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 smoothly-varying positive-definite inner product g_p on the tangent space T_pM at each point p.

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

A Riemannian metric makes it possible to define many geometric notions, including angles, lengths of curves, areas of surfaces, higher-dimensional analogues of area (volumes, etc.), extrinsic curvature of submanifolds, and the intrinsic curvature of the manifold itself.

The requirement that g_p is smoothly-varying amounts to that for any smooth coordinate chart (U, x) on M, the n² functions

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

are smooth functions, i.e., they are infinitely differentiable. The section Riemannian manifolds with continuous metrics handles the case where the $g_{ij}$ are merely continuous.

History

Riemannian manifolds were first conceptualized by Bernhard Riemann.

In 1827, Carl Friedrich Gauss made a distinction between the intrinsic and extrinsic properties for surfaces in 3-dimensional space. A property of a surface is extrinsic if might change when a map that preserves local distance measurements, called a local isometry, is applied to it. However, the abstract surface itself has properties that remain unchanged under local isometries. These properties are called intrinsic. Gauss's Theorema Egregium ("remarkable theorem" in Latin) says that the Gaussian curvature of a surface does not change if a local isometry is applied to the surface. In other words, the Gaussian curvature of a surface is an intrinsic property.

Riemannian manifolds were first defined by Bernhard Riemann. They generalized the classical theory of surfaces to higher-dimensional manifolds.

Elie Cartan introduced the Cartan connection, one of the first concepts of a 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

The tangent plane of the sphere with two vectors in it. A Riemannian metric allows one to take the inner product of these vectors.

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 $g_{p}:T_{p}M\times T_{p}M\to \mathbb {R}$ in a smooth way (see the section on regularity below). 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

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

These $n^{2}$ functions $g_{ij}:U\to \mathbb {R}$ can be put together into an $n\times n$ matrix-valued function on $U$. The requirement that $g_{p}$ is a positive-definite inner product then 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 its components $g_{ij}:U\to \mathbb {R}$ are continuous in any smooth coordinate chart $(U,x).$ The Riemannian metric $g$ is smooth if its components $g_{ij}$ are smooth in any smooth coordinate chart. One can consider many other types of Riemannian metrics in this spirit, such as Lipschitz Riemannian metrics or measurable Riemannian metrics.

There are 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). The section Riemannian manifolds with continuous metrics handles the case where $g$ is merely continuous, but $g$ is assumed to be smooth in this article 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 $\mathbb {R} ^{n}.$ The (canonical) Euclidean metric $g^{\text{can}}$ is given by

$g^{\text{can}}\left(\sum _{i}a_{i}{\frac {\partial }{\partial x^{i}}},\sum _{j}b_{j}{\frac {\partial }{\partial x^{j}}}\right)=\sum _{i}a_{i}b_{i}$

or equivalently

$g^{\text{can}}=(dx^{1})^{2}+\cdots +(dx^{n})^{2}$

or equivalently by its coordinate functions

$g_{ij}^{\text{can}}=\delta _{ij}$ where $\delta _{ij}$ is the Kronecker delta.

The Riemannian manifold $(\mathbb {R} ^{n},g^{\text{can}})$ is called Euclidean space.

Submanifolds

Main article: Riemannian submanifold

The sphere $S^{n}$ with the round metric is an embedded Riemannian submanifold of $\mathbb {R} ^{n+1}$.

Let $(M,g)$ be a Riemannian manifold and let $i:N\to M$ be an immersed submanifold or an embedded submanifold of $M$. The pullback $i^{*}g$ of $g$ is a Riemannian metric on $N$, and $(N,i^{*}g)$ is said to be a Riemannian submanifold of $(M,g)$.

In the case where $N\subseteq M$, the map $i:N\to M$ is given by $i(x)=x$ and the metric $i^{*}g$ is just the restriction of $g$ to vectors tangent along $N$. In general, the formula for $i^{*}g$ is

$i^{*}g_{p}(v,w)=g_{i(p)}{\big (}di_{p}(v),di_{p}(w){\big )}$,

where $di_{p}(v)$ is the pushforward of $v$ by $i.$

Examples:

• The $n$-sphere $S^{n}=\{x\in \mathbb {R} ^{n+1}:(x^{1})^{2}+\cdots +(x^{n+1})^{2}=1\}$ is a smooth embedded submanifold of $\mathbb {R} ^{n+1}$ with its standard metric. The Riemannian metric this induces on $S^{n}$ is called the round metric or standard metric.
• An ellipsoid in $\mathbb {R} ^{3}$ is a smooth embedded submanifold of $\mathbb {R} ^{3}$ with its standard metric.
• The graph of a smooth function $f:\mathbb {R} ^{n}\to \mathbb {R}$ is a smooth embedded submanifold of $\mathbb {R} ^{n+1}$ with its standard metric.
• If $(M,g)$ is not simply connected, there is a covering map ${\widetilde {M}}\to M$, where ${\widetilde {M}}$ is the universal cover of $M$. This is an immersion, 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.

On the other hand, if $N$ already has a Riemannian metric ${\tilde {g}}$, then the immersion (or embedding) $i:N\to M$ is called an isometric immersion (or isometric embedding) if ${\tilde {g}}=i^{*}g$. Hence isometric immersions and isometric embeddings are Riemannian submanifolds.

Products

Let $(M,g)$ and $(N,h)$ be two Riemannian manifolds, and consider the product manifold $M\times N$. 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

  ${\widetilde {g}}_{p,q}((u_{1},u_{2}),(v_{1},v_{2}))=g_{p}(u_{1},v_{1})+h_{q}(u_{2},v_{2}).$

• If $(U,x)$ is a smooth coordinate chart on $M$ and $(V,y)$ is a smooth coordinate chart on $N$, then $(U\times V,(x,y))$ is a smooth coordinate chart on $M\times N.$ Let $g_{U}$ be the representation of $g$ in the chart $(U,x)$ and let $h_{V}$ be the representation of $h$ in the chart $(V,y)$. The representation of ${\widetilde {g}}$ in the coordinates $(U\times V,(x,y))$ is

  ${\widetilde {g}}=\sum _{ij}{\widetilde {g}}_{ij}dx^{i}dx^{j}$ where $({\widetilde {g}}_{ij})={\begin{pmatrix}g_{U}&0\\0&h_{V}\end{pmatrix}}.$

For example, the n-torus $T^{n}$ is defined 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, then one can consider the product Riemannian metric on $T^{n}.$ It is called a flat torus.

Positive combinations of metrics

Let $g_{1},\ldots ,g_{k}$ be Riemannian metrics on $M.$ If $a_{1},\ldots ,a_{k}$ are any positive numbers, then $a_{1}g_{1}+\ldots +a_{k}g_{k}$ is another Riemannian metric on $M.$

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 using only that a smooth manifold is a locally Euclidean topological space, for this result it is necessary to use that smooth manifolds are Hausdorff and paracompact. The reason is that the proof makes use of a partition of unity.

Proof that every smooth manifold admits a Riemannian metric

Let $M$ be a smooth manifold and $\{(U_{\alpha },\varphi _{\alpha })\}_{\alpha \in A}$ 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 $\{\tau _{\alpha }\}_{\alpha \in A}$ be a differentiable partition of unity subordinate to the given atlas, i.e. such that $\operatorname {supp} (\tau _{\alpha })\subseteq U_{\alpha }$ for all $\alpha \in A$. Define a Riemannian metric $g$ on $M$ by $g:=\sum _{\alpha \in A}\tau _{\alpha }\cdot {\tilde {g_{\alpha }}}$ where ${\tilde {g_{\alpha }}}:=\varphi _{\alpha }^{*}g^{\text{can}}$. Here $g^{\text{can}}$ is the Euclidean metric on $\mathbb {R} ^{n}$ and $\varphi _{\alpha }^{*}g^{\mathrm {can} }$ is its pullback along $\varphi _{\alpha }$. While ${\tilde {g_{\alpha }}}$ is only defined on $U_{\alpha }$, the product $\tau _{\alpha }\cdot {\tilde {g_{\alpha }}}$ is defined and smooth on $M$ since $\operatorname {supp} (\tau _{\alpha })\subseteq U_{\alpha }$. It takes the value 0 outside of $U_{\alpha }$. Because the atlas is locally finite, at every point the sum contains only finitely many nonzero terms, so the sum converges. Now one needs to check that $g$ is actually a Riemannian metric. That is, one needs to check that $g_{p}$ is smooth at $p\in M$ and that for fixed $p$ it is positive-definite, symmetric, and linear in the first argument. Symmetry is verified as follows: $g_{p}(u,v)=\sum _{\alpha \in A}\tau _{\alpha }(p)\cdot ({\tilde {g_{\alpha }}})_{p}(u,v)=\sum _{\alpha \in A}\tau _{\alpha }(p)\cdot ({\tilde {g_{\alpha }}})_{p}(v,u)=g_{p}(v,u)$. Linearity in the first argument is verified as follows: $g_{p}(au+bw,v)=\sum _{\alpha \in A}\tau _{\alpha }(p)\cdot ({\tilde {g_{\alpha }}})_{p}(au+bw,v)=a\sum _{\alpha \in A}\tau _{\alpha }(p)\cdot ({\tilde {g_{\alpha }}})_{p}(u,v)+b\sum _{\alpha \in A}\tau _{\alpha }(p)\cdot ({\tilde {g_{\alpha }}})_{p}(w,v)=ag_{p}(u,v)+bg_{p}(w,v)$. Positive-definiteness is verified as follows: $g_{p}(u,u)=\sum _{\alpha \in A}\tau _{\alpha }(p)\cdot ({\tilde {g_{\alpha }}})_{p}(u,u)>0$ for $u\neq 0$. Because the atlas is locally finite, $p$ has a neighborhood on which the sum $\sum \tau _{\alpha }\cdot {\tilde {g_{\alpha }}}$ contains only finitely many nonzero terms. On this neighborhood, $g=\sum \tau _{\alpha }\cdot {\tilde {g_{\alpha }}}$ is a finite sum of smooth functions. Therefore $g$ is smooth at $p$.

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. The nonnegative function $t\mapsto |\gamma '(t)|_{\gamma (t)}$ is defined on the interval $[0,1]$ except for at finitely many points. The length $L(\gamma )$ of an admissible curve $\gamma :[0,1]\to M$ is defined as

$L(\gamma )=\int _{0}^{1}|\gamma '(t)|_{\gamma (t)}\,dt$.

The integrand is bounded and continuous except at finitely many points, so it is integrable. For $(M,g)$ a connected Riemannian manifold, define $d_{g}:M\times M\to [0,\infty )$ by

$d_{g}(p,q)=\inf\{L(\gamma ):\gamma {\text{ an admissible curve with }}\gamma (0)=p,\gamma (1)=q\}.$

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

Proof sketch that $(M,d_{g})$ is a metric space, and the metric topology on $(M,d_{g})$ agrees 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 $d_{g}(p,q)>0$. Verification of the other metric space axioms is omitted. 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 admissible curve $\gamma :[0,1]\to M$ from p to q, there must be some minimal $\delta >0$ such that $\gamma (\delta )\notin V;$ clearly $\gamma (\delta )\in \partial V.$ The length of $\gamma$ is at least as large as the restriction of $\gamma$ to $[0,\delta ].$ So $L(\gamma )\geq {\sqrt {\lambda }}\int _{0}^{\delta }\|\gamma '(t)\|\,dt.$ 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 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, 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.

Diameter

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, 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.

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 $M$. An (affine) 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

In Euclidean space $\mathbb {R} ^{n}$, the maximal geodesics are straight lines.

In the round sphere $S^{n}$, the maximal geodesics are great circles.

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.

Examples

• The nonconstant maximal geodesics of $\mathbb {R} ^{2}$ with its standard Riemannian metric are exactly the straight lines.
• The nonconstant maximal geodesics of $S^{n}$ with the round metric are exactly the great circles.

Hopf–Rinow theorem

Main article: Hopf–Rinow theorem

The punctured plane $\mathbb {R} ^{2}\backslash \{(0,0)\}$ is not geodesically complete because the maximal geodesic with initial conditions $p=(1,1)$, $v=(1,1)$ does not have domain $\mathbb {R}$.

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 the maximal geodesic with initial conditions $p=(1,1)$, $v=(1,1)$ does not have domain $\mathbb {R}$.

The Hopf–Rinow theorem characterizes geodesically complete manifolds.

Theorem: Let $(M,g)$ be a connected Riemannian manifold. 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

Parallel transport of a tangent vector along a curve in the sphere.

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 along a curve in the setting of a general Riemannian manifold. Given a fixed connection, there is a unique way to do parallel transport.

Riemann curvature tensor

Main article: Riemann curvature tensor

The Riemann curvature tensor measures precisely the extent to which parallel transporting vectors around a small rectangle is not the identity map. The Riemann curvature tensor is 0 at every point if and only if the manifold is locally isometric to Euclidean space.

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

The Ricci curvature tensor $Ric$ plays a defining role in the theory of Einstein manifolds. Specifically, a (pseudo-)Riemannian metric $g$ is called an Einstein metric if Einstein's equation

$Ric=\lambda g$ for some constant $\lambda$

holds.

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. The Ricci curvature tensor is a covariant 2-tensor field.

Scalar curvature

Main article: Scalar curvature

Riemannian manifolds with continuous metrics

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

• Isometries between Riemannian manifolds with continuous metrics are defined the same as in the smooth case.
• One can consider Riemannian submanifolds of Riemannian manifolds with continuous metrics. The pullback metric of a continuous metric through a smooth function is still a continuous metric.
• The product of Riemannian manifolds with continuous metrics is defined the same as in the smooth case and yields a Riemannian manifold with a continuous metric.
• The positive combination of continuous Riemannian metrics is a continuous Riemannian metric.
• The length of an admissible curve is defined exactly the same as in the case when the metric is smooth.
• The metric $d_{g}:M\times M\to [0,\infty )$ is defined exactly the same as in the case when the metric is smooth. As before, $(M,d_{g})$ is a metric space, and the metric topology on $(M,d_{g})$ coincides with the topology on $M$.

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

Sources

• 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