Riemannian Metric
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Riemannian Metric

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space is a real, smooth manifold M equipped with an inner product gp on the tangent space TpM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then is a smooth function. The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

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

Introduction

In 1828, Carl Friedrich Gauss proved his Theorema Egregium (remarkable theorem in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, curvature 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. Albert Einstein used the theory of Riemannian manifolds (formally pseudo-Riemannian manifolds) to develop his general theory of relativity. In particular, his equations for gravitation are constraints on the curvature of space-time.

Overview

The tangent bundle of a smooth manifold M assigns to each fixed point of M a vector space called the tangent space, and each tangent space can be equipped with an inner product. If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold. For example, a smooth curve has tangent vector in the tangent space TM(?(t0)) at any point , and each such vector has length ??′(t0)?, where ?·? denotes the norm induced by the inner product on TM(?(t0)). The integral of these lengths gives the length of the curve ?:

${\displaystyle L(\alpha )=\int _{0}^{1}\|\alpha '(t)\|\,\mathrm {d} t.}$

Smoothness of ?(t) for t in guarantees that the integral L(?) exists and the length of this curve is defined.

In many instances, in order to pass from a linear-algebraic concept to a differential-geometric one, the smoothness requirement is very important.

Every smooth submanifold of Rn with a Euclidean metric has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on Rn. In fact, as follows from the Nash embedding theorem, all Riemannian manifolds can be realized this way. In particular one could define Riemannian manifold as a metric space which is isometric to a smooth submanifold of Rn with the induced intrinsic metric, where isometry here is meant in the sense of preserving the length of curves. This definition might theoretically not be flexible enough, but it is quite useful to build the first geometric intuitions in Riemannian geometry.

Riemannian manifolds as metric spaces

Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of the positive-definite quadratic forms on the tangent bundle. This definition allows the construction of an accompanying metric space:

If is a continuously differentiable curve in the Riemannian manifold M, then we define its length L(?) in analogy with the example above by

${\displaystyle L(\gamma )=\int _{a}^{b}\|\gamma '(t)\|\,\mathrm {d} t.}$

With this definition of length, every connected Riemannian manifold M becomes a metric space (and even a length metric space) in a natural fashion: the distance between the points x and y of M is defined as

${\displaystyle d(x,y)=\inf\{L(\gamma ):\gamma {\text{ a continuously differentiable curve joining }}x{\text{ and }}y\}.}$

Even though Riemannian manifolds are usually "curved", there is still a notion of "straight line" on them: the geodesics. These are curves which locally join their points along shortest paths.

Assuming the manifold is complete, any two points x and y can be connected with a geodesic whose length is . Without completeness, this need not be true. For example, in the punctured plane the distance between the points and is 2, but there is no geodesic realizing this distance.

Properties

In Riemannian manifolds, the notions of geodesic completeness and metric space completeness are the same: that each implies the other is the content of the Hopf-Rinow theorem.

Riemannian metrics

Let M be a differentiable manifold of dimension n. A Riemannian metric on M is a family of (positive-definite) inner products

${\displaystyle g_{p}\colon T_{p}M\times T_{p}M\rightarrow \mathbf {R} ,\qquad p\in M}$

such that, for every pair of differentiable vector fields X, Y on M,

${\displaystyle p\mapsto g_{p}(X|_{p},Y|_{p})}$

defines a smooth function .

We can also regard a Riemannian metric g as a symmetric -tensor field that is positive-definite at every point (i.e. whenever ).

In a system of local coordinates on the manifold M given by n real-valued functions x1, x2, ..., xn, the vector fields

${\displaystyle \left\{{\frac {\partial }{\partial x^{1}}},\dotsc ,{\frac {\partial }{\partial x^{n}}}\right\}}$

give a basis of tangent vectors at each point of M. Relative to this coordinate system, the components of the metric tensor are, at each point p,

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

Equivalently, the metric tensor can be written in terms of the dual basis {dx1, ..., dxn} of the cotangent bundle as

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

The differentiable manifold M endowed with this metric g is a Riemannian manifold, denoted .

Examples

• With ${\displaystyle {\frac {\partial }{\partial x^{i}}}}$ identified with , the standard metric over an open subset is defined by
${\displaystyle g_{p}^{\mathrm {can} }\colon T_{p}U\times T_{p}U\longrightarrow \mathbf {R} ,\qquad \left(\sum _{i}a_{i}{\frac {\partial }{\partial x^{i}}},\sum _{j}b_{j}{\frac {\partial }{\partial x^{j}}}\right)\longmapsto \sum _{i}a_{i}b_{i}.}$
Then g is a Riemannian metric, and
${\displaystyle g_{ij}^{\mathrm {can} }=\langle e_{i},e_{j}\rangle =\delta _{ij}.}$
Equipped with this metric, Rn is called Euclidean space of dimension n and gijcan is called the (canonical) Euclidean metric.
• Let be a Riemannian manifold and be a submanifold of M. Then the restriction of g to vectors tangent along N defines a Riemannian metric over N.
• More generally, be an immersion. Then, if N has a Riemannian metric, f induces a Riemannian metric on M via pullback:
${\displaystyle g_{p}^{M}\colon T_{p}M\times T_{p}M\longrightarrow \mathbf {R} ,}$
${\displaystyle (u,v)\longmapsto g_{p}^{M}(u,v):=g_{f(p)}^{N}(df_{p}(u),df_{p}(v)).}$
This is then a metric; the positive definiteness follows from the injectivity of the differential of an immersion.
• Let be a Riemannian manifold, be a differentiable map and be a regular value of h (the differential dh(p) is surjective for all ). Then is a submanifold of M of dimension n. Thus h−1(q) carries the Riemannian metric induced by inclusion.
• In particular, consider the following map :
${\displaystyle h\colon \mathbf {R} ^{n}\longrightarrow \mathbf {R} ,\qquad (x^{1},\dotsc ,x^{n})\longmapsto \sum _{i=1}^{n}(x^{i})^{2}-1.}$
Then 0 is a regular value of h and
${\displaystyle h^{-1}(0)=\left\{x\in \mathbf {R} ^{n}\,\left\vert \,\sum _{i=1}^{n}(x^{i})^{2}=1\right.\right\}=\mathbf {S} ^{n-1}}$
is the unit sphere . The metric induced from Rn on Sn−1 is called the canonical metric of Sn−1.
• Let M1 and M2 be two Riemannian manifolds and consider the cartesian product with the product structure. Furthermore, let and be the natural projections. For , a Riemannian metric on can be introduced as follows :
${\displaystyle g_{(p,q)}^{M_{1}\times M_{2}}\colon T_{(p,q)}(M_{1}\times M_{2})\times T_{(p,q)}(M_{1}\times M_{2})\longrightarrow \mathbf {R} ,}$
${\displaystyle (u,v)\longmapsto g_{p}^{M_{1}}(T_{(p,q)}\pi _{1}(u),T_{(p,q)}\pi _{1}(v))+g_{q}^{M_{2}}(T_{(p,q)}\pi _{2}(u),T_{(p,q)}\pi _{2}(v)).}$
The identification
${\displaystyle T_{(p,q)}(M_{1}\times M_{2})\cong T_{p}M_{1}\oplus T_{q}M_{2}}$
allows us to conclude that this defines a metric on the product space.
The torus possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from R2 on the circle and then taking the product metric. The torus Tn endowed with this metric is called the flat torus.
• Let g0, g1 be two metrics on M. Then,
${\displaystyle {\tilde {g}}:=\lambda g_{0}+(1-\lambda )g_{1},\qquad \lambda \in [0,1],}$
is also a metric on M.

The induced (pullback) metric

If is a differentiable map and a Riemannian manifold, then the pullback of gN along f is the induced metric on the manifold M, in other words it is a bilinear form on the tangent bundle of M. The pullback is the form f*gN on TM defined for by

${\displaystyle (f^{*}g^{N})(v,w)=g^{N}(df(v),df(w))\,,}$

where df(v) is the pushforward of v by f.

The form f*gN is in general only semidefinite, because df can have a kernel. If f is a diffeomorphism, or more generally an immersion, then f*gN is a Riemannian metric on M called the pullback metric. In particular, every embedded smooth submanifold inherits a metric from being embedded in a Riemannian manifold, and every covering space inherits a metric from covering a Riemannian manifold.

Existence of a metric

Every paracompact differentiable manifold admits a Riemannian metric.

Isometries

Let and be two Riemannian manifolds, and be a diffeomorphism. Then, f is called an isometry, if

${\displaystyle g^{M}=f^{*}g^{N}\,,}$

or pointwise

${\displaystyle g_{p}^{M}(u,v)=g_{f(p)}^{N}(df(u),df(v))\qquad \forall p\in M,\forall u,v\in T_{p}M.}$

Moreover, a differentiable mapping is called a local isometry at if there is a neighbourhood , , such that is a diffeomorphism satisfying the previous relation.

Riemannian manifolds as metric spaces

A connected Riemannian manifold carries the structure of a metric space whose distance function is the arc length of a minimizing geodesic. Moreover, this metric space's natural topology agrees with the manifold's topology.[1]

Specifically, let be a connected Riemannian manifold. Let be a parametrized curve in M, which is differentiable with velocity vector c′. The length of c is defined as

${\displaystyle L_{a}^{b}(c):=\int _{a}^{b}{\sqrt {g(c'(t),c'(t))}}\,\mathrm {d} t=\int _{a}^{b}\|c'(t)\|\,\mathrm {d} t.}$

By change of variables, the arclength is independent of the chosen parametrization. In particular, a curve can be parametrized by its arc length. A curve is parametrized by arclength if and only if for all .

The distance function is defined by

${\displaystyle d(p,q)=\inf L(\gamma )}$

where the infimum extends over all differentiable curves ? beginning at and ending at .

This function d satisfies the properties of a distance function for a metric space. The only property which is not completely straightforward is to show that implies that . For this property, one can use a normal coordinate system, which also allows one to show that the topology induced by d is the same as the original topology on M.

Diameter

The diameter of a Riemannian manifold M is defined by

${\displaystyle \operatorname {diam} (M):=\sup _{p,q\in M}d(p,q)\in \mathbf {R} _{\geq 0}\cup \{+\infty \}.}$

The diameter is invariant under global isometries. Furthermore, the Heine-Borel property holds for (finite-dimensional) Riemannian manifolds: M is compact if and only if it is complete and has finite diameter.

Geodesic completeness

A Riemannian manifold M is geodesically complete if for all , the exponential map expp is defined for all , i.e. if any geodesic ?(t) starting from p is defined for all values of the parameter . The Hopf-Rinow theorem asserts that M is geodesically complete if and only if it is complete as a metric space.

If M is complete, then M 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 which are not complete.

References

1. ^ Lee, John M. (2013). Introduction to Smooth Manifolds (2nd ed.). New York: Springer. Theorem 13.29. ISBN 978-1-4419-9981-8.

Further reading

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