=(as+b)g$, $g$ being the invariant metric and $a>0$ and $b$ constants
independent on $s$. For example, this follows easily from the calculations in
[CU22] and [CaSc22]. Applied to our case and using (1.39) this shows that

$\displaystyle f_{n}\leq
e^{-\frac{t^{2}}{2}(an+b)}\chi_{(\Sigma-M^{\varepsilon_{n}}_{n})}$ (1.43)

for some constants $a>0$ and $b$. Since the integral is extended in
$t\geq\varepsilon_{n}$, we get that the integrand goes to zero uniformly at
worst as $e^{-\frac{a}{2}\varepsilon^{2}_{n}n}$ when $n$ diverges. Since
$\lim_{n\to\infty}\varepsilon^{2}_{n}n=0$, the assert is proved. ∎

Remark: The assumption for the codimension to be 1 has been made to keep the
proof technically simple. We believe that the same proposition is true for
constant codimension $q>1$ and we expect it to hold also for codimension
$q_{n}$, if it doesn’t grow too fast with $n$. We leave the investigation of
this point for future work.

##  2\. Characterization through wasserstein and box distance

Let $(M,\mu,g)$ be a Riemann manifold and $C\subseteq M$ a submanifold. The
projection map $proj_{C}:M\rightarrow C$ is defined as a map such that
$d(x,proj_{C}(x))=d(x,C)$, where $d$ is the geodetic distance associated with
$g$. For the existence and, more in general, a theory of distance functions in
$\mathbb{R}^{n}$ context see e.g. [CanSin]; for an extension of this theory to
a Riemannian context see e.g.[ManteMen], [Fath].

######  Proposition 5.

Let $((N_{n},\mu_{n},g_{n}),M_{n})$ a sequence of Riemannian manifolds, with
$M_{n}\subsetneq N_{n}$ Riemannian submanifold with endowed measure
$\\#proj_{N_{n},M_{n}}\mu_{n}$. Let $d_{W_{2}}$ be the Wasserstein distance of
order 2. If $d_{W_{2}}(\mu_{n},\\#proj_{N_{n},M_{n}}\mu_{n})\rightarrow 0$
then $M_{n}$ is a concentration locus for $N_{n}$

###### Proof.

If $\pi_{n}$ is the geodesic projection on $M_{n}$ and $d_{n}$ the geodesic
distance generated by the invariant metric $g_{n}$, then, the cost to
transport the mass $m_{n}=\mu_{n}(N_{n}\setminus M_{n}^{\varepsilon_{n}})$ in
terms of $d_{W_{2}}^{2}$ is at least

$\displaystyle\int_{N_{n}\setminus
M_{n}^{\varepsilon_{n}}}d_{n}^{2}(x,\pi_{n}(x))d\mu_{n}(x)>\varepsilon_{n}^{2}m_{n},$

for $M_{n}^{\varepsilon_{n}}$ a tubular neighbourhood of radius
$\varepsilon_{n}$ of $M_{n}$. In particular, for any fixed choice
$\varepsilon_{n}=\varepsilon>0$, since
$d_{W_{2}}(\mu_{n},\\#proj_{N_{n},M_{n}}\mu_{n})\rightarrow 0$, we get that
$m_{n}\rightarrow 0$. ∎

Proposition 5 cannot be reversed. Indeed, in case of a concentration locus
$M_{n}\subseteq N_{n}$, if the diameter of $N_{n}$ is unbounded $d_{W}$ could
not converge to 0.

Nevertheless the following holds.

######  Proposition 6.

Let $(N_{n},M_{n})$ a sequence as in the previous Proposition such that
volumes and diameters of $N_{n}$ are bounded by $h>0$. If $M_{n}$ is a
concentration locus for $N_{n}$ then for all $n\in\mathbb{N}$
$d_{W_{1}}(\mu_{n},\\#proj\mu_{n})\rightarrow 0$

###### Proof.

By contradiction, suppose that for infinitely many $n$
$d_{W_{1}}(\mu_{n},\\#proj\mu_{n})\geq k>0$. Since the distance function
determines the optimal transport through the project function,
$\int_{N_{n}}d_{n}(x,\pi_{n}(x))d\mu_{n}(x)\geq k$, where $\pi_{n}$ is the
geodetic projection over $M_{n}$ and $d_{n}$ the geodetic distance generated
by the invariant metric $g_{n}$ on $N_{n}$. Since $M_{n}$ is a concentration
locus for $N_{n}$ we can set $n^{\prime}$ in such a way for all $n>n^{\prime}$
$\epsilon_{n},\varepsilon_{n}<\frac{k}{4h}$ and, $\mu_{n}(N_{n}\setminus
M_{n}^{\varepsilon_{n}})<\epsilon_{n}$. Therefore,

$\displaystyle k$ $\displaystyle\leq\int_{N_{n}\setminus
M_{n}^{\epsilon_{n}}}d_{n}(x,\pi_{n}(x))d\mu_{n}(x)+\int_{M_{n}^{\epsilon_{n}}}d_{n}(x,\pi_{n}(x))d\mu_{n}(x)$
$\displaystyle\leq\int_{N_{n}\setminus
M_{n}^{\epsilon_{n}}}hd\mu_{n}(x)+\int_{M_{n}^{\epsilon_{n}}}\frac{k}{4h}d\mu_{n}(x)\leq
2h\frac{k}{4h}=\frac{k}{2},$

which is a contradiction.

∎

######  Remark 1.

Observe that convergence in Wasserstein distance (Strassen’s Theorem [Shi]
implies Prohorov distance convergence) implies box distance convergence
(Proposition 4.12 [Shi]), which in turn implies $d_{conc}$ convergence
(Proposition 5.5 (2) [Shi]).

Therefore, let $M$ be a complete separable metric space and ($\mu_{i}$) a
sequence of Borel probability measures on $M$. Consider the following three
conditions: (1) $\mu_{i}$ converges weakly to $\mu$. (2) $(M,\mu_{i})$ box-
converges to $(M,\mu)$. (3) $(M,\mu_{i})$ $d_{conc}$-converges (or
concentrates) to $(M,\mu)$.

Then, the following implications hold: (1) $\Rightarrow$ (2) $\Rightarrow$
(3).

A counterexample of (2) $\Rightarrow$ (1) is easy. Just take a sequence
$x_{i}$ in $M$ and let $\mu_{i}$ be the Dirac measure at $x_{i}$. Then, all
$(M,\mu_{i})$ are mm-isomorphic to each other, so that (2) holds. However (1)
does not hold if $x_{i}$ does not converge in $M$.

A counterexample of (3) $\Rightarrow$ (2) is the sequence of unit spheres
$S^{n}(1)$ with dimension $n$ going to infinity. $S^{n}(1)$
$d_{conc}$-converges to one-point space, but it is divergent for box-distance
(see Cor. 5.20 [Shi]).

######  Theorem 2.

Let $(N_{n},\mu_{n},g_{n})$ a sequence of Riemann manifolds with haar measure
$\mu_{n}$, geodesic distance $d_{n}$ generated by the invariant metric $g_{n}$
(they are in particular mm-spaces), and $M_{n}\subseteq N_{n}$ sub manifolds
with measures $\sigma_{n}=\\#proj\mu_{n}$ and metrics
$g^{\prime}_{n}=g_{n}|_{M_{n}}$.
Provided that $d_{box}((N_{n},\mu_{n}),(M_{n},\sigma_{n}))\rightarrow 0$ and
assuming
$(M_{n},\sigma_{n},g^{\prime}_{n})\rightarrow_{d_{conc}}(M,\sigma,g^{\prime})$
then
$(N_{n},\mu_{n},g_{n})\rightarrow_{d_{conc}}(M,\sigma,g^{\prime})$.

###### Proof.

By Proposition 5.5 [Shi],
$d_{conc}((N_{n},\mu_{n},g_{n}),(M_{n},\sigma_{n},g^{\prime}_{n}))\leq
d_{box}((N_{n},\mu_{n},g_{n}),(M_{n},\sigma_{n},g^{\prime}_{n}))$. Hence,
$d_{conc}((N_{n},\mu_{n},g_{n}),(M_{n},\sigma_{n},g^{\prime}_{n}))\rightarrow
0$.
The following chain of inequalities

$d_{conc}((N_{n},\mu_{n},g_{n}),(M,\sigma,g))\leq
d_{conc}((N_{n},\mu_{n},g_{n}),(M_{n},\sigma_{n},g^{\prime}_{n}))\ +\
d_{conc}((M_{n},\mu_{n},g_{n}),(M,\sigma,g))$

plainly drives to the thesis. ∎

######  Remark 2.

The above Theorem holds also with the hypothesis
$d_{W_{1}}(\mu_{n},\sigma_{n})\rightarrow 0$. Indeed, we can consider
$\sigma_{n}$ as a measure $\sigma^{\prime}_{n}$ on $N_{n}$ having support
$spt(\sigma^{\prime}_{n})=M_{n}$ where it is equal to $\sigma_{n}$. Without
loss in generality, we can consider $(N_{n},\sigma^{\prime}_{n},g_{n})$ equal,
as mm-spaces, to $(M_{n},\sigma_{n},g^{\prime}_{n})$, since they are mm-
isomorphic. Indeed, two mm-spaces are mm-isomorphic if there exists an
isometry between their supports of their respective measures (see [Gro99]
p.117).

Now the following Corollary follows

######  Corollary 1.

Let $(N_{n},M_{n})$ a sequence of Riemannian manifolds $M_{n}\subsetneq N_{n}$
such that volumes and diameters of $N_{n}$ are bounded by $h\in\mathbb{R}$. If
$M_{n}$ is a concentration locus for $N_{n}$ and
$(M_{n},\sigma_{n},g^{\prime}_{n})\rightarrow_{d_{conc}}(M,\sigma,g^{\prime})$
then $(N_{n},\mu_{n},g_{n})\rightarrow_{d_{conc}}(M,\sigma,g^{\prime})$.

Now we move towards a $d_{box}$-characterization.

######  Proposition 7.

Let $M_{n}$ be a concentration locus for $N_{n}$, $M_{n}$ totally geodesic
submanifolds, with the properties that the geodesic distance on $M_{n}$ is the
same as in $N_{n}$. Then, $d_{box}(N_{n},M_{n})$ converges to 0.

###### Proof.

Let us first observe that if $M_{n}^{\epsilon_{n}}$ is the tubular
neighbourhood of radius $\epsilon_{n}$, then, for $s,t\in
M_{n}^{\epsilon_{n}}$ we have

$|d_{n}(s,t)-d^{\prime}_{n}(proj_{M_{n}}(s),proj_{M_{n}}(t))|\leq
O(\epsilon_{n}),$

where $d_{n}^{\prime}$ is the geodesic distance in $M_{n}$. Indeed, inside the
tube by elementary distance inequalities
$|d_{n}(s,t)-d_{n}(proj_{M_{n}}(s),proj_{M_{n}}(t))|\leq 2\epsilon$, and since
by hypothesis
$d_{n}(proj_{M_{n}}(s),proj_{M_{n}}(t))=d^{\prime}_{n}(proj_{M_{n}}(s),proj_{M_{n}}(t))$.
Now define two parameters, $\phi,\psi$ for $N_{n}$ and $M_{n}$, respectively,
$\phi:[0,\epsilon_{n})\rightarrow N_{n}\setminus M_{n}^{\epsilon_{n}}$ and
$\phi:[\epsilon_{n},1)\rightarrow M_{n}^{\epsilon_{n}}$, where $\epsilon_{n}$
is chosen so that $\mu_{n}(N_{n}\setminus
M_{n}^{\epsilon_{n}})\leq\epsilon_{n}$. This is always possible, since if we
have two sequences $a_{n}$ and $b_{n}$ positive, converging to zero and such
that $\mu_{n}(N_{n}\setminus M_{n}^{a_{n}})\leq b_{n}$, then we can choose
$\epsilon_{n}=\max\\{a_{n},b_{n}\\}$. Let $\psi:=proj_{M_{n}}\circ\phi$.
From the above inequality, it follows:

$\displaystyle|d_{n}(\phi(s),\phi(t))-d_{n}^{\prime}(\psi(s),\psi(t))|\leq
O(\epsilon_{n}).$

This implies the thesis.

∎

Combining Proposition 7 and Theorem 2 we get the following Corollary

######  Corollary 2.

Let $(N_{n},\mu_{n},g_{n})$ be a sequence of mm-spaces, with $M_{n}$ totally
geodesic and with the same geodesic distance as in $N_{n}$. Suppose that
$M_{n}$ define a concentration locus for $N_{n}$ with a sequence of radii
$\epsilon_{n}\rightarrow 0$, and
$(M_{n},\sigma_{n},g^{\prime}_{n})\rightarrow_{d_{conc}}(M,\sigma,g^{\prime})$.
Then, $(N_{n},\mu_{n},g_{n})\rightarrow_{d_{conc}}(M,\sigma,g^{\prime})$.

In [CU22], by using the Macdonald formula [M], it is shown that
$SU(n),Spin(n),Usp(n)$ have all totally geodesic concentration loci.

Observe that even under hypothesis $d_{box}(N_{n},H_{n})\rightarrow 0$, for
any $N_{n}\in SO(n),SU(n),Spin(n)$, by Corollary 5.20 [Shi], all of them don’t
determine $d_{box}$ divergent sequences.

Observe that Corollary 2 applies even if $N_{n}$ is not a Levy family. In this
case, the sequence concentrates anyway to $M$ which is not necessarily a
point. This unveils that the concentration phenomenon is far from being
exhausted by the Levy families. In particular, if $N_{n}$ are topological
groups, by a Schneider result [S], $M$ should be not only the concentration
set but also an $N$-invariant subspace of $S(N)$, where $N$ is a second-
countable topological group completion of $\bigcup N_{n}$, and $S(N)$ is the
Samuel compactification of $N$. If $M$ is minimal, then it is the universal
minimal flow of $N$. Since concretely describable universal minimal flows are
rather rare, this could be a way to construct them.
These constructions can find interesting applications to the sequences of
$U(N)$ with different rescaled geometries.

## Acknowledgments

We are extremely grateful to Alessio Figalli for his insightful suggestions
and precise remarks. We also thank Carlo Mantegazza and Takeshi Shioya for
some helpful discussions.

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