Riemannian manifold

In differential geometry, a Riemannian manifold (or Riemannian space) ${\displaystyle (M,g)}$, so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ${\displaystyle M}$ equipped with a smoothly-varying family ${\displaystyle g}$ of positive-definite inner products ${\displaystyle g_{p}}$ on the tangent spaces ${\displaystyle T_{p}M}$ at each point ${\displaystyle p}$.^[1]

The family ${\displaystyle g}$ of inner products is called a Riemannian metric (or a Riemannian metric tensor, or just a metric).^[1] 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 ${\displaystyle g}$ is smoothly-varying amounts to that for any smooth coordinate chart ${\displaystyle (U,x)}$ on ${\displaystyle M}$, the 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.^[1]

History

In 1827, Carl Friedrich Gauss discovered that the Gaussian curvature of a surface embedded in 3-dimensional space only depends on local measurements made within the surface (the first fundamental form).^[2] This result is known as the Theorema Egregium ("remarkable theorem" in Latin).

A map that preserves the local measurements of a surface is called a local isometry. Call a property of a surface an intrinsic property if it is preserved by local isometries and call it an extrinsic property if it is not. In this language, the Theorema Egregium says that the Gaussian curvature is an intrinsic property of surfaces.

Riemannian manifolds and their curvature were first introduced non-rigorously by Bernhard Riemann in 1854.^[3] However, they would not be formalized until much later. In fact, the more primitive concept of a smooth manifold was first explicitly defined only in 1913 in a book by Hermann Weyl.^[3]

Élie 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

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, a measuring stick that gives 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 puts a measuring stick on every tangent space.

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).^[1] 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 ${\displaystyle (M,g)}$.^[1] 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

${\displaystyle g_{ij}|_{p}:=g_{p}\left(\left.{\frac {\partial }{\partial x^{i}}}\right|_{p},\left.{\frac {\partial }{\partial x^{j}}}\right|_{p}\right)}$.^[4]

These ${\displaystyle n^{2}}$ functions ${\displaystyle g_{ij}:U\to \mathbb {R} }$ can be put together into an ${\displaystyle n\times n}$ matrix-valued function on ${\displaystyle U}$. The requirement that ${\displaystyle g_{p}}$ is a positive-definite inner product then 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

${\displaystyle g=\sum _{i,j}g_{ij}\,dx^{i}\otimes dx^{j}.}$^[4]

Regularity of the Riemannian metric

The Riemannian metric ${\displaystyle g}$ is continuous if its components ${\displaystyle g_{ij}:U\to \mathbb {R} }$ are continuous in any smooth coordinate chart ${\displaystyle (U,x).}$ The Riemannian metric ${\displaystyle g}$ is smooth if its components ${\displaystyle g_{ij}}$ 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 geometric analysis in which one wants to consider non-smooth Riemannian metrics. See for instance (Gromov 1999) and (Shi and Tam 2002). The section Riemannian manifolds with continuous metrics handles the case where ${\displaystyle g}$ is merely continuous, but ${\displaystyle g}$ is assumed to be smooth in this article unless stated otherwise.

Musical isomorphism

In analogy to how an inner product on a vector space induces an isomorphism between a vector space and its dual given by ${\displaystyle v\mapsto \langle v,\cdot \rangle }$, a Riemannian metric induces an isomorphism of bundles between the tangent bundle and the cotangent bundle. Namely, if ${\displaystyle g}$ is a Riemannian metric, then

${\displaystyle v\mapsto g_{p}(v,\cdot )}$

is a bundle isomorphism from the tangent bundle ${\displaystyle TM}$ to the cotangent bundle ${\displaystyle T^{*}M}$.^[5]

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 ${\displaystyle g=f^{\ast }h}$,^[6] 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 Euclidean space (to be defined soon) 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).^[6]

Volume

An oriented ${\displaystyle n}$-dimensional Riemannian manifold ${\displaystyle (M,g)}$ has a unique ${\displaystyle n}$-form ${\displaystyle dV_{g}}$ called the Riemannian volume form.^[7] The Riemannian volume form is preserved by orientation-preserving isometries.^[8] The volume form gives rise to a measure on ${\displaystyle M}$ which allows measurable functions to be integrated.^{[citation needed]} If ${\displaystyle M}$ is compact, the volume of ${\displaystyle M}$ is ${\displaystyle \int _{M}dV_{g}}$.^[7]

Examples

Euclidean space

Let ${\displaystyle x^{1},\ldots ,x^{n}}$ denote the standard coordinates on ${\displaystyle \mathbb {R} ^{n}.}$ The (canonical) Euclidean metric ${\displaystyle g^{\text{can}}}$ is given by^[9]

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

or equivalently

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

or equivalently by its coordinate functions

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

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

Submanifolds

Let ${\displaystyle (M,g)}$ be a Riemannian manifold and let ${\displaystyle i:N\to M}$ be an immersed submanifold or an embedded submanifold of ${\displaystyle M}$. The pullback ${\displaystyle i^{*}g}$ of ${\displaystyle g}$ is a Riemannian metric on ${\displaystyle N}$, and ${\displaystyle (N,i^{*}g)}$ is said to be a Riemannian submanifold of ${\displaystyle (M,g)}$.^[10]

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

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

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

Examples:

• The ${\displaystyle n}$-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 Euclidean space ${\displaystyle \mathbb {R} ^{n+1}}$.^[11] The Riemannian metric this induces on ${\displaystyle S^{n}}$ is called the round metric or standard metric.

• Fix real numbers ${\displaystyle a,b,c}$. The ellipsoid

  ${\displaystyle \left\{x\in \mathbb {R} ^{3}:{\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1\right\}}$

is a smooth embedded submanifold of Euclidean space ${\displaystyle \mathbb {R} ^{3}}$.

• The graph of a smooth function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ is a smooth embedded submanifold of ${\displaystyle \mathbb {R} ^{n+1}}$ with its standard metric.
• If ${\displaystyle (M,g)}$ is not simply connected, there is a covering map ${\displaystyle {\widetilde {M}}\to M}$, where ${\displaystyle {\widetilde {M}}}$ is the universal cover of ${\displaystyle M}$. This is an immersion (since it is locally a diffeomorphism), so ${\displaystyle {\widetilde {M}}}$ automatically inherits a Riemannian metric. By the same principle, any smooth covering space of a Riemannian manifold inherits a Riemannian metric.

On the other hand, if ${\displaystyle N}$ already has a Riemannian metric ${\displaystyle {\tilde {g}}}$, then the immersion (or embedding) ${\displaystyle i:N\to M}$ is called an isometric immersion (or isometric embedding) if ${\displaystyle {\tilde {g}}=i^{*}g}$. Hence isometric immersions and isometric embeddings are Riemannian submanifolds.^[10]

Products

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

  ${\displaystyle {\widetilde {g}}_{p,q}((u_{1},u_{2}),(v_{1},v_{2}))=g_{p}(u_{1},v_{1})+h_{q}(u_{2},v_{2}).}$^[12]

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

  ${\displaystyle {\widetilde {g}}=\sum _{ij}{\widetilde {g}}_{ij}\,dx^{i}\,dx^{j}}$ where ${\displaystyle ({\widetilde {g}}_{ij})={\begin{pmatrix}g_{U}&0\\0&h_{V}\end{pmatrix}}.}$^[12]

For example, consider the ${\displaystyle n}$-torus ${\displaystyle T^{n}=S^{1}\times \cdots \times S^{1}}$. If each copy of ${\displaystyle S^{1}}$ is given the round metric, the product Riemannian manifold ${\displaystyle T^{n}}$ is called the 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.^[13]

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.

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 ${\displaystyle M}$.^[14]

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.^{[citation needed]}

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. It is also not true that any complete metric space of finite diameter must be compact; it matters that the metric space came 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 ${\displaystyle M}$. An (affine) 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 every 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.^[15]

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

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.

A connection ${\displaystyle \nabla }$ is said to preserve the metric if

${\displaystyle X{\bigl (}g(Y,Z){\bigr )}=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)}$

A connection ${\displaystyle \nabla }$ is torsion-free if

${\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,Y],}$

where ${\displaystyle [\cdot ,\cdot ]}$ is the Lie bracket.

A Levi-Civita connection is a torsion-free connection that preserves the metric. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection.^[16]

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.^[17] 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:^[18]

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

Geodesics

Geodesics are curves with no intrinsic acceleration. They are the generalization of straight lines in Euclidean space to arbitrary Riemannian manifolds. An ant living in a Riemannian manifold walking straight ahead without making any effort to accelerate or turn would trace out a geodesic.

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.^[19]

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.^[20] 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.^[21]

Every curve ${\displaystyle \gamma :[0,1]\to M}$ that has the shortest length of any admissible curve with the same endpoints as ${\displaystyle \gamma }$ is a geodesic (in a unit-speed reparameterization).^[22]

Examples

• The nonconstant maximal geodesics of the Euclidean plane ${\displaystyle \mathbb {R} ^{2}}$ are exactly the straight lines.^[21] This agrees the fact from Euclidean geometry that the shortest path between two points is a straight line segment.
• The nonconstant maximal geodesics of ${\displaystyle S^{2}}$ with the round metric are exactly the great circles.^[23] Since the Earth is approximately a sphere, this means that the shortest path a plane can fly between two locations on Earth is a segment of a great circle.

Hopf–Rinow theorem

The Riemannian manifold ${\displaystyle M}$ with its Levi-Civita connection is geodesically complete if the domain of every maximal geodesic is ${\displaystyle (-\infty ,\infty )}$.^[24]. 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 the maximal geodesic with initial conditions ${\displaystyle p=(1,1)}$, ${\displaystyle v=(1,1)}$ does not have domain ${\displaystyle \mathbb {R} }$.

The Hopf–Rinow theorem characterizes geodesically complete manifolds.

Theorem: Let ${\displaystyle (M,g)}$ be a connected Riemannian manifold. The following are equivalent:^[25]

• The metric space ${\displaystyle (M,d_{g})}$ is complete (every ${\displaystyle d_{g}}$-Cauchy sequence converges),
• All closed and bounded subsets of ${\displaystyle M}$ are compact,
• ${\displaystyle M}$ 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 along a curve in the setting of a general Riemannian manifold. Given a fixed connection, there is a unique way to do parallel transport.^[26]

Specifically, call a smooth vector field ${\displaystyle V}$ along a smooth curve ${\displaystyle \gamma }$ parallel along ${\displaystyle \gamma }$ if ${\displaystyle D_{t}V=0}$ identically.^[21] Fix a curve ${\displaystyle \gamma :[0,1]\to M}$ with ${\displaystyle \gamma (0)=p}$ and ${\displaystyle \gamma (1)=q}$. to parallel transport a vector ${\displaystyle v\in T_{p}M}$ to a vector in ${\displaystyle T_{q}M}$ along ${\displaystyle \gamma }$, first extend ${\displaystyle v}$ to a vector field parallel along ${\displaystyle \gamma }$, and then take the value of this vector field at ${\displaystyle q}$.

The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the punctured plane ${\displaystyle \mathbb {R} ^{2}\smallsetminus \{0,0\}}$. The curve the parallel transport is done along is the unit circle. In polar coordinates, the metric on the left is the standard Euclidean metric ${\displaystyle dx^{2}+dy^{2}=dr^{2}+r^{2}\,d\theta ^{2}}$, while the metric on the right is ${\displaystyle dr^{2}+d\theta ^{2}}$. This second metric has a singularity at the origin, so it does not extend past the puncture, but the first metric extends to the entire plane.

Warning: This is parallel transport on the punctured plane along the unit circle, not parallel transport on the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.

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.^[27] The Riemann curvature tensor is 0 at every point if and only if the manifold is locally isometric to Euclidean space.^[28]

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. The Riemann curvature tensor is a ${\displaystyle (1,3)}$-tensor field.^[29]

Ricci curvature tensor

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

${\displaystyle Ric=\lambda g}$ for some constant ${\displaystyle \lambda }$

holds.^[30]

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. The Ricci curvature tensor is a covariant 2-tensor field.^[31]

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.^[32]
• The Riemannian distance function ${\displaystyle d_{g}:M\times M\to [0,\infty )}$ 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 ${\displaystyle M}$.^[33]

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 ${\displaystyle T_{x}M.}$^{[citation needed]}
• A strong Riemannian metric on ${\displaystyle M}$ is a weak Riemannian metric such that ${\displaystyle g_{x}}$ induces the topology on ${\displaystyle T_{x}M}$. If ${\displaystyle g}$ is a strong Riemannian metric, then ${\displaystyle M}$ must be a Hilbert manifold.^{[citation needed]}

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 ${\displaystyle T_{x}H.}$ The metric ${\displaystyle g_{x}(u,v)=\langle u,v\rangle }$ for all ${\displaystyle x,u,v\in H}$ is a strong Riemannian metric.^{[citation needed]}
• 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.^{[citation needed]} Its tangent bundle at the identity is the set of smooth vector fields on ${\displaystyle M.}$^{[citation needed]} Let ${\displaystyle \mu }$ be a volume form on ${\displaystyle M.}$ The ${\displaystyle L^{2}}$ weak Riemannian metric on ${\displaystyle \operatorname {Diff} (M)}$, denoted ${\displaystyle G}$, is defined as follows. Let ${\displaystyle f\in \operatorname {Diff} (M),}$ ${\displaystyle u,v\in T_{f}\operatorname {Diff} (M).}$ Then for ${\displaystyle x\in M,u(x)\in T_{f(x)}M}$,

  ${\displaystyle G_{f}(u,v)=\int _{M}g_{f(x)}(u(x),v(x))\,d\mu (x)}$.^{[citation needed]}

Metric space structure

Length of curves and the Riemannian distance function ${\displaystyle d_{g}:M\times M\to [0,\infty )}$ are defined in a way similar to the finite-dimensional case. The distance function ${\displaystyle d_{g}}$, called the geodesic distance, is always a pseudometric (a metric that does not separate points), but it may not be a metric.^[34] In the finite-dimensional case, the proof that the Riemannian distance function separates points uses the existence of a pre-compact open set around any point. In the infinite case, open sets are no longer pre-compact, so the proof fails.

• 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.^{[citation needed]}
• If ${\displaystyle g}$ is a weak Riemannian metric, ${\displaystyle d_{g}}$ may fail to separate points. In fact, it may even be identically 0.^[34] For example, if ${\displaystyle (M,g)}$ is a compact Riemannian manifold, then the ${\displaystyle L^{2}}$ weak Riemannian metric on ${\displaystyle \operatorname {Diff} (M)}$ induces vanishing geodesic distance.^[35]

Hopf–Rinow theorem

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

Theorem: Let ${\displaystyle (M,g)}$ be a strong Riemannian manifold. Then metric completeness (in the metric ${\displaystyle d_{g}}$) implies geodesic completeness.^{[citation needed]}

However, a geodesically complete strong Riemannian manifold might not be metrically complete and it might have closed and bounded subsets that are not compact.^{[citation needed]} Further, a strong Riemannian manifold for which all closed and bounded subsets are compact might not be geodesically complete.^{[citation needed]}

If ${\displaystyle g}$ is a weak Riemannian metric, then no notion of completeness implies the other in general.^{[citation needed]}

See also

References

Notes

Sources

External links

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