On the Geometry of Metric Spaces

1. Introduction

The measurement of distances because of practical reasons, as the delimitation of parcels of farmland after floodings or those related to construction problems, lies behind the origins of mathematics in ancient civilizations. It was Euclid, in Proposition 20 of the first book of the Elements, who proved that the sum of two sides of a triangle is greater than the remaining one. This fact was probably used by Archimedes to state as assumption, in his work On the sphere and cylinder, that the straight line is the shortest between two points. Both achievements are now proven in basic courses on Linear Algebra.

Presently, the notion of distance on a nonempty set $X$ is formalized as a function $d:X\times X\to {\mathbb{R}}$ satisfying

1.

$d(x,y)=0$ if and only if $x=y$,

2.

$d(x,y)=d(y,x)$ for all $x,y\in X$,

3.

$d(x,y)\leqslant d(x,z)+d(z,y)$ for all $x,y,z\in X$.

These conditions imply $d\geqslant 0$. The second property is the symmetry one. In Gromov’s words this assumption “[…] unpleaseantly limits many applications.” Removing this symmetry condition gives rise to the concept of asymmetric distance, see Busemann’s Local metric geometry (1958). The third property is the well-known triangle inequality.

The pair $(X,d)$, where $X$ is a nonempty set and $d$ a distance on $X$, is called a metric space.

Metric spaces were introduced by Fréchet in 1906 in the context of functional analysis. Shortly after, a comprehensive treatment of the theory was presented by Hausdorff in the second part of his monograph Hau14 published in 1914, where the notion of topological space was introduced.

Metric spaces appear everywhere in modern mathematics. They are an invaluable tool in functional analysis, geometry of manifolds, graph theory, artificial intelligence, and many other fields. The aim of this note is to give a glance of the role of metric spaces in several fields of mathematics without any intention of being exhaustive. We focus on different notions of curvature in metric spaces, compactness results for Riemannian manifolds, and an application of metric graph theory to group theory.

2. Length Spaces

Given a subset of a Euclidean space we may consider the restriction of the Euclidean distance to the set. For instance, the distance between two antipodal points on a unit sphere is $2$. However, this distance is not intrinsic in the sense that cannot be measured inside the subset.

Given a continuous curve $\gamma :[a,b]\to X$ on a metric space, its length $L(\gamma )$ can be defined as the supremum of the quantities

$$\begin{equation*} \sum _{i=1}^k d(\gamma (t_{i-1}),\gamma (t_i)), \end{equation*}$$

over all partitions $a=t_0<t_1<\cdots <t_k=b$, $k\in \mathbb{N}$, of the interval $[a,b]$. When $L(\gamma )$ is finite the curve $\gamma$ is said to be rectifiable. Hence the length of curves, sometimes infinite, can be measured in metric spaces.

A metric space $(X,d)$ is a length space if the distance $d(x,y)$ between any pair of points $x,y\in X$ can be computed as

$$\begin{equation*} \inf _\gamma L(\gamma ), \end{equation*}$$

where $\gamma :[a,b]\to X$ is any rectifiable curve connecting $x$ and $y$ (i.e., such that $\gamma (a)=x$, $\gamma (b)=y$). A length space $(X,d)$ is geodesic if for every pair of points $x,y\in X$, there exists a rectifiable curve $\gamma$ connecting $x,y$, such that $d(x,y)=L(\gamma )$. A geodesic in $X$ is a rectifiable curve $\gamma :[a,b]\to X$ such that $d(\gamma (t),\gamma (s))=L(\gamma |_{[s,t]})$ for all $a\leqslant s<t\leqslant b$. The interested reader is referred to Burago, Burago and Ivanov’s BBI01 for a complete presentation of length spaces.

Examples of length spaces are Riemannian, Finsler and sub-Riemannian manifolds. A Riemannian manifold $(M,g)$ is a manifold $M$ together with an scalar product $g$ on the fiber bundle (a continuously varying scalar product on each tangent space). In a Finsler manifold the scalar product is replaced by a smooth norm. A sub-Riemannian manifold $(M,\mathcal{H},h)$ is a manifold $M$ together with a nonintegrable horizontal distribution $\mathcal{H}$ and a positive definite scalar product $h$ on $\mathcal{H}$. The classical example of a sub-Riemannian manifold is the Heisenberg group ${\mathbb{H}}^1$: the $3$-dimensional Euclidean space ${\mathbb{R}}^3$ together with a noncommutative product and the left-invariant vector fields

$$\begin{equation*} X=\frac{\partial }{\partial x}+y\frac{\partial }{\partial z},\quad Y=\frac{\partial }{\partial y}-x\frac{\partial }{\partial z},\quad Z=\frac{\partial }{\partial z} \end{equation*}$$

induce a left-invariant Riemannian metric $g_L$ making $X,Y,Z$ an orthonormal basis. The vectors fields $X,Y$ generate a nonintegrable distribution $\mathcal{H}$ that, together with the restriction of $g_L$ to $\mathcal{H}$, provide a sub-Riemannian structure on ${\mathbb{H}}^1$. On a connected sub-Riemannian manifold $M$, Chow–Rashevskii theorem, see §0.4 and §1.1, §1.2 in Gromov Gro96, implies that every pair of points can be connected by a smooth horizontal curve on $M$ (i.e., everywhere tangent to the horizontal distribution).

To define a distance we consider, in the Riemannian and Finsler cases, the class of piecewise smooth curves and, in the sub-Riemannian case, the class of piecewise smooth horizontal curves. In all cases, the length of a curve $\gamma :I\to M$ can be computed as

$$\begin{equation*} \int _I \|\gamma '(s) \|\,ds, \end{equation*}$$

where $\|\cdot \|$ is the norm associated to $g$ in the Riemannian case, the Finsler norm, or the one associated to $h$ in the sub-Riemannian case. Associated to this length we can define a distance between two points as the infimum of the length of the curves joining both points. This way we obtain the Riemannian distance, the Finsler distance, or the Carnot-Carathéodory distance. Geodesics are curves of minimum distance connecting two given points.

Figure 1.

Geodesics in the Riemannian and sub-Riemannian Heisenberg group ${\mathbb{H}}^1$ connecting two points in the same vertical line. The vertical line is a Riemannian geodesic. The helicoidal curve is a sub-Riemannian geodesic, forced to be tangent to the horizontal distribution.

3. Metric Spaces with Bounded Curvature

Curvature in Riemannian geometry was introduced as a measure of how a Riemannian space deviates from Euclidean space, see Gromov Gro91 for an enlightening discussion. For surfaces embedded in Euclidean space, the curvature coincides with the product of the principal curvatures of the surface and it is referred to as the Gauß curvature. The classical Gauß–Bonnet theorem provides a first hint on how the curvature $K$ affects the behavior of geodesic triangles on a surface. If $T$ is a triangle delimited by geodesics making inner angles $\alpha ,\beta$ and $\gamma$ then

$$\begin{equation*} \int _T K=(\alpha +\beta +\gamma )-\pi . \end{equation*}$$

Hence the sum of the inner angles of a triangle on a surface depends on the sign of the Gauß curvature: it is equal to $\pi$ on surfaces with $K=0$, larger than $\pi$ on surfaces with $K\geqslant 0$ and smaller than $\pi$ when $K\leqslant 0$.

Figure 2.

Behavior of geodesics triangles according to Gauß–Bonnet theorem.

According to the historical account in the preface of Alexander, Kapovitch, and Petrunin’s monograph, see arXiv:1903.08539, the first synthetic description of curvature is due to A. Wald in a paper published in 1936, followed by a paper by Alexandrov in 1941. It is generally considered that Rauch’s comparison results for Jacobi fields in the early 1950s can be thought of as an infinitesimal triangle comparison result which was soon followed by the celebrated Toponogov’s comparison theorem in the late 1950s.

A version of triangle comparison not involving angles can be stated as follows. Assume we have a point $p\in M$ in a Riemannian manifold and a unit-speed geodesic $\gamma :[0,T]\to M$ in a small neighborhood of $p$. We consider a second unit-speed geodesic $\bar{\gamma }:[0,T]\to {\mathbb{R}}^2$ in the plane and a point $\bar{p}\in {\mathbb{R}}^2$ such that $|\bar{p}-\bar{\gamma }(0)|=d(p,\gamma (0))$, $|\bar{p}-\bar{\gamma }(T)|=d(p,\gamma (T))$. This way we construct a comparison triangle whose vertices $\bar{\gamma }(0)$, $\bar{\gamma }(T)$, $\bar{p}$ lie at the same distance as the ones of the original triangle $\gamma (0),\gamma (T),p$. We define the functions $f(t)=d(p,\gamma (t))$, $\bar{f}(t)=|\bar{p}-\bar{\gamma }(t)|$. Then we have the following.

Theorem.

Under the conditions above

(1)

If $M$ has nonnegative sectional curvatures then $f(t)\geqslant \bar{f}(t)$.

(2)

If $M$ has nonpositive curvature then $f(t)\leqslant \bar{f}(t)$.

This result is typically proven by means of Jacobi fields, see Cheeger and Ebin (1975), but perhaps the reader could find useful the following alternative argument. Assume that the sectional curvatures of $M$ are no larger than $0$. Take $p\in M$, a normal neighborhood $U$ of $p$ where the exponential map is a diffeomorphism, and a geodesic $\gamma :[0,T]\to U$ parameterized by arc length. A basic result on Jacobi fields, similar to the one used in the proof of Bishop’s volume comparison, implies that the Hessian $\nabla ^2$ of the function $d^2/2$, where $d$ is the distance to $p$, satisfies

$$\begin{equation} \nabla ^2\big (\tfrac{1}{2}d^2\big )(v,v)\geqslant 1 {\tag{1}} \end{equation}$$

for any $v\in T_qM$, $q\in U$, and $|v|=1$. In Euclidean space we would have equality in 1 replacing $d$ by the Euclidean distance $\bar{d}$ to $\bar{p}$. Then the functions $h(t)=d^2(\gamma (t))/2$ and $\bar{h}(t)=\bar{d}^2(\bar{\gamma }(t))/2$ satisfy

$$\begin{multline*} h''(t)=\nabla ^2\big (\tfrac{1}{2}d^2\big )(\gamma '(t),\gamma '(t)) \\ \geqslant \nabla ^2\big (\tfrac{1}{2}\bar{d}^2\big )(\bar{\gamma }'(t),\bar{\gamma }'(t))=\bar{h}''(t). \end{multline*}$$

Hence $(h-\bar{h})''\geqslant 0$ on $[0,T]$ and, since the function $h-\bar{h}$ vanishes at the endpoints of the interval, we get $h\leqslant \bar{h}$, which implies $f\leqslant \bar{f}$.

Figure 3.

Illustration of triangle comparison. The one to the right lies in a Riemannian manifold $M$. The one to the left is a comparison triangle in the Euclidean plane. If the sectional curvature of $M$ is nonnegative then $d(p,\gamma (t))\geqslant |\bar{p}-\bar{\gamma }(t)|$. The opposite inequality holds when $M$ has nonpositive sectional curvature.

This comparison can be stated also in terms of angles. Consider a small geodesic triangle in a Riemannian manifold $M$. If $M$ has nonnegative curvature then, at every vertex, the angle made by the sides of the triangle is larger than or equal to the comparison angle at the vertex (i.e., the Euclidean angle of a comparison triangle). For nonpositive curvature we have the opposite inequality. Similar comparisons with a simply connected surface of constant curvature $\kappa$ hold when the sectional curvature of $M$ is no smaller or no larger than $\kappa$.

The triangle comparison for Riemannian manifolds allows to introduce a synthetic notion of metric space $X$ with curvature bounded, above or below, in terms of properties of geodesic triangles on the metric space.

Note that angles can be defined in a metric space using the law of cosines. Given three different points $x,y,z$ in a metric space $X$, we can build a comparison Euclidean triangle of vertices $\bar{x},\bar{y},\bar{z}$ by requiring that the distances $d(x,y),d(y,z),d(x,z)$ equal $|\bar{x}-\bar{y}|$, $|\bar{y}-\bar{z}|$, $|\bar{x}-\bar{z}|$. The comparison angle $\tilde{\measuredangle } xyz$ is defined as the Euclidean angle $\measuredangle \bar{x}\bar{y}\bar{z}$, that is,

$$\begin{equation*} \tilde{\measuredangle } xyz=\arccos \frac{d(y,x)^2+d(y,z)^2-d(x,z)^2}{2\,d(y,x)\,d(y,z)}. \end{equation*}$$

If we have two curves $\alpha ,\beta :[0,\varepsilon )\to X$ emanating from the same point $x$, the angle $\measuredangle (\alpha ,\beta )$ is

$$\begin{equation*} \measuredangle (\alpha ,\beta )=\lim _{t,s\to 0} \measuredangle \alpha (s)x \beta (t) \end{equation*}$$

when the limit exists.

The term CAT$(k)$ spaces is used for metric spaces with $k$ as an upper curvature bound (CAT stands for Cartan–Alexandrov–Topogonov) and the term Alexandrov spaces for spaces with lower curvature bounds. The reader should be aware that sometimes a different terminology is used depending on the direction of the bound.

The theory of CAT$(k)$ spaces and the one of Alexandrov spaces differ in many respects. In the case of curvature bounded above, Hadamard manifolds, complete simply connected length spaces of nonpositive curvature, play a central role. Almost all classical results for Riemannian manifolds have been extended to Hadamard manifolds, like uniqueness of geodesics connecting two given points, the validity of triangle comparison for large triangles, the fact that the squared distance function is globally concave. Essential references on CAT$(k)$ spaces are Bridson and Haeffliger BH99, and Ballman, Gromov, and Schroeder BGS85. As for Alexandrov spaces, there are two main points that make the theory quite different from the one of CAT$(k)$ spaces. The first is that triangle comparison holds for arbitrarily large triangles without additional assumptions. This is the content of Toponogov’s theorem. The second one concerns the local structure: in an Alexandrov space the Hausdorff dimension coincides with the topological dimension, which is an integer or infinite. An Alexandrov space is a manifold except on a small set of points. Examples of Alexandrov spaces are the boundaries of convex sets in Euclidean spaces. Numerous results valid for Riemannian manifolds also hold in the class of Alexandrov spaces with curvature bounded from below by $k$. To cite just a few, an upper bound on the diameter when $k>0$, a splitting theorem, Gromov–Bishop inequalities for the volume of balls. Petrunin also proved a Levy–Gromov type isoperimetric inequality (see the author’s monograph Rit23 for the Riemannian case). The reader is referred to Burago, Burago and Ivanov BBI01, the already mentioned monograph by Alexander, Kapovitch, and Petrunin, and the references cited therein, for excellent introductions to the theory of metric spaces with bounds on curvature.

4. Convergence of Metric Spaces

The introduction of curvature bounds on metric spaces might seem an abstract construction with no special interest. However, it is related to the behavior of sequences of Riemannian manifolds. One can consider the space of all Riemannian manifolds with some topology and asks whether some given subset is compact or precompact with this topology. In many cases the limit of the sequence is not a Riemannian manifold. This is one of the situations where metric spaces with bounds on curvature play a role.

A standard way of measuring the distance between two sets $A,B$ in a metric space $(X,d)$ is the Hausdorff distance $d_H(A,B)$, defined by

$$\begin{equation*} d_H(A,B)= \inf \big \{\varepsilon >0: A\subset B_\varepsilon , B\subset A_\varepsilon \big \}, \end{equation*}$$

where for any set $C\subset X$, $C_\varepsilon =\big \{x\in X: d(x,C)<\varepsilon \big \}$ is the open tubular neighborhood of $C$ of radius $\varepsilon$.

Gromov, see Gro81Gro07 introduced a way of measuring the distance between two metric spaces $X, Y$, nowadays known as the Gromov–Hausdorff distance, denoted by $d_{GH}(X,Y)$. To compute it, we take all metric spaces $Z$ which contain isometric copies $X'$ of $X$ and $Y'$ of $Y$, compute their Hausdorff distance $d_H(X',Y')$ on $Z$, and take the infimum over all possible metric spaces $Z$.

Gromov–Hausdorff distance is really a distance on the space of isometric classes of metric spaces since two isometric metric spaces have $d_{GH}$-distance equal to $0$. To measure the Gromov–Hausdorff distance using its definition is not very practical. However, for compact metric spaces $X$, $Y$, should we have two $\varepsilon$-nets $\{x_1,\ldots ,x_n\}$ on $X$, and $\{y_1,\ldots ,y_n\}$ on $Y$ such that

$$\begin{equation*} |d_X(x_i,x_j)-d_Y(y_i,y_j)|<\delta \quad \text{for all }i,j=1,\ldots , n, \end{equation*}$$

then we would have $d_{GH}(X,Y)<2\varepsilon +\delta$.

Important classes of Riemannian manifolds which are precompact in the Gromov–Hausdorff distance are the following:

•

For $m\in \mathbb{N}$ and $R,V>0$, the class of $m$-dimensional Riemannian manifolds with volume $|M|\leqslant V$ and injectivity radius $\text{inj}(M)\geqslant R$.

•

For any $m\in \mathbb{N}$ and $\kappa \in {\mathbb{R}}$, $D>0$, the class of $m$-dimensional Riemannian manifolds with $\text{diam} M\leqslant D$ and sectional curvature $\geqslant \kappa$. The same result holds assuming instead that the Ricci curvature is $\geqslant \kappa$.

While the Gromov–Hausdorff distance is useful when considering compact metric spaces, it is usually a very strong convergence for unbounded metric spaces. In this case it should be replaced by the pointed Gromov–Hausdorff convergence.

Natural measures which can be considered on metric spaces are the Hausdorff measures. Given $s\geqslant 0$, the $s$-dimensional Hausdorff measure associated to a set $A\subset X$ is given, up to a constant, by

$$\begin{equation*} \mathcal{H}^s(A)=\inf _{\delta >0}\mathcal{H}_\delta ^s(A), \end{equation*}$$

where $\mathcal{H}^s_\delta (A)$ is the supremum of $\sum _{i}\text{diam}(E_i)^s$ taken over all coverings of $A$ by sets $E_i$ satisfying $\text{diam}(E_i)\leqslant \delta$. For a metric space $X$ there exists a constant $N\in [0,+\infty ]$ such that $\mathcal{H}^s(X)=0$ for $s>N$ and $\mathcal{H}^s(X)=\infty$ for all $s<N$. The quantity $N$ is called the Hausdorff dimension of $X$ and is denoted by $\dim _H(X)$.

It is also important to note that Gromov–Hausdorff limits (see next section) of Alexandrov spaces of curvature $\geqslant \kappa$ are themselves Alexandrov spaces of curvature $\geqslant \kappa$. There are also compactness results for Alexandrov spaces. Gromov himself proved that the space $\mathcal{M}(n,\kappa ,D)$ composed of the Alexandov spaces with curvature $\geqslant \kappa$, diameter $\leqslant D$ and Hausdorff (topological) dimension $\leqslant n$ is compact in the Gromov–Hausdorff topology.

Although the Hausdorff measure $\mathcal{H}^N$ on a metric space $(X,d)$, with $N=\dim _H(X)$, is a natural measure to consider, sometimes it is more interesting to take a different one, see Appendix 2 in CC97. This leads to the notion of metric measure space $(X,d,\mu )$, which is nothing but a metric space $(X,d)$ together with a Borel measure $\mu$ on $X$. If $(X,d)$ is a length (geodesic) space we refer to $(X,d,\mu )$ as a length (geodesic) measure space.

A sequence $(X_i,d_i,\mu _i)$ of metric measure spaces converges in measured Gromov–Hausdorff topology to a metric measure space $(X,d,\mu )$ if there is a sequence of measurable maps $f_i:X_i\to X$ and a sequence of real numbers $\varepsilon _i$ converging to $0$ such that the maps $f_i$ are $(1,\varepsilon _i)$-quasi-isometries:

$$\begin{equation*} \big |\,d(f_i(x),f_i(x'))-d_i(x,x')\big |<\varepsilon _i\qquad x,x'\in X_i, \end{equation*}$$

the tubular neighborhood of radius $\varepsilon _i$ of $f_i(X_i)$ is $X$, and the push-forward measures $(f_i)_*\mu _i$ converge in weak topology to $\mu$, cf. Fukaya (1987).

5. Metric Spaces with a Lower Bound on Ricci Curvature

That the curvature of a Riemannian manifold implies metric properties of triangles was the key to introduce a synthetic notion of metric spaces with bounded curvature. An analogous approach should be possible for the Ricci curvature on a Riemannian manifold.

It looks like the first ones to discuss the possibility of such generalization were Cheeger and Colding in Appendix 2 in CC97, who suggested to use lower bounds on the volume growth of sectors. See also §5.44 in Gromov Gro07. In a Riemannian manifold $M$, the classical Bishop–Gromov theorem implies that if $\mathbb{M}_\kappa$ is a model Riemannian manifold (complete simply connected with constant sectional curvature $\kappa$) of the same dimension as $M$ and satisfying $\text{Ric}_M\geqslant \text{Ric}_{\mathbb{M}_\kappa }$, we have

$$\begin{equation*} \frac{|A(p,r_2,r_3)|}{|A(p,r_1,r_2)|}\leqslant \frac{|A_0(r_2,r_3)|}{|A_0(r_1,r_2)|} \end{equation*}$$

for arbitrary $r_1\leqslant r_2\leqslant r_3$, where $A(p,r,s)$ is the annulus $\big \{q\in M: r\leqslant d(q,p)\leqslant s\big \}$, and $A_0(r,s)$ the corresponding annulus in $M_0$, whose volume is independent of the base point. The symbol $|\cdot |$ denotes the Riemannian volumes. This inequality is also valid if we consider a geodesic sector $S\subset M$ with focal point $p$, which is nothing but a set $S$ such that, for each $q\in S$, there is a geodesic segment connecting $p$ and $q$. Let $S(r,s)=S\cap A(p,r,s)$. Then we have

$$\begin{equation*} \frac{|S(r_2,r_3)|}{|S(r_1,r_2)|}\leqslant \frac{|A_0(r_2,r_3)|}{|A_0(r_1,r_2)|} \end{equation*}$$

for arbitrary $r_1<r_2<r_3$.

Figure 4.

Comparison of volume of geodesic sectors.

Another synthetic approach to Ricci curvature mentioned in Cheeger and Colding CC97 is the use of the Laplacian of the distance function. The classical Bishop volume comparison for balls in an $m$-dimensional manifold $M$ with $\text{Ric}\geqslant (m-1)\kappa$ follows from comparison of the mean curvatures of geodesic spheres with the same radii in $M$ and the space form $\mathbb{M}_\kappa$ of constant sectional curvature $\kappa$. The mean curvature of a geodesic sphere is the Laplacian of the distance function $d$ to a fixed point. So, for instance, when $\kappa =0$, this translates into the equation

$$\begin{equation} \Delta d\leqslant (m-1)\frac{1}{d} {\tag{2}} \end{equation}$$

at least for small radius. Note that $(m-1)/d$ is the mean curvature of a ball of radius $d$ in the Euclidean space ${\mathbb{R}}^m$. Equation 2 was shown to hold in a weak sense in Riemannian manifolds by Calabi (1958). In order for this approach to work we need the extension of the notion of Laplacian to a metric space. Observe that for a Lipschitz function on a metric measure space $(X,d,\mu )$ we can define

$$\begin{equation*} |\nabla f|(x)=\limsup _{y\to x} \frac{|f(y)-f(x)|}{d(y,x)}, \end{equation*}$$

and the Cheeger energy

$$\begin{equation*} \text{Ch}(f)=\int _X |\nabla f|^2\,d\mu . \end{equation*}$$

By approximation of measurable functions by Lipschitz functions, this energy can be extended to wider classes. In general, $\text{Ch}$ is not a quadratic form. We say that a metric measure space is infinitesimally Hilbertian if the Cheeger energy is a quadratic form in $L^2(X,\mu )$. On such spaces it is possible to define a weak notion of Laplacian, thus giving sense to inequality 2. This extension of differential calculus to metric spaces has been carried out by Gigli Gig15. See also Ambrosio’s lecture at ICM2018 Amb18. A recent monograph by Heinonen, Koskela, Shanmugalingam, and Tyson (2015) is a very good introduction to analysis on metric spaces.

Yet another generalization of the notion of Ricci curvature comes from the well-known Bochner’s formula. Recall that Bochner’s formula in a Riemannian manifold reads

$$\begin{equation} \frac{1}{2}\Delta |\nabla u|^2=\nabla u(\Delta u)+|\nabla ^2u|^2+\text{Ric}(\nabla u,\nabla u). {\tag{3}} \end{equation}$$

If we assume $\text{Ric}\geqslant K$ and use the estimate $|\nabla ^2u|^2\geqslant (\Delta u)^2/m$ we arrive at Bochner’s inequality

$$\begin{equation*} \frac{1}{2}\Delta |\nabla u|^2\geqslant \nabla u(\Delta u)+\frac{(\Delta u)^2}{m}+K|\nabla u|^2. \end{equation*}$$

Here $m=\dim M$ and, of course, can be replaced by a larger number. All the operators appearing in this formula can be defined weakly on a metric measure space assuming some extra hypotheses, thus providing a notion of Ricci curvature bounded below.

5.1. An approach based on mass transport

In the first decade of the 21st century, Lott and Villani LV09 and Sturm Stu06aStu06b introduced independently equivalent notions of metric measure spaces with Ricci curvature bounded below. These notions were based on mass transportation properties on Riemannian manifolds with Ricci curvature bounded below.

Let us introduce first a few concepts, see Villani Vil09 for an excellent exposition. Given a metric space $(X,d)$ we consider the space $\mathcal{P}(X)$ of probability measures on $X$ and $\mathcal{P}_2(X)\subset \mathcal{P}(X)$ the subset of those $\mu \in \mathcal{P}(X)$ satisfying

$$\begin{equation*} \int _X d^2(x,y)\,d\mu (y)<\infty \end{equation*}$$

for some (all) $x\in X$. Given $\mu _0,\mu _1\in \mathcal{P}_2(X)$, a transport plan between $\mu _0$ and $\mu _1$ is a probability measure $\mu$ on $X\times X$ with marginals $\mu _0,\mu _1$ (i.e., the push-forward measures $(p_1)_*(\mu )$, $(p_2)_*(\mu )$ by the projections $p_i:X\times X\to X$, $i=1,2$, are $\mu _0$ and $\mu _1$). We denote the set of transport plans between $\mu _0$ and $\mu _1$ by Plan$(\mu _0,\mu _1)$. The Wasserstein distance $\text{W}_2$ on $\mathcal{P}_2(X)$ is defined by

$$\begin{equation*} \text{W}_2^2(\mu _0,\mu _1)=\min _{\mu \in \text{Plan}(\mu _0,\mu _1)}\int _{X\times X} d^2(x_0,x_1)\,d\mu (x_0,x_1). \end{equation*}$$

A minimizer of this problem is called an optimal transport plan between $\mu _0$ and $\mu _1$. The metric space $(\mathcal{P}_2(X),\text{W}_2)$ inherits many properties of $(X,d)$. The equivalence between geodesics in Wasserstein space and certain optimal transport plans was established by Lott and Villani LV09, §2.3 and Sturm Stu06a, §2.3.

Given $(X,d,\mu )$, we define the Shannon and Rényi entropy functionals for a measure $\nu$ by

$$\begin{equation*} \text{Ent}(\nu )= \begin{cases} \int _X\rho \log (\rho )\,d\mu ,&\text{if the integral exists}, \\ +\infty , & \text{otherwise}, \end{cases} \end{equation*}$$

and

$$\begin{equation*} \text{Ent}_N(\nu )=-\int _X\rho ^{1-1/N}\,d\mu , \end{equation*}$$

for $N\in [1,\infty )$. Here $\nu =\rho \mu +\nu _s$ is the decomposition of $\nu$ as the sum of a measure absolutely continuous with respect to $\mu$ and a singular measure $\nu _s\perp \mu$.

We say that a function $f:\mathcal{P}_2(X)\to {\mathbb{R}}$ is $K$-convex if for any Wasserstein geodesic $\Gamma :[0,1]\to \mathcal{P}_2(X)$,

$$\begin{equation*} \begin{split} f(\Gamma (t))&\leqslant (1-t)f(\Gamma (0)) \\ &+tf(\Gamma (1))-\frac{K}{2}(1-t) \text{W}_2^2(\Gamma (0),\Gamma (1)). \end{split} \end{equation*}$$

Observe that $0$-convex just means convex. Then we say that $(X,d,\mu )$ satisfies the CD$(0,\infty )$ property (or that has curvature $\geqslant 0$) if, for any pair $\mu _0,\mu _1\in \mathcal{P}_2(X)$, there is a Wasserstein geodesic $\Gamma :[0,1]\to \mathcal{P}_2(X)$ connecting $\mu _0$, $\mu _1$ such that $\text{Ent}\circ \Gamma$ is convex. We say that $(X,d,\mu )$ satisfy the CD$(0,N)$ property for $N\in [1,\infty )$ if, for any $\mu _0,\mu _1\in \mathcal{P}_2(X)$, there exists a Wasserstein geodesic $\Gamma :[0,1]\to \mathcal{P}_2(X)$ connecting $\mu _0$, $\mu _1$ such that $\text{Ent}_{N'}\circ \Gamma$ is convex for all $N'\geqslant N$.

Property CD$(K,\infty )$ is defined the same way as CD$(0,\infty )$ replacing convexity by $K$-convexity of $\text{Ent}\circ \Gamma$ for any Wasserstein geodesic. Property CD$(K,N)$, for $N\in [1,\infty )$, is defined by replacing $K$-convexity by a more involved condition in which the functions $1-t$, $t$, are replaced by functions $\tau _{K,N'}^{(1-t)}$, $\tau _{K,N'}^{(t)}$, see Definition 5.4 in Ambrosio Amb18.

The CD$(K,N)$ property is satisfied for $N$-dimensional Riemannian manifolds with $\text{Ric}\geqslant K$ and is stable under measured Gromov–Hausdorff convergence.

Several refinements were made later to the theory of CD$(K,N)$ spaces. The spaces $\text{CD}^*(K,N)$ satisfy locally the CD condition. Equivalence of CD and $\text{CD}^*$ notions for nonbranching metric measure spaces has been proved by Cavalletti and Milman CM21. A geodesic space $(X,d)$ is nonbranching if the map $f_t:\text{Geo}(X)\to X^2$, taking any geodesic $\gamma :[0,1]\to X$ to the pair $(\gamma (0),\gamma (t))$, is injective for all $t\in (0,1]$. Finally RCD$(K,N)$ and RCD$^*(K,N)$ spaces (the R stands for Riemannian) mean that the infinitesimally Hilbertian hypothesis is added. Recently Erbar, Kuwada, and Sturm EKS15 and Ambrosio, Mondino and Savaré AMS19 obtained the validity of Bochner’s inequality in a RCD$(K,\infty )$ metric measure space.

Petrunin (2011) checked the compatibility of these notions with the one of curvature bounded below by proving that $m$-dimensional Alexandrov spaces with nonnegative curvature satisfy the curvature dimension condition CD$(0,m)$ for $m<\infty$.

Another approach to a synthetic notion of lower bound on the Ricci curvature was introduced by Ohta (2007) in terms of the metric contraction property. This property is also equivalent, on a Riemannian manifold, to having a a lower bound on the Ricci curvature. Also Alexandrov spaces satisfy this property. Roughly speaking, the measure contraction property implies that for every set $A\subset X$ with $0<\mu (A)<\infty$ and for every $x\in X$, the normalized restriction of the measure $\mu$ to $A$ (i.e., $\mu |_A/\mu (A)$) can be transported in a controlled way along geodesics to the Dirac measure $\delta _x$.

It is worth mentioning that the notion of CD$(K,N)$ and the metric contraction property do not hold in sub-Riemannian manifolds, even in the simplest case of the Heisenberg groups, as shown by Juillet (2009). Recently Milman (2021), inspired by Barilari and Rizzi (2019), has suggested a quasi-convex relaxation QCD$(Q,K,N)$ of the CD$(K,N)$ condition for sub-Riemannian manifolds which coincides with the latter when $Q=1$.

6. Metric Structures on Graphs

A graph $\mathcal{G}=(V,E)$ is composed of a set of vertices $V$ and a set of edges $E$ connecting two given vertices. We assume that no more than one edge connects two given vertices and that there are no loops, that is, edges connecting the same vertex. A graph can be undirected, meaning that every edge has no orientation, and in this case we denote an edge connecting the vertices $e,v$ by $\{e,v\}$. A graph can also be directed, meaning that every edge has an initial and a final point. In this case, it is denoted by $(e,v)$. The vertex $e$ is the origin of the edge and $v$ the endpoint. We say that $e,v$ are incident to $(e,v)$ (or $\{e,v\}$). Of course $(e,v)$ and $(v,e)$ are different edges in a directed graph.

Given two vertices $e,v$ in a graph, a path connecting $e$ and $v$ is a sequence of vertices $x_0=e$, $x_1,\ldots ,x_k=v$ so that $\{x_{i-1},x_i\}$ (or $(x_{i-1},x_i)$) is an edge for all $i=1,\ldots ,k$. A weight function $\omega :E\to {\mathbb{R}}^+$ assigns to each edge $\ell$ a positive measure $\omega (\ell )$ of the edge. A graph is connected if any pair of vertices can be connected by a path.

We can define a distance on the set of vertices of an undirected connected graph with an arbitrary weight function by defining

$$\begin{equation*} d(e,v)=\inf \sum _{i=1}^k \omega (\{x_{i-1},x_i\}), \end{equation*}$$

where the infimum is taken over all paths $x_0,\ldots ,x_k$ connecting $e$ and $v$. Geodesics or shortest length paths are defined as the ones that realize the distance between its extreme points. The portion of a path connecting two vertices in a geodesic is also a geodesic.

6.1. Asymmetric distances on graphs

If $\mathcal{G}=(V,E)$ is a directed graph with a weight function $\omega :E\to {\mathbb{R}}$, we can define a “distance” between two vertices $e,v$ simply by setting

$$\begin{equation*} d(e,v)=\inf \sum _{i=1}^k \omega ((x_{i-1},x_i)) \end{equation*}$$

in case there is a path connecting $e$ and $v$. The infimum is taken over all paths $x_0,\ldots ,x_k$ connecting $e$ and $v$. If there is no such path we define

$$\begin{equation*} d(e,v)=\infty . \end{equation*}$$

We also define $d(e,e)=0$ for all $e\in V$. This way we have defined a map $d:V\times V\to \overline{R}={\mathbb{R}}\cup \{\infty \}$ such that the following properties hold

(1)

$d(e,v)\geqslant 0$ for all $e,v\in V$,

(2)

$d(e,v)\leqslant d(e,w)+d(w,v)$ for all $e,v,w\in V$.

In the triangle inequality we have used $\infty +a=a+\infty =\infty +\infty =\infty$ for all $a\geqslant 0$. In case $\omega (\ell )\geqslant \varepsilon >0$ for all $\ell \in E$ or some another alternative hypothesis is assumed we also have

1’.

$d(e,v), d(v,e)>0$ if $e\neq v$.

A map $d:V\times V\to \overline{R}$ satisfying 1, $1'$, and 2 is called an asymmetric distance on $V$. The pair $(V,d)$ is called an asymmetric metric space.

Figure 5.

Geodesics or shortest length paths between vertices $1$ and $5$ in a directed graph depicted in red. The weight function here is $\omega (\ell )=1$ for any $\ell \in E$.

Metric structures on graphs are commonly used in computer science. For instance, the breadth-first search algorithm, an exploratory algorithm in graphs, consist essentially on the computation of the metric balls on an undirected graph, where the weight function is $\omega (\ell )=1$ for any edge $\ell$. Dijkstra’s algorithm computes shortest length paths between two vertices in a directed graph with positive weights using metric projections on metric balls recursively calculated, see CLRS09.

6.2. Curvature-dimension conditions on graphs

A notion of curvature-dimension curvature on graphs has been introduced by Lin and Yau LY10. See also Ollivier (2009) for a notion of Ricci curvature on Markov chains on graphs, and Erbar and Maas (2012) for finite Markov chains. For the Lin and Yau notion, consider an undirected graph $\mathcal{G}=(V,E)$ with weight function $\omega$. We assume that $\mathcal{G}$ is locally finite in the sense that the number of edges connecting every vertex to others is finite. Define

$$\begin{equation*} \mu (x)=\sum _{\{x,y\}\in E}\omega (\{x,y\}), \end{equation*}$$

and a measure of a set $\Omega \subset V$ by

$$\begin{equation*} \mu (\Omega )=\sum _{x\in \Omega } \omega (x). \end{equation*}$$

For a function $f:V\to {\mathbb{R}}$ the Laplacian $\Delta f$ and squared gradient $|\nabla f|^2$ are defined by

$$\begin{align*} \Delta f (x)&=\frac{1}{\mu (x)}\sum _{\{x,y\}\in E} \omega (\{x,y\})(f(y)-f(x)), \\ |\nabla f|^2 (x)&=\frac{1}{\mu (x)}\sum _{\{x,y\}\in E} \omega (\{x,y\})(f(y)-f(x))^2, \end{align*}$$

for every $x\in V$. In addition, for $f,g:V\to {\mathbb{R}}$, we define the operators

$$\begin{align*} \Gamma (f,g)&=\frac{1}{2}\big \{\Delta (fg)-f\Delta g-g\Delta f\big \}, \\ \Gamma _2(f,g)&=\frac{1}{2}\big \{\Delta \Gamma (f,g)-\Gamma (f,\Delta g)-\Gamma (g,\Delta f)\big \}. \end{align*}$$

According to Lin and Yaus’s paper LY10 the graph $\mathcal{G}$ satisfies the curvature-dimension condition CD$(m,K)$ if

$$\begin{equation*} \Gamma _2(f,f)\geqslant \frac{1}{m}(\Delta f)^2+K\Gamma (f,f), \end{equation*}$$

and the CD$(\infty ,K)$ condition if

$$\begin{equation*} \Gamma _2(f,f)\geqslant K\Gamma (f,f). \end{equation*}$$

The CD$(m,K)$ curvature-dimension condition is the discrete counterpart of Bochner’s inequality. Lin and Yan used these notions to prove that a locally finite graph satisfies the CD$(2,\tfrac{1}{d}-1)$ curvature condition, where $d$ is the weighted degree of the graph, given by

$$\begin{equation*} d=\sup _{x\in V}\sup _{\{x,y\}\in E} \frac{\mu (x)}{\omega (\{x,y\})}. \end{equation*}$$

They also obtained a lower bound of the first nonzero eigenvalue of the Laplacian on the graph in terms of the degree $d$ and the diameter of the graph.

6.3. The Cayley graph

An interesting metric structure can be introduced on groups, allowing the use of geometric arguments to obtain results on the algebraic structure. Assume we have a group $G$ with a set of generators $S$ satisfying $S^{-1}=S$, that is, for every $s\in S$ we have $s^{-1}\in S$. We define the undirected Cayley graph associated to $G$ as $(G,E)$, where

$$\begin{equation*} E=\big \{\{e,v\}:e^{-1}v\in S\big \}. \end{equation*}$$

Hence the vertices are the elements of the group and the edges incident to $e\in G$ are of the form $\big \{\{e,es\}: s\in S\big \}$. We take as weight function $\omega (\ell )=1$ for any edge $\ell$. If $x_0=e,x_1,\ldots ,x_k=v$ is a path connecting $e$ and $v$ then there exist $s_1,\ldots ,s_k$ such that $x_{i-1}s_i=x_i$ for $i=1,\ldots ,k$. Hence $v=x_k=e(s_1\cdots s_k)$ and

$$\begin{equation*} e^{-1}v=s_1\cdots s_k. \end{equation*}$$

Reciprocally, if we can express $e^{-1}v$ as a product of $k$ elements of $S$ then there is a path $e=x_0,x_1,\ldots ,x_k=v$ connecting $e$ and $v$. Hence the associated distance is the minimum number of generators needed to express $e^{-1}v$. This is called the word metric on $G$.

Figure 6.

Unit balls for the group $\mathbb{Z}^2$ with set of generators $S=\{\pm (1,0),\pm (0,1)\}$, and $S=\{\pm (1,0),\pm (0,1),\pm (1,1)\}$, respectively.

Since left-translations are isometries of the word metric, all metric balls $B(r)$ of a fixed radius $r>0$ have the same number of elements. A group has polynomial growth if there are constants $C>0$ and $d>0$ such that

$$\begin{equation*} \text{card } B(r)\leqslant C r^d. \end{equation*}$$

The first striking application of metric theory to group theory was the following result by Gromov

Theorem (Gro81).

If a finitely generated group has polynomial growth then it contains a nilpotent subgroup of finite index.

This result is usually considered as the starting point of Geometric group theory. See also Gromov’s Hyperbolic groups (1987) and Asymptotic invariants of infinite groups (1993) for influential works on the subject.

Final Comments

The theory of metric spaces is a lively research subject in current Mathematics. The techniques developed to face the problems in this area have had a deep, unifying and simplifying influence in related areas of Mathematics. Despite the fact we have only covered in this note only a few of the geometric aspects, there is an enormous development in progress of the theory of analysis on metric spaces. The reader is referred to Semmes’s papers (2003) in these Notices for an overview of the subject.

A version of this manuscript with full references can be found on the author’s personal webpage: http://www.ugr.es/~ritore.