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Riemannian manifold From Wikipedia, the free encyclopedia

In Riemannian and the of surfaces, a or Riemannian ( M,g) is a real M in which each is equipped with an inner product g, a Riemannian , which varies smoothly from to point. The terms are named after German mathematician .

A Riemannian metric makes it possible to define various geometric notions on a Riemannian manifold, such as , lengths of , (or ), , of functions and of vector fields.

Riemannian should not be confused with Riemann surfaces, manifolds that locally are patches of the complex plane. Contents

 1 Introduction  2 Overview  2.1 Riemannian manifolds as metric spaces  2.2 Properties

 3 Riemannian metrics  3.1 Examples  3.2 The metric  3.3 Existence of a metric  3.4

 4 Riemannian manifolds as metric spaces  4.1  4.2 completeness

 5 See also  6 References  7 External links

Introduction

In 1828, proved his Theorema Egregium ( remarkable theorem in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a can be determined entirely by measuring 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 was intrinsic to the manifold and not dependent upon its in higher-dimensional spaces. used the theory of Riemannian manifolds to develop his General Theory of Relativity. In particular, his equations for gravitation are restrictions on the curvature of space. Overview

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The tangent of a smooth manifold M assigns to each fixed point of M a called the tangent space, and each tangent space can be equipped with an inner product. If such a collection of inner products on the 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 (t): [0, 1] M has tangent  vector (t0) in the tangent space T M(t0) at any point t0 ∈ (0, 1), and each such vector has length (t0)    , where · denotes the induced by the inner product on TM(t0). The of these lengths gives the length of the curve :

Smoothness of (t) for t in [0, 1] 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 requirement is very important.

Every smooth of Rn 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 , all Riemannian manifolds can be realized this way. In particular one could define Riemannian manifold as a which is isometric to a smooth submanifold of Rn with the induced , where 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 manifolds as metric spaces

Usually a Riemannian manifold is defined as a smooth manifold with a smooth of the positive- definite quadratic forms on the tangent bundle. Then one has to work to show that it can be turned to a metric space:

If : [ a, b] M is a continuously in the Riemannian manifold M, then we define its length L() in analogy with the example above by

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 d(x, y) between the points x and y of M is defined as

d(x,y) = inf{ L( ) : is a continuously differentiable curve joining x and y}.

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

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Assuming the manifold is compact, any two points x and y can be connected with a geodesic whose length is d(x,y). Without compactness, this need not be true. For example, in the punctured plane R2 \ {0}, the distance between the points (−1, 0) and (1, 0) is 2, but there is no geodesic realizing this distance.

Properties

In Riemannian manifolds, the notions of geodesic completeness, topological completeness and metric 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 n. A Riemannian metric on M is a family of (positive definite) inner products

such that, for all differentiable vector fields X,Y on M,

defines a smooth M R.

More formally, a Riemannian metric g is a section of S2(T*M), the symmetric square of the .

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

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

Equivalently, the can be written in terms of the dual basis {d x1, …, dxn} of the cotangent bundle as

Endowed with this metric, the differentiable manifold ( M,g) is a Riemannian manifold .

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Examples

 With identified with , the standard metric over an open is defined by

Then g is a Riemannian metric, and

n can Equipped with this metric, R is called of dimension n and gij is called the Euclidean metric .

 Let ( M,g) 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. n n+k  More generally, let f:M N be an . Then, if N has a Riemannian metric, f induces a Riemannian metric on M via pullback:

This is then a metric; the positive definiteness follows of the injectivity of the differential of an immersion.

M n+k k  Let ( M,g ) be a Riemannian manifold, h:M N be a differentiable map and q∈N be a regular value of h (the differential dh (p) is surjective for all p∈h-1(q)). Then h-1(q)∈M 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 :

Then, 0 is a regular value of h and

is the unit . The metric induced from on Sn − 1 is called the canonical metric of Sn − 1.

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 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 :

The identification

allows us to conclude that this defines a metric on the product space.

The possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from 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,

is also a metric on M.

The pullback metric

If f:MN is a differentiable map and ( N,gN) a Riemannian manifold, then the pullback of gN along f is a on the tangent space of M. The pullback is the quadratic form f*gN on TM defined for v, w ∈ TpM by

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

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

Existence of a metric

Every paracompact differentiable manifold admits a Riemannian metric. To prove this result, let M be a manifold and {( U, (U))| ∈I} a locally finite of open U of M and onto open subsets of Rn

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Let be a differentiable subordinate to the given atlas. Then define the metric g on M by

where gcan is the Euclidean metric. This is readily seen to be a metric on M.

Isometries

Let (M,gM) and (N,gN) be two Riemannian manifolds, and be a diffeomorphism. Then, f is called an isometry , if

or pointwise

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 arclength of a minimizing geodesic.

Specifically, let ( M,g) 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

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

The distance function d : M×M [0, ) is defined by

where the infimum extends over all differentiable curves beginning at p∈M and ending at q∈M.

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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 d(p,q)=0 implies that p=q. For this property, one can use a 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

The diameter is 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 exp p 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. See also

 Riemannian geometry   sub-Riemannian manifold  pseudo-Riemannian manifold  Metric tensor   Space (mathematics) References

 Jost, Jürgen (2008), Riemannian Geometry and Geometric Analysis (5th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3540773405  do Carmo, Manfredo (1992), Riemannian geometry , Basel, Boston, Berlin: Birkhäuser, ISBN 978- 0-8176-3490-2 [1] External links

 L.A. Sidorov (2001), "Riemannian metric" , in Hazewinkel, Michiel, Encyclopaedia of

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Mathematics , Springer, ISBN 978 -1556080104 , http://eom.springer.de/R/r082180.htm Retrieved from "http://en.wikipedia.org/wiki/Riemannian_manifold" Categories: Riemannian geometry | Structures on manifolds

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