Riemannian manifold

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 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.

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

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

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

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

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

The Riemannian metric in coordinates

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

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

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

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

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

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

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

Regularity of the Riemannian metric

The Riemannian metric ${\displaystyle g}$ is continuous if ${\displaystyle g_{ij}:U\to \mathbb {R} }$ are continuous in any smooth coordinate chart ${\displaystyle (U,x).}$ The Riemannian metric ${\displaystyle g}$ is smooth if ${\displaystyle g_{ij}}$ 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 the ${\displaystyle g_{ij}}$ are merely continuous, but ${\displaystyle g}$ is 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 ${\displaystyle (M,g)}$ and ${\displaystyle (N,h)}$ are two Riemannian manifolds, a diffeomorphism ${\displaystyle f:M\to N}$ is called an isometry if ${\displaystyle g=f^{\ast }h}$, that is, if

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

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

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

Examples

Euclidean space

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

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

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

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

Embedded submanifolds

Let ${\displaystyle (M,g)}$ be a Riemannian manifold and let ${\displaystyle N\subset M}$ be an embedded submanifold of ${\displaystyle M}$. The restriction of g to vectors tangent along N defines a Riemannian metric on N.

• The n-sphere ${\displaystyle S^{n}=\{x\in \mathbb {R} ^{n+1}:(x^{1})^{2}+\cdots +(x^{n+1})^{2}=1\}}$ is a smooth embedded submanifold of ${\displaystyle \mathbb {R} ^{n+1}}$ with its standard metric. The Riemannian metric this induces on ${\displaystyle S^{n}}$ is called the round metric.
• An ellipsoid in ${\displaystyle \mathbb {R} ^{3}}$ is an embedded submanifold, hence it has a Riemannian metric.
• The graph of a smooth function ${\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} }$ is an embedded submanifold, so it has a Riemannian metric.

Immersions

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

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

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

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

• An important example occurs when ${\displaystyle (M,g)}$ is not simply connected, so that there is a covering map ${\displaystyle {\widetilde {M}}\to M.}$ 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.

Products

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

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

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

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

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

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

Positive combinations of metrics

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

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 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.

An alternative proof uses the Whitney embedding theorem to embed ${\displaystyle M}$ into Euclidean space and then pulls back the metric from Euclidean space to ${\displaystyle M}$. On the other hand, the Nash embedding theorem states that, given any smooth Riemannian manifold ${\displaystyle (M,g),}$ there is an embedding ${\displaystyle F:M\to \mathbb {R} ^{N}}$ for some ${\displaystyle N}$ such that the pullback by ${\displaystyle F}$ of the standard Riemannian metric on ${\displaystyle \mathbb {R} ^{N}}$ is ${\displaystyle g.}$ 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 ${\displaystyle \gamma :[0,1]\to M}$ whose velocity ${\displaystyle \gamma '(t)\in T_{\gamma (t)}M}$ is nonzero everywhere it is defined. The nonnegative function ${\displaystyle t\mapsto |\gamma '(t)|_{\gamma (t)}}$ is defined on the interval ${\displaystyle [0,1]}$ except for at finitely many points. The length ${\displaystyle L(\gamma )}$ of an admissible curve ${\displaystyle \gamma :[0,1]\to M}$ is defined as

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

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

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

Theorem: ${\displaystyle (M,d_{g})}$ is a metric space, and the metric topology on ${\displaystyle (M,d_{g})}$ coincides with the topology on ${\displaystyle M}$.^[1]

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

Proof sketch that ${\displaystyle p\neq q}$ implies ${\displaystyle d_{g}(p,q)>0}$ and that the metric topology of ${\displaystyle (M,d_{g})}$ coincides with the topology on ${\displaystyle M}$.

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 ${\displaystyle (U,x)}$ be a smooth coordinate chart with ${\displaystyle x(p)=0}$ and ${\displaystyle q\notin U.}$ Let ${\displaystyle V\ni x}$ be an open subset of ${\displaystyle U}$ with ${\displaystyle {\overline {V}}\subset U.}$ By continuity of ${\displaystyle g}$ and compactness of ${\displaystyle {\overline {V}},}$ there is a positive number ${\displaystyle \lambda }$ such that ${\displaystyle g(X,X)\geq \lambda \|X\|^{2}}$ for any ${\displaystyle r\in V}$ and any ${\displaystyle X\in T_{r}M,}$ where ${\displaystyle \|\cdot \|}$ denotes the Euclidean norm induced by the local coordinates. Let R denote ${\displaystyle \sup\{r>0:B_{r}(0)\subset x(V)\},}$ to be used at the final step of the proof. Now, given any admissible curve ${\displaystyle \gamma :[0,1]\to M}$ from p to q, there must be some minimal ${\displaystyle \delta >0}$ such that ${\displaystyle \gamma (\delta )\notin V;}$ clearly ${\displaystyle \gamma (\delta )\in \partial V.}$ The length of ${\displaystyle \gamma }$ is at least as large as the restriction of ${\displaystyle \gamma }$ to ${\displaystyle [0,\delta ].}$ So ${\displaystyle L(\gamma )\geq {\sqrt {\lambda }}\int _{0}^{\delta }\|\gamma '(t)\|\,dt.}$ The integral which appears here represents the Euclidean length of a curve from 0 to ${\displaystyle x(\partial V)\subset \mathbb {R} ^{n}}$, and so it is greater than or equal to R. So we conclude ${\displaystyle L(\gamma )\geq {\sqrt {\lambda }}R.}$ 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 ${\displaystyle (M,d_{g})}$ coincides with the original topological space structure of ${\displaystyle M}$.

Although the length of a curve is given by an explicit formula, it is generally impossible to write out the distance function ${\displaystyle d_{g}}$ by any explicit means. In fact, if ${\displaystyle M}$ is compact, there always exist points where ${\displaystyle d_{g}:M\times M\to \mathbb {R} }$ 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 ${\displaystyle (M,g)}$ is an ellipsoid.

Diameter

The diameter of the metric space ${\displaystyle (M,d_{g})}$ is

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

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

If ${\displaystyle (M,d_{g})}$ 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

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

with the uniform metric

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

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

Connections, geodesics, and curvature

Connections

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

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

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

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

The expression ${\displaystyle \nabla _{X}Y}$ is called the covariant derivative of ${\displaystyle Y}$ with respect to ${\displaystyle X}$.^[1]

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

A torsion-free connection for which ${\displaystyle g}$ is parallel with respect to ${\displaystyle \nabla }$ is called a Levi-Civita connection. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection.^[1]

Covariant derivative along a curve

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

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

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

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

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

Geodesics

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

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

For every ${\displaystyle p\in M}$ and ${\displaystyle v\in T_{p}M}$, there exists a geodesic ${\displaystyle \gamma :I\to M}$ defined on some open interval ${\displaystyle I}$ containing 0 such that ${\displaystyle \gamma (0)=p}$ and ${\displaystyle \gamma '(0)=v}$. Any two such geodesics agree on their common domain.^[1] Taking the union over all open intervals ${\displaystyle I}$ containing 0 on which a geodesic satisfying ${\displaystyle \gamma (0)=p}$ and ${\displaystyle \gamma '(0)=v}$ exists, one obtains a geodesic called a maximal geodesic of which every geodesic satisfying ${\displaystyle \gamma (0)=p}$ and ${\displaystyle \gamma '(0)=v}$ is a restriction.

Examples

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

Geodesics of a metric space

There is also a notion of a geodesic of a metric space. Relative to the metric space ${\displaystyle (M,d_{g})}$, a path ${\displaystyle c:[a,b]\to M}$ is a unit-speed geodesic if for every ${\displaystyle t_{0}\in [a,b]}$ there is an interval ${\displaystyle J\subset [a,b]}$ containing it such that

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

Informally, one may say that one is asking for ${\displaystyle c}$ to locally 'stretch itself out' as much as it can, subject to the unit-speed constraint. The idea is that if ${\displaystyle c:[a,b]\to M}$ is admissible and ${\displaystyle |c'(t)|_{c(t)}=1}$ for all ${\displaystyle t}$ where the derivative exists, then one automatically has ${\displaystyle d_{g}(c(s),c(t))\leq |s-t|}$ by applying the triangle inequality to a Riemann sum approximation of the integral defining the length of ${\displaystyle c.}$ So the unit-speed geodesic condition as given above is requiring ${\displaystyle c(s)}$ and ${\displaystyle c(t)}$ 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 ${\displaystyle (M,g)}$ may force even the most innocuous geodesics to bend back and intersect themselves.

Unit-speed geodesics, as defined here, are necessarily continuous, and in fact Lipschitz, but they are not necessarily differentiable or piecewise differentiable.

The Hopf–Rinow theorem

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

If ${\displaystyle M}$ 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.^{[citation needed]}

The Hopf–Rinow theorem characterizes geodesically complete manifolds.

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

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

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.^[1]

Riemann curvature tensor

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

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

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

Ricci curvature tensor

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

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

where ${\displaystyle \operatorname {tr} }$ is the trace.^[1]

Scalar curvature

Riemannian manifolds with continuous metrics

Throughout this section, Riemannian metrics ${\displaystyle g}$ 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 ${\displaystyle d_{g}:M\times M\to [0,\infty )}$ is defined exactly the same as in the case when the metric is smooth. As before, ${\displaystyle (M,d_{g})}$ is a metric space, and the metric topology on ${\displaystyle (M,d_{g})}$ coincides with the topology on ${\displaystyle M}$.
• The metric space geodesics of a Riemannian manifold can be considered just as in the case when the metric is smooth.

Infinite-dimensional manifolds

The statements and theorems above are for finite-dimensional manifolds—manifolds whose charts map to open subsets of ${\displaystyle \mathbb {R} ^{n}.}$ 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 ${\displaystyle M}$ is a smooth function ${\displaystyle g:TM\times TM\to \mathbb {R} ,}$ such that for any ${\displaystyle x\in M}$ the restriction ${\displaystyle g_{x}:T_{x}M\times T_{x}M\to \mathbb {R} }$ is an inner product on ${\displaystyle T_{x}M.}$
• A strong Riemannian metric on ${\displaystyle M}$ is a weak Riemannian metric, such that ${\displaystyle g_{x}}$ induces the topology on ${\displaystyle T_{x}M.}$ Note that if ${\displaystyle M}$ is not a Hilbert manifold then ${\displaystyle g}$ cannot be a strong metric.

Examples

• If ${\displaystyle (H,\langle \,\cdot ,\cdot \,\rangle )}$ is a Hilbert space, then for any ${\displaystyle x\in H,}$ one can identify ${\displaystyle H}$ with ${\displaystyle T_{x}H.}$ By setting for all ${\displaystyle x,u,v\in H}$ ${\displaystyle g_{x}(u,v)=\langle u,v\rangle }$ one obtains a strong Riemannian metric.
• Let ${\displaystyle (M,g)}$ be a compact Riemannian manifold and denote by ${\displaystyle \operatorname {Diff} (M)}$ 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 ${\displaystyle M.}$ Let ${\displaystyle \mu }$ be a volume form on ${\displaystyle M.}$ Then one can define ${\displaystyle G,}$ the ${\displaystyle L^{2}}$ weak Riemannian metric, on ${\displaystyle \operatorname {Diff} (M).}$ Let ${\displaystyle f\in \operatorname {Diff} (M),}$ ${\displaystyle u,v\in T_{f}\operatorname {Diff} (M).}$ Then for ${\displaystyle x\in M,u(x)\in T_{f(x)}M}$ and define ${\displaystyle G_{f}(u,v)=\int _{M}g_{f(x)}(u(x),v(x))d\mu (x).}$ The ${\displaystyle L^{2}}$ weak Riemannian metric on ${\displaystyle \operatorname {Diff} (M)}$ 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 ${\displaystyle d_{g}:M\times M\to [0,\infty )}$ 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 ${\displaystyle g}$ is a strong Riemannian metric on ${\displaystyle M}$, then ${\displaystyle d_{g}}$ separates points (hence is a metric) and induces the original topology.
• If ${\displaystyle g}$ is a weak Riemannian metric but not strong, ${\displaystyle d_{g}}$ 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 ${\displaystyle (M,g)}$ be a strong Riemannian manifold. Then metric completeness (in the metric ${\displaystyle d_{g}}$) 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 ${\displaystyle g}$ is a weak Riemannian metric, then no notion of completeness implies the other in general.

See also

Footnotes

References

External links

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