Riemannian manifold

This article is about a concept from differential geometry. For the algebraic concept, see Zariski–Riemann space.

Not to be confused with Riemann surface.

In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product g_p on the tangent space T_pM at each point p and with a proper connection.

The family g_p of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds.

A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n² functions

$g\left({\frac {\partial }{\partial x^{i}}},{\frac {\partial }{\partial x^{j}}}\right):U\to \mathbb {R}$

are smooth functions, i.e., they are infinitely differentiable. These functions are commonly designated as $g_{ij}$. This convention will be used in this article except where otherwise stated.

With further restrictions on the $g_{ij}$, one could also consider Lipschitz Riemannian metrics or measurable Riemannian metrics, among many other possibilities.

A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle at an intersection, length of a curve, area of a surface and higher-dimensional analogues (volume, etc.), extrinsic curvature of submanifolds, and intrinsic curvature of the manifold itself.

History

In 1828, Carl Friedrich Gauss proved the Theorema Egregium ("remarkable theorem" in Latin), which says that the Gaussian curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, the Gaussian curvature of a surface is an intrinsic property that does not depend on how the surface might be embedded in 3-dimensional space. See Differential geometry of surfaces.

Bernhard Riemann extended Gauss's theory to higher-dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that is intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces.

Elie Cartan introduced the Cartan connection, one of the first concepts of connection. Levi-Civita defined the Levi-Civita connection, a special connection on a Riemannian manifold.

Albert Einstein used the theory of pseudo-Riemannian manifolds (a generalization of Riemannian manifolds) to develop general relativity. In particular, his equations for gravitation are constraints on the curvature of spacetime. Other applications of Riemannian geometry include computer graphics and artificial intelligence.

Definition

Riemannian metrics and Riemannian manifolds

Let $M$ be a smooth manifold. For each point $p\in M$, there is an associated vector space $T_{p}M$ called the tangent space of $M$ at $p$. Vectors in $T_{p}M$ are thought of as the vectors tangent to $M$ at $p$.

However, $T_{p}M$ does not come equipped with an inner product, which would give tangent vectors a concept of length and angle. This is an important deficiency because calculus teaches that to calculate the length of a curve, the length of vectors tangent to the curve must be defined.

A Riemannian metric $g$ on $M$ assigns to each $p$ a positive-definite inner product $g_{p}:T_{p}M\times T_{p}M\to \mathbb {R}$. This induces a norm $|\cdot |_{p}:T_{p}M\to \mathbb {R}$ defined by $|v|_{p}={\sqrt {g_{p}(v,v)}}$. A smooth manifold $M$ endowed with a Riemannian metric $g$ is a Riemannian manifold, denoted $(M,g)$. A Riemannian metric is a special case of a metric tensor.

The Riemannian metric in coordinates

If $(x^{1},\ldots ,x^{n}):U\to \mathbb {R} ^{n}$ are smooth local coordinates on $M$, the vectors

$\left\{{\frac {\partial }{\partial x^{1}}}{\Big |}_{p},\dotsc ,{\frac {\partial }{\partial x^{n}}}{\Big |}_{p}\right\}$

form a basis of the vector space $T_{p}M$ for any $p\in U$. Relative to this basis, one can define the Riemannian metric's components at each point $p$ by

$g_{ij}|_{p}:=g_{p}\left(\left.{\frac {\partial }{\partial x^{i}}}\right|_{p},\left.{\frac {\partial }{\partial x^{j}}}\right|_{p}\right).$

One could consider these as $n^{2}$ individual functions $g_{ij}:U\to \mathbb {R}$ or as a single $n\times n$ matrix-valued function on $U$. The requirement that $g_{p}$ is a positive-definite inner product says exactly that this matrix-valued function is a symmetric positive-definite matrix at $p$.

In terms of the tensor algebra, the Riemannian metric can be written in terms of the dual basis $\{dx^{1},\ldots ,dx^{n}\}$ of the cotangent bundle as

$g=\sum _{i,j}g_{ij}\,\mathrm {d} x^{i}\otimes \mathrm {d} x^{j}.$

Regularity of the Riemannian metric

The Riemannian metric $g$ is continuous if $g_{ij}:U\to \mathbb {R}$ are continuous in any smooth coordinate chart $(U,x).$ The Riemannian metric $g$ is smooth if $g_{ij}$ are smooth in any smooth coordinate chart. One can consider many other types of Riemannian metrics in this spirit.

In most expository accounts of Riemannian geometry, Riemannian metrics are always taken to be smooth. However, there are important situations in which the metrics are not smooth. Riemannian metrics produced by methods of geometric analysis, in particular, can be less than smooth. See for instance (Gromov 1999) and (Shi and Tam 2002). This article will consider the case where $g$ is not smooth, but $g$ will be smooth unless stated otherwise.

Isometries

An isometry is a function between Riemannian manifolds which preserves all of the structure of Riemannian manifolds. If two Riemannian manifolds have an isometry between them, they are called isometric, and they are considered to be the same manifold for the purpose of Riemannian geometry.

Specifically, if $(M,g)$ and $(N,h)$ are two Riemannian manifolds, a diffeomorphism $f:M\to N$ is called an isometry if $g=f^{\ast }h$, that is, if

$g_{p}(u,v)=h_{f(p)}(df_{p}(u),df_{p}(v))$

for all $p\in M$ and $u,v\in T_{p}M.$ For example, translations and rotations are both isometries from $\mathbb {R} ^{n}$ to itself.

One says that a smooth map $f:M\to N,$ not assumed to be a diffeomorphism, is a local isometry if every $p\in M$ has an open neighborhood $U$ such that $f:U\to f(U)$ is an isometry (and thus a diffeomorphism).

Examples

Euclidean space

Let $x^{1},\ldots ,x^{n}$ denote the standard coordinates on $\mathbb {R} ^{n}.$ Then define $g_{p}^{\mathrm {can} }:T_{p}\mathbb {R} ^{n}\times T_{p}\mathbb {R} ^{n}\to \mathbb {R}$ by

$\left(\sum _{i}a_{i}{\frac {\partial }{\partial x^{i}}},\sum _{j}b_{j}{\frac {\partial }{\partial x^{j}}}\right)\longmapsto \sum _{i}a_{i}b_{i}.$

Phrased differently: relative to the standard coordinates, the local representation $g_{ij}:U\to \mathbb {R}$ is given by the constant value $\delta _{ij}.$

This is clearly a Riemannian metric, and is called the standard Riemannian structure on $\mathbb {R} ^{n}.$ It is also referred to as Euclidean space of dimension n and g_ij^can is also called the (canonical) Euclidean metric.

Embedded submanifolds

Main article: Riemannian submanifold

Let $(M,g)$ be a Riemannian manifold and let $N\subset M$ be an embedded submanifold of $M,$ which is at least $C^{1}.$ Then the restriction of g to vectors tangent along N defines a Riemannian metric over N.

• For example, consider $S^{n-1}=\{x\in \mathbb {R} ^{n}:(x^{1})^{2}+\cdots +(x^{n})^{2}=1.\},$ which is a smooth embedded submanifold of the Euclidean space with its standard metric. The Riemannian metric this induces on $S^{n-1}$ is called the standard metric or canonical metric on $S^{n-1}.$
• There are many similar examples. For example, every ellipsoid in $\mathbb {R} ^{3}$ has a natural Riemannian metric. The graph of a smooth function $f:\mathbb {R} ^{3}\to \mathbb {R}$ is an embedded submanifold, and so has a natural Riemannian metric as well.

Immersions

Main article: Riemannian submanifold

Let $(M,g)$ be a Riemannian manifold and let $f:\Sigma \to M$ be a differentiable map. Then one may consider the pullback of $g$ via $f$, which is a symmetric 2-tensor on $\Sigma$ defined by

$(f^{\ast }g)_{p}(v,w)=g_{f(p)}{\big (}df_{p}(v),df_{p}(w){\big )},$

where $df_{p}(v)$ is the pushforward of $v$ by $f.$

In this setting, generally $f^{\ast }g$ will not be a Riemannian metric on $\Sigma ,$ since it is not positive-definite. For instance, if $f$ is constant, then $f^{\ast }g$ is zero. In fact, $f^{\ast }g$ is a Riemannian metric if and only if $f$ is an immersion, meaning that the linear map $df_{p}:T_{p}\Sigma \to T_{f(p)}M$ is injective for each $p\in \Sigma .$

• An important example occurs when $(M,g)$ is not simply connected, so that there is a covering map ${\widetilde {M}}\to M.$ This is an immersion, and so the universal cover of any Riemannian manifold automatically inherits a Riemannian metric. More generally, but by the same principle, any covering space of a Riemannian manifold inherits a Riemannian metric.
• Also, an immersed submanifold of a Riemannian manifold inherits a Riemannian metric.

Product metrics

Let $(M,g)$ and $(N,h)$ be two Riemannian manifolds, and consider the Cartesian product $M\times N$ with the usual product smooth structure. The Riemannian metrics $g$ and $h$ naturally put a Riemannian metric ${\widetilde {g}}$ on $M\times N,$ which can be described in a few ways.

• Considering the decomposition $T_{(p,q)}(M\times N)\cong T_{p}M\oplus T_{q}N,$ one may define

  ${\widetilde {g}}_{p,q}(u\oplus x,v\oplus y)=g_{p}(u,v)+h_{q}(x,y).$

• Let $(U,x)$ be a smooth coordinate chart on $M$ and let $(V,y)$ be a smooth coordinate chart on $N.$ Then $(U\times V,(x,y))$ is a smooth coordinate chart on $M\times N.$ For convenience let $\operatorname {Sym} _{n\times n}^{+}$ denote the collection of positive-definite symmetric $n\times n$ real matrices. Denote the coordinate representation of $g$ relative to $(U,x)$ by $g_{U}:U\to \operatorname {Sym} _{m\times m}^{+}$ and denote the coordinate representation of $h$ relative to $(V,y)$ by $h_{V}:V\to \operatorname {Sym} _{n\times n}^{+}.$ Then the local coordinate representation of ${\widetilde {g}}$ relative to $(U\times V,(x,y))$ is ${\widetilde {g}}_{U\times V}:U\times V\to \operatorname {Sym} _{(m+n)\times (m+n)}^{+}$ given by

  $(p,q)\mapsto {\begin{pmatrix}g_{U}(p)&0\\0&h_{V}(q)\end{pmatrix}}.$

A standard example is to consider the n-torus $T^{n},$ define as the n-fold product $S^{1}\times \cdots \times S^{1}.$ If one gives each copy of $S^{1}$ its standard Riemannian metric, considering $S^{1}\subset \mathbb {R} ^{2}$ as an embedded submanifold (as above), then one can consider the product Riemannian metric on $T^{n}.$ It is called a flat torus.

Convex combinations of metrics

Let $g_{0}$ and $g_{1}$ be two Riemannian metrics on $M.$ Then, for any number $\lambda \in [0,1],$

${\tilde {g}}:=\lambda g_{0}+(1-\lambda )g_{1}$

is also a Riemannian metric on $M.$ More generally, if $a$ and $b$ are any two positive numbers, then $ag_{0}+bg_{1}$ is another Riemannian metric.

Every smooth manifold admits a Riemannian metric

Theorem: Every smooth manifold admits a (non-canonical) Riemannian metric.

This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed by only using that a smooth manifold is locally Euclidean, for this result it is necessary to include in the definition of "smooth manifold" that it is Hausdorff and paracompact. The reason is that the proof makes use of a partition of unity.

Proof

Let $M$ be a differentiable manifold and $\{(U_{\alpha },\varphi _{\alpha })\}_{\alpha \in I}$ a locally finite atlas so that $U_{\alpha }\subseteq M$ are open subsets and $\varphi _{\alpha }\colon U_{\alpha }\to \varphi _{\alpha }(U_{\alpha })\subseteq \mathbf {R} ^{n}$ are diffeomorphisms. Such an atlas exists because the manifold is paracompact.

Let $\{\tau _{\alpha }\}_{\alpha \in I}$ be a differentiable partition of unity subordinate to the given atlas, i.e. such that $\operatorname {supp} \,\tau _{\alpha }\subseteq U_{\alpha }$ for all $\alpha \in I$.

Then define the metric $g$ on $M$ by

$g:=\sum _{\beta \in I}\tau _{\beta }\cdot {\tilde {g}}_{\beta },\qquad {\text{with}}\qquad {\tilde {g}}_{\beta }:=\varphi _{\beta }^{*}g^{\mathrm {can} }\,\,{\text{on}}\,\,U_{\beta },$

where $g^{\mathrm {can} }$ is the Euclidean metric on $\mathbb {R} ^{n}$ and $\varphi _{\beta }^{*}g^{\mathrm {can} }$ is its pullback along $\varphi _{\beta }$.

This is readily seen to be a metric on $M$.

An alternative proof uses the Whitney embedding theorem to embed $M$ into Euclidean space and then pulls back the metric from Euclidean space to $M$. On the other hand, the Nash embedding theorem states that, given any smooth Riemannian manifold $(M,g),$ there is an embedding $F:M\to \mathbb {R} ^{N}$ for some $N$ such that the pullback by $F$ of the standard Riemannian metric on $\mathbb {R} ^{N}$ is $g.$ That is, the entire structure of a smooth Riemannian manifold can be encoded by a diffeomorphism to a certain embedded submanifold of some Euclidean space. Therefore, one could argue that nothing can be gained from the consideration of abstract smooth manifolds and their Riemannian metrics. However, there are many natural smooth Riemannian manifolds, such as the set of rotations of three-dimensional space and the hyperbolic space, of which any representation as a submanifold of Euclidean space will fail to represent their remarkable symmetries and properties as clearly as their abstract presentations do.

Metric space structure

An admissible curve is a piecewise smooth curve $\gamma :[0,1]\to M$ whose velocity $\gamma '(t)\in T_{\gamma (t)}M$ is nonzero everywhere it is defined.

For $(M,g)$ a connected Riemannian manifold, define $d_{g}:M\times M\to [0,\infty )$ by

$d_{g}(p,q)=\inf\{L(\gamma ):\gamma {\text{ a piecewise admissible curve from }}p{\text{ to }}q\}.$

Theorem: $(M,d_{g})$ is a metric space, and the metric topology on $(M,d_{g})$ coincides with the topology on $M$.

Connections, geodesics, and curvature

Connections

Main article: Affine connection

An (affine) connection is an additional structure on a Riemannian manifold that defines differentiation of one vector field with respect to another. Connections contain geometric data, and two Riemannian manifolds with different connections have different geometry.

Let ${\mathfrak {X}}(M)$ denote the space of vector fields on $M$. A connection

$\nabla :{\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\to {\mathfrak {X}}(M)$

on $M$ is a bilinear map $(X,Y)\mapsto \nabla _{X}Y$ such that

1. For any function $f\in C^{\infty }(M)$, $\nabla _{f_{1}X_{1}+f_{2}X_{2}}Y=f_{1}\nabla _{X_{1}}Y+f_{2}\nabla _{X_{2}}Y$,
2. The product rule $\nabla _{X}fY=X(f)Y+f\nabla _{X}Y$ holds.

The expression $\nabla _{X}Y$ is called the covariant derivative of $Y$ with respect to $X$.

Levi-Civita connection

Main article: Levi-Civita connection

Two Riemannian manifolds with different connections have different geometry. Thankfully, there is a natural connection associated to a Riemannian manifold, called the Levi-Civita connection.

There are two extra conditions a connection could satisfy:

1. $g$ is parallel with respect to $\nabla$ if $\nabla g=0$,
2. $\nabla$ is torsion-free if $\nabla _{X}Y-\nabla _{Y}X=[X,Y]$, where $[\cdot ,\cdot ]$ is the Lie bracket.

A torsion-free connection for which $g$ is parallel with respect to $\nabla$ is called a Levi-Civita connection. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection.

Covariant derivative along a curve

If $\gamma :[0,1]\to M$ is a smooth curve, a smooth vector field along $\gamma$ is a smooth map $X:[0,1]\to TM$ such that $X(t)\in T_{\gamma (t)}M$ for all $t\in [0,1]$. The set ${\mathfrak {X}}(\gamma )$ of smooth vector fields along $\gamma$ is a vector space under pointwise vector addition and scalar multiplication. One can also pointwise multiply a smooth vector field along $\gamma$ by a smooth function $f:[0,1]\to \mathbb {R}$:

$(fX)(t)=f(t)X(t)$ for $X\in {\mathfrak {X}}(\gamma )$.

Let $X$ be a smooth vector field along $\gamma$. If ${\tilde {X}}$ is a smooth vector field on a neighborhood of the image of $\gamma$ such that $X(t)={\tilde {X}}_{\gamma (t)}$, then ${\tilde {X}}$ is called an extension of $X$.

Given a fixed connection $\nabla$ on $M$ and a smooth curve $\gamma :[0,1]\to M$, there is a unique operator $D_{t}:{\mathfrak {X}}(\gamma )\to {\mathfrak {X}}(\gamma )$, called the covariant derivative along $\gamma$, such that:

1. $D_{t}(aX+bY)=aD_{t}X+bD_{t}Y$,
2. $D_{t}(fX)=f'X+fD_{t}X$,
3. If ${\tilde {X}}$ is an extension of $X$, then $D_{t}X(t)=\nabla _{\gamma '(t)}{\tilde {X}}$.

Geodesics

Main article: Geodesic

Geodesics are curves with no intrinsic acceleration. They are the generalization of straight lines in Euclidean space to arbitrary Riemannian manifolds.

Fix a connection $\nabla$ on $M$. Let $\gamma :[0,1]\to M$ be a smooth curve. The acceleration of $\gamma$ is the vector field $D_{t}\gamma '$ along $\gamma$. If $D_{t}\gamma '=0$ for all $t$, $\gamma$ is called a geodesic.

For every $p\in M$ and $v\in T_{p}M$, there exists a geodesic $\gamma :I\to M$ defined on some open interval $I$ containing 0 such that $\gamma (0)=p$ and $\gamma '(0)=v$. Any two such geodesics agree on their common domain. Taking the union over all open intervals $I$ containing 0 on which a geodesic satisfying $\gamma (0)=p$ and $\gamma '(0)=v$ exists, one obtains a geodesic called a maximal geodesic of which every geodesic satisfying $\gamma (0)=p$ and $\gamma '(0)=v$ is a restriction.

The Hopf–Rinow theorem

Main article: Hopf–Rinow theorem

The Riemannian manifold $M$ with its Levi-Civita connection is geodesically complete if the domain of every maximal geodesic is $(-\infty ,\infty )$. The plane $\mathbb {R} ^{2}$ is geodesically complete. On the other hand, the punctured plane $\mathbb {R} ^{2}\smallsetminus \{(0,0)\}$ with the restriction of the Riemannian metric from $\mathbb {R} ^{2}$ is not geodesically complete as there is no unit-speed geodesic from $(1,0)$ to $(-1,0)$.

If $M$ is geodesically complete, then it is "non-extendable" in the sense that it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse is not true, however: there exist non-extendable manifolds that are not complete.

The Hopf–Rinow theorem characterizes geodesically complete manifolds.

Theorem: Let $(M,g)$ be a connected Riemannian manifold with a smooth metric. The following are equivalent:

• The metric space $(M,d_{g})$ is complete (every $d_{g}$-Cauchy sequence converges),
• A subset of $M$ is compact if and only if it is closed and $d_{g}$-bounded,
• $M$ is geodesically complete.

Parallel transport

Main article: Parallel transport

In Euclidean space, all tangent spaces are canonically identified with each other via translation, so it is easy to move vectors from one tangent space to another. Parallel transport is a way of moving vectors from one tangent space to another in the setting of a general manifold. Given a fixed connection, there is a way to do parallel transport.

Riemann curvature tensor

Main article: Riemann curvature tensor

Fix a connection $\nabla$ on $M$. The Riemann curvature tensor is the map $R:{\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\to {\mathfrak {X}}(M)$ defined by

$R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z$

where $[X,Y]$ is the Lie bracket of vector fields. The Riemann curvature tensor is a $(1,3)$-tensor field.

Ricci curvature tensor

Main article: Ricci curvature

Fix a connection $\nabla$ on $M$. The Ricci curvature tensor is

$Ric(X,Y)=\operatorname {tr} (Z\mapsto R(Z,X)Y)$

where $\operatorname {tr}$ is the trace.

Scalar curvature

Main article: Scalar curvature

Riemannian manifolds with continuous metrics

Throughout this section, $g$ will be assumed to be continuous but not necessarily smooth.

The length of piecewise $C^{1}$ curves

If $\gamma :[a,b]\to M$ is differentiable, then it assigns to each $t\in (a,b)$ a vector $\gamma '(t)$ in the vector space $T_{\gamma (t)}M,$ the size of which can be measured by the norm $|\cdot |_{\gamma (t)}.$ So $t\mapsto |\gamma '(t)|_{\gamma (t)}$ defines a nonnegative function on the interval $(a,b).$ The length is defined as the integral of this function; however, as presented here, there is no reason to expect this function to be integrable. It is typical to suppose $g$ to be continuous and $\gamma$ to be continuously differentiable, so that the function to be integrated is nonnegative and continuous, and hence the length of $\gamma ,$

$L(\gamma )=\int _{a}^{b}|\gamma '(t)|_{\gamma (t)}\,dt,$

is well-defined. This definition can easily be extended to define the length of any piecewise-continuously differentiable curve.

In many instances, such as in defining the Riemann curvature tensor, it is necessary to require that $g$ has more regularity than mere continuity. For this section, $g$ continuous is enough to use the length defined above in order to endow a connected Riemannian manifold $M$ with the structure of a metric space.

Metric space structure

For $(M,g)$ a connected Riemannian manifold with $g$ continuous, define $d_{g}:M\times M\to [0,\infty )$ by

$d_{g}(p,q)=\inf\{L(\gamma ):\gamma {\text{ a piecewise }}C^{1}{\text{ curve from }}p{\text{ to }}q\}.$

Theorem: $(M,d_{g})$ is a metric space and the metric topology on $(M,d_{g})$ coincides with the topology on $M$.

In verifying that $(M,d_{g})$ satisfies all of the axioms of a metric space, the most difficult part is checking that $p\neq q$ implies $d_{g}(p,q)>0$.

(Sketched) Proof that $p\neq q$ implies $d_{g}(p,q)>0.$ Briefly: there must be some precompact open set around p which every curve from p to q must escape. By selecting this open set to be contained in a coordinate chart, one can reduce the claim to the well-known fact that, in Euclidean geometry, the shortest curve between two points is a line. In particular, as seen by the Euclidean geometry of a coordinate chart around p, any curve from p to q must first pass though a certain "inner radius." The assumed continuity of the Riemannian metric g only allows this "coordinate chart geometry" to distort the "true geometry" by some bounded factor.

To be precise, let $(U,x)$ be a smooth coordinate chart with $x(p)=0$ and $q\notin U.$ Let $V\ni x$ be an open subset of $U$ with ${\overline {V}}\subset U.$ By continuity of $g$ and compactness of ${\overline {V}},$ there is a positive number $\lambda$ such that $g(X,X)\geq \lambda \|X\|^{2}$ for any $r\in V$ and any $X\in T_{r}M,$ where $\|\cdot \|$ denotes the Euclidean norm induced by the local coordinates. Let R denote $\sup\{r>0:B_{r}(0)\subset x(V)\},$ to be used at the final step of the proof.

Now, given any piecewise continuously differentiable path $\gamma :[a,b]\to M$ from p to q, there must be some minimal $\delta >0$ such that $\gamma (a+\delta )\notin V;$ clearly $\gamma (a+\delta )\in \partial V.$

The length of $\gamma$ is at least as large as the restriction of $\gamma$ to $[a,a+\delta ].$ So

$L(\gamma )\geq {\sqrt {\lambda }}\int _{a}^{a+\delta }\|\gamma '(t)\|\,dt.$

The integral which appears here represents the Euclidean length of a curve from 0 to $x(\partial V)\subset \mathbb {R} ^{n}$, and so it is greater than or equal to R. So we conclude $L(\gamma )\geq {\sqrt {\lambda }}R.$

The observation that underlies the above proof, about comparison between lengths measured by g and Euclidean lengths measured in a smooth coordinate chart, also verifies that the metric space topology of $(M,d_{g})$ coincides with the original topological space structure of $M$.

Although the length of a curve is given by an explicit formula, it is generally impossible to write out the distance function $d_{g}$ by any explicit means. In fact, if $M$ is compact then, even when g is smooth, there always exist points where $d_{g}:M\times M\to \mathbb {R}$ is non-differentiable, and it can be remarkably difficult to even determine the location or nature of these points, even in seemingly simple cases such as when $(M,g)$ is an ellipsoid.

Geodesics

Let $(M,g)$ be a connected Riemannian manifold with $g$ continuous and consider the associated metric space $(M,d_{g}).$ Relative to this metric space structure, a path $c:[a,b]\to M$ is a unit-speed geodesic if for every $t_{0}\in [a,b]$ there is an interval containing it $J\subset [a,b]$ such that

$d_{g}(c(s),c(t))=|s-t|\qquad \forall s,t\in J.$

Informally, one may say that one is asking for $c$ to locally 'stretch itself out' as much as it can, subject to the unit-speed constraint. The idea is that if $c:[a,b]\to M$ is piecewise continuously differentiable and $|c'(t)|_{c(t)}=1$ for all $t,$ then one automatically has $d_{g}(c(s),c(t))\leq |s-t|$ by applying the triangle inequality to a Riemann sum approximation of the integral defining the length of $c.$ So the unit-speed geodesic condition as given above is requiring $c(s)$ and $c(t)$ to be as far from one another as possible. The fact that we are only looking for curves to locally stretch themselves out is reflected by the first two examples given below; the global shape of $(M,g)$ may force even the most innocuous geodesics to bend back and intersect themselves.

• Consider $(M,g)$ the circle $S^{1}$ with its standard Riemannian metric and $c:\mathbb {R} \to S^{1}$ given by $t\mapsto (\cos t,\sin t).$ Recall that $d_{g}$ is measured by the lengths of curves along $S^{1}$, not by the straight-line paths in the plane. This example also exhibits the necessity of selecting out the subinterval $J,$ since the curve $c$ repeats back on itself in a particularly natural way.
• Likewise, if $(M,g)$ is the round sphere $S^{2}$ with its standard Riemannian metric, then a unit-speed path along an equatorial circle will be a geodesic. A unit-speed path along the other latitudinal circles will not be geodesic.
• Consider the case that $(M,g)$ is $\mathbb {R} ^{2}$ with its standard Riemannian metric. Then a unit-speed line such as $t\mapsto (2^{-1/2}t,2^{-1/2}t)$ is a geodesic but the curve $c$ from the first example above is not.

Note that unit-speed geodesics, as defined here, are by necessity continuous, and in fact Lipschitz, but they are not necessarily differentiable or piecewise differentiable.

Hopf–Rinow theorem

Let $(M,g)$ be a connected Riemannian manifold with $g$ continuous. The Hopf–Rinow theorem, in this setting, says (Gromov 1999):

• If the metric space $(M,d_{g})$ is complete (every $d_{g}$-Cauchy sequence converges) then
  • every closed and bounded subset of $M$ is compact,
  • given any $p,q\in M$ and any real numbers a, b with a < b, there is a unit-speed geodesic $c:[a,b]\to M$ from $p$ to $q$ such that $d_{g}(c(s),c(t))=|s-t|$ for all $s,t\in [a,b].$

The essence of the proof is that once the first half is established, one may directly apply the Arzelà–Ascoli theorem, in the context of the compact metric space ${\overline {B_{2d_{g}(p,q)}(p)}},$ to a sequence of piecewise continuously differentiable unit-speed curves from $p$ to $q$ whose lengths approximate $d_{g}(p,q).$ The resulting subsequential limit is the desired geodesic.

Diameter

Let $(M,g)$ be a connected Riemannian manifold with with $g$ continuous. The diameter of the metric space $(M,d_{g})$ is

$\operatorname {diam} (M,d_{g})=\sup\{d_{g}(p,q):p,q\in M\}.$

The Hopf–Rinow theorem shows that if $(M,d_{g})$ is complete and has finite diameter, then it is compact. Conversely, if $(M,d_{g})$ is compact, then the function $d_{g}:M\times M\to \mathbb {R}$ has a maximum, since it is a continuous function on a compact metric space. This proves the following.

If $(M,d_{g})$ is complete, then it is compact if and only if it has finite diameter.

This is not the case without the completeness assumption; for counterexamples one could consider any open bounded subset of a Euclidean space with the standard Riemannian metric.

More generally, and with the same one-line proof, every compact metric space has finite diameter. However, it is not true that a complete metric space of finite diameter must be compact. For an example of a complete and non-compact metric space of finite diameter, consider

$X={\Big \{}{\text{continuous functions }}f:[0,1]\to \mathbb {R} {\text{ with }}\sup _{x\in [0,1]}|f(x)|\leq 1{\Big \}}$

with the uniform metric

$d(f,g)=\sup _{x\in [0,1]}|f(x)-g(x)|.$

So, although all of the terms in the above corollary of the Hopf–Rinow theorem involve only the metric space structure of $(M,g),$ it is important that the metric is induced from a Riemannian manifold.

Infinite-dimensional manifolds

The statements and theorems above are for finite-dimensional manifolds—manifolds whose charts map to open subsets of $\mathbb {R} ^{n}.$ These can be extended, to a certain degree, to infinite-dimensional manifolds; that is, manifolds that are modeled after a topological vector space; for example, Fréchet, Banach and Hilbert manifolds.

Definitions

Riemannian metrics are defined in a way similar to the finite-dimensional case. However there is a distinction between two types of Riemannian metrics:

• A weak Riemannian metric on $M$ is a smooth function $g:TM\times TM\to \mathbb {R} ,$ such that for any $x\in M$ the restriction $g_{x}:T_{x}M\times T_{x}M\to \mathbb {R}$ is an inner product on $T_{x}M.$
• A strong Riemannian metric on $M$ is a weak Riemannian metric, such that $g_{x}$ induces the topology on $T_{x}M.$ Note that if $M$ is not a Hilbert manifold then $g$ cannot be a strong metric.

Examples

• If $(H,\langle \,\cdot ,\cdot \,\rangle )$ is a Hilbert space, then for any $x\in H,$ one can identify $H$ with $T_{x}H.$ By setting for all $x,u,v\in H$ $g_{x}(u,v)=\langle u,v\rangle$ one obtains a strong Riemannian metric.
• Let $(M,g)$ be a compact Riemannian manifold and denote by $\operatorname {Diff} (M)$ its diffeomorphism group. The latter is a smooth manifold (see here) and in fact, a Lie group. Its tangent bundle at the identity is the set of smooth vector fields on $M.$ Let $\mu$ be a volume form on $M.$ Then one can define $G,$ the $L^{2}$ weak Riemannian metric, on $\operatorname {Diff} (M).$ Let $f\in \operatorname {Diff} (M),$ $u,v\in T_{f}\operatorname {Diff} (M).$ Then for $x\in M,u(x)\in T_{f(x)}M$ and define $G_{f}(u,v)=\int _{M}g_{f(x)}(u(x),v(x))d\mu (x).$ The $L^{2}$ weak Riemannian metric on $\operatorname {Diff} (M)$ induces vanishing geodesic distance, see Michor and Mumford (2005).

Metric space structure

Length of curves is defined in a way similar to the finite-dimensional case. The function $d_{g}:M\times M\to [0,\infty )$ is defined in the same manner and is called the geodesic distance. In the finite-dimensional case, the proof that this function is a metric uses the existence of a pre-compact open set around any point. In the infinite case, open sets are no longer pre-compact and so this statement may fail.

• If $g$ is a strong Riemannian metric on $M$, then $d_{g}$ separates points (hence is a metric) and induces the original topology.
• If $g$ is a weak Riemannian metric but not strong, $d_{g}$ may fail to separate points or even be degenerate.

For an example of the latter, see Valentino and Daniele (2019).

Hopf–Rinow theorem

In the case of strong Riemannian metrics, part of the finite-dimensional Hopf–Rinow still works.

Theorem: Let $(M,g)$ be a strong Riemannian manifold. Then metric completeness (in the metric $d_{g}$) implies geodesic completeness.

A proof can be found in (Lang 1999, Chapter VII, Section 6). The other statements of the finite-dimensional case may fail. An example can be found here.

If $g$ is a weak Riemannian metric, then no notion of completeness implies the other in general.

See also

• Riemannian geometry
• Finsler manifold
• Sub-Riemannian manifold
• Pseudo-Riemannian manifold
• Metric tensor
• Hermitian manifold
• Space (mathematics)
• Wave maps equation

References

• Lee, John M. (2018). Introduction to Riemannian Manifolds. Springer-Verlag. ISBN 978-3-319-91754-2.
• do Carmo, Manfredo (1992). Riemannian geometry. Basel: Birkhäuser. ISBN 978-0-8176-3490-2.
• Gromov, Misha (1999). Metric structures for Riemannian and non-Riemannian spaces (Based on the 1981 French original ed.). Birkhäuser Boston, Inc., Boston, MA. ISBN 0-8176-3898-9.
• Jost, Jürgen (2008). Riemannian Geometry and Geometric Analysis (5th ed.). Berlin: Springer-Verlag. ISBN 978-3-540-77340-5.
• Shi, Yuguang; Tam, Luen-Fai (2002). "Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature". J. Differential Geom. 62 (1): 79–125. arXiv:math/0301047. doi:10.4310/jdg/1090425530. S2CID 13841883.
• Lang, Serge (1999). Fundamentals of differential geometry. New York: Springer-Verlag. ISBN 978-1-4612-0541-8.
• Magnani, Valentino; Tiberio, Daniele (2020). "A remark on vanishing geodesic distances in infinite dimensions". Proc. Amer. Math. Soc. 148 (1): 3653–3656. arXiv:1910.06430. doi:10.1090/proc/14986. S2CID 204578276.
• Michor, Peter W.; Mumford, David (2005). "Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms". Documenta Math. 10: 217–245. arXiv:math/0409303. doi:10.4171/dm/187. S2CID 69260.

External links

• L.A. Sidorov (2001) [1994], "Riemannian metric", Encyclopedia of Mathematics, EMS Press