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In differential geometry, a Riemannian manifold or Riemannian space is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. A common convention is to take ''g'' to be smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :$g\left\left(\frac,\frac\right\right):U\to\mathbb$ are smooth functions. In the same way, one could also consider Lipschitz Riemannian metrics or measurable Riemannian metrics, among many other possibilities. The family ''g''''p'' 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 pseudo-Riemannian manifolds (a generalization of Riemannian manifolds) to develop his general theory of relativity. In particular, his equations for gravitation are constraints on the curvature of spacetime.

Definition

The tangent bundle of a smooth manifold $M$ assigns to each point $p$ of $M$ a vector space $T_pM$ called the tangent space of $M$ at $p.$ A Riemannian metric (by its definition) assigns to each $p$ a positive-definite inner product $g_p:T_pM\times T_pM\to\mathbb,$ along with which comes a norm $|\cdot|_p:T_pM\to\mathbb$ defined by $|v|_p=\sqrt.$ The smooth manifold $M$ endowed with this metric $g$ is a Riemannian manifold, denoted $\left(M,g\right)$. When given a system of smooth local coordinates on $M,$ given by $n$ real-valued functions $\left(x^1,\ldots,x^n\right):U\to\mathbb^n,$ the vectors :$\left\$ form a basis of the vector space $T_pM,$ for any $p\in U.$ Relative to this basis, one can define metric tensor "components" at each point $p$ by :$g_|_p:=g_p\left\left(\left.\frac\_p,\left.\frac\_p\right\right).$ One could consider these as $n^2$ individual functions $g_:U\to\mathbb$ or as a single $n\times n$ matrix-valued function on $U;$ note that the "Riemannian" assumption says that it is valued in the subset consisting of symmetric positive-definite matrices. In terms of tensor algebra, the metric tensor can be written in terms of the dual basis of the cotangent bundle as :$g=\sum_g_ \, \mathrm d x^i\otimes \mathrm d x^j.$

Isometries

If $\left(M,g\right)$ and $\left(N,h\right)$ are two Riemannian manifolds, with $f:M\to N$ a diffeomorphism, then $f$ is called an isometry if $g=f^\ast h,$ i.e. if : $g_p\left(u,v\right)=h_\left(df_p\left(u\right),df_p\left(v\right)\right)$ for all $p\in M$ and $u,v\in T_pM.$ One says that a map $f:M\to N,$ not assumed to be a diffeomorphism, is a local isometry if every $p\in M$ has an open neighborhood $U\ni p$ such that $f:U\to f\left(U\right)$ is a diffeomorphism and isometry.

Regularity of a Riemannian metric

One says that the Riemannian metric $g$ is continuous if $g_:U\to\mathbb$ are continuous when given any smooth coordinate chart $\left(U,x\right).$ One says that $g$ is smooth if these functions are smooth when given any smooth coordinate chart. One could also consider many other types of Riemannian metrics in this spirit. In most expository accounts of Riemannian geometry, the metrics are always taken to be smooth. However, there can be important reasons to consider metrics which are less 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).

Overview

Examples of Riemannian manifolds will be discussed below. A famous theorem of John Nash states that, given any smooth Riemannian manifold $\left(M,g\right),$ there is a (usually large) number $N$ and an embedding $F:M\to\mathbb^N$ such that the pullback by $F$ of the standard Riemannian metric on $\mathbb^n$ is $g.$ Informally, the entire structure of a smooth Riemannian manifold can be encoded by a diffeomorphism to a certain embedded submanifold of some Euclidean space. In this sense, it is arguable 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.

Examples

Euclidean space

Let $x^1,\ldots,x^n$ denote the standard coordinates on $\mathbb^n.$ Then define $g^_p: T_p\mathbb^n\times T_p\mathbb^n\to\mathbb$ by : $\left\left(\sum_ia_i\frac,\sum_jb_j\frac\right\right)\longmapsto \sum_i a_ib_i.$ Phrased differently: relative to the standard coordinates, the local representation $g_:U\to\mathbb$ is given by the constant value $\delta_.$ This is clearly a Riemannian metric, and is called the standard Riemannian structure on $\mathbb^n.$ 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 $\left(M,g\right)$ be a Riemannian manifold and let $N\subset M$ be an embedded submanifold of $M,$ which is at least $C^1.$ Then the restriction of ''g'' to vectors tangent along ''N'' defines a Riemannian metric over ''N''. * For example, consider $S^=\,$ which is a smooth embedded submanifold of the Euclidean space with its standard metric. The Riemannian metric this induces on $S^$ is called the standard metric or canonical metric on $S^.$ * There are many similar examples. For example, every ellipsoid in $\mathbb^3$ has a natural Riemannian metric. The graph of a smooth function $f:\mathbb^3\to\mathbb$ is an embedded submanifold, and so has a natural Riemannian metric as well.

Immersions

Let $\left(M,g\right)$ be a Riemannian manifold and let $f:\Sigma\to M$ be a differentiable map. Then one may consider the pullback of $g$ via $f$, which is a symmetric 2-tensor on $\Sigma$ defined by : $\left(f^\ast g\right)_p\left(v,w\right)=g_\big\left(df_p\left(v\right),df_p\left(w\right)\big\right),$ where $df_p\left(v\right)$ is the pushforward of $v$ by $f.$ In this setting, generally $f^\ast g$ will not be a Riemannian metric on $\Sigma,$ since it is not positive-definite. For instance, if $f$ is constant, then $f^\ast g$ is zero. In fact, $f^\ast g$ is a Riemannian metric if and only if $f$ is an immersion, meaning that the linear map $df_p:T_p\Sigma\to T_M$ is injective for each $p\in\Sigma.$ * An important example occurs when $\left(M,g\right)$ is not simply-connected, so that there is a covering map $\widetilde\to M.$ 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.

Product metrics

Let $\left(M,g\right)$ and $\left(N,h\right)$ be two Riemannian manifolds, and consider the cartesian product $M\times N$ with the usual product smooth structure. The Riemannian metrics $g$ and $h$ naturally put a Riemannian metric $\widetilde$ on $M\times N,$ which can be described in a few ways. * Considering the decomposition $T_\left(M\times N\right)\cong T_pM\oplus T_qN,$ one may define : $\widetilde_\left(u\oplus x,v\oplus y\right)=g_p\left(u,v\right)+h_q\left(x,y\right).$ * Let $\left(U,x\right)$ be a smooth coordinate chart on $M$ and let $\left(V,y\right)$ be a smooth coordinate chart on $N.$ Then $\left(U\times V,\left(x,y\right)\right)$ is a smooth coordinate chart on $M\times N.$ For convenience let $\operatorname_^+$ denote the collection of positive-definite symmetric $n\times n$ real matrices. Denote the coordinate representation of $g$ relative to $\left(U,x\right)$ by $g_U:U\to\operatorname_^+$ and denote the coordinate representation of $h$ relative to $\left(V,y\right)$ by $h_V:V\to\operatorname_^+.$ Then the local coordinate representation of $\widetilde$ relative to $\left(U\times V,\left(x,y\right)\right)$ is $\widetilde_:U\times V\to\operatorname_^+$ given by :: $\left(p,q\right)\mapsto \beging_U\left(p\right)&0\\ 0&h_V\left(q\right)\end.$ A standard example is to consider the n-torus $T^n,$ define 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 $S^1\subset\mathbb^2$ as an embedded submanifold (as above), then one can consider the product Riemannian metric on $T^n.$ It is called a flat torus.

Convex combinations of metrics

Let $g_0$ and $g_1$ be two Riemannian metrics on $M.$ Then, for any number $\lambda\in,1$ :$\tilde g:=\lambda g_0 + \left(1-\lambda\right)g_1$ is also a Riemannian metric on $M.$ More generally, if $a$ and $b$ are any two positive numbers, then $ag_0+bg_1$ is another Riemannian metric.

Every smooth manifold has a 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.

The metric space structure of continuous connected Riemannian manifolds

The length of piecewise continuously-differentiable curves

If $\gamma:,bto M$ is differentiable, then it assigns to each $t\in\left(a,b\right)$ a vector $\gamma\text{'}\left(t\right)$ in the vector space $T_M,$ the size of which can be measured by the norm $|\cdot|_.$ So $t\mapsto|\gamma\text{'}\left(t\right)|_$ defines a nonnegative function on the interval $\left(a,b\right).$ The length is defined as the integral of this function; however, as presented here, there is no reason to expect this function to be integrable. It is typical to suppose ''g'' to be continuous and $\gamma$ to be continuously differentiable, so that the function to be integrated is nonnegative and continuous, and hence the length of $\gamma,$ : $L\left(\gamma\right)=\int_a^b|\gamma\text{'}\left(t\right)|_\,dt,$ is well-defined. This definition can easily be extended to define the length of any piecewise-continuously differentiable curve. In many instances, such as in defining the Riemann curvature tensor, it is necessary to require that ''g'' has more regularity than mere continuity; this will be discussed elsewhere. For now, continuity of ''g'' will be enough to use the length defined above in order to endow ''M'' with the structure of a metric space, provided that it is connected.

The metric space structure

Precisely, define $d_g:M\times M\to\left[0,\infty\right)$ by :$d_g\left(p,q\right) = \inf \.$ It is mostly straightforward to check the well-definedness of the function $d_g,$ its symmetry property $d_g\left(p,q\right)=d_g\left(q,p\right),$ its reflexivity property $d_g\left(p,p\right)=0,$ and the triangle inequality $d_g\left(p,q\right)+d_g\left(q,r\right)\geq d_g\left(p,r\right),$ although there are some minor technical complications (such as verifying that any two points can be connected by a piecewise-differentiable path). It is more fundamental to understand that $p\neq q$ ensures $d_g\left(p,q\right)>0,$ and hence that $d_g$ satisfies all of the axioms of a metric. The observation that underlies the above proof, about comparison between lengths measured by ''g'' and Euclidean lengths measured in a smooth coordinate chart, also verifies that the metric space topology of $\left(M,d_g\right)$ 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 then, even when ''g'' is smooth, there always exist points where $d_g:M\times M\to\mathbb$ 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 $\left(M,g\right)$ is an ellipsoid.

Geodesics

As in the previous section, let $\left(M,g\right)$ be a connected and continuous Riemannian manifold; consider the associated metric space $\left(M,d_g\right).$ Relative to this metric space structure, one says that a path $c:\left[a,b\right]\to M$ is a unit-speed [[geodesic if for every $t_0\in\left[a,b\right]$ there exists an interval $J\subset\left[a,b\right]$ which contains $t_0$ and such that : $d_g\left(c\left(s\right),c\left(t\right)\right)=|s-t|\qquad\forall s,t\in J.$ Informally, one may say that one is asking for $c$ to locally 'stretch itself out' as much as it can, subject to the (informally considered) unit-speed constraint. The idea is that if $c:,bto M$ is (piecewise) continuously differentiable and $|c\text{'}\left(t\right)|_=1$ for all $t,$ then one automatically has $d_g\left(c\left(s\right),c\left(t\right)\right)\leq |s-t|$ by applying the triangle inequality to a Riemann sum approximation of the integral defining the length of $c.$ So the unit-speed geodesic condition as given above is requiring $c\left(s\right)$ and $c\left(t\right)$ 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 $\left(M,g\right)$ may force even the most innocuous geodesics to bend back and intersect themselves. * Consider the case that $\left(M,g\right)$ is the circle $S^1$ with its standard Riemannian metric, and $c:\mathbb\to S^1$ is given by $t\mapsto\left(\cos t,\sin t\right).$ Recall that $d_g$ is measured by the lengths of curves along $S^1$, not by the straight-line paths in the plane. This example also exhibits the necessity of selecting out the subinterval $J,$ since the curve $c$ repeats back on itself in a particularly natural way. * Likewise, if $\left(M,g\right)$ is the round sphere $S^2$ with its standard Riemannian metric, then a unit-speed path along an equatorial circle will be a geodesic. A unit-speed path along the other latitudinal circles will not be geodesic. * Consider the case that $\left(M,g\right)$ is $\mathbb^2$ with its standard Riemannian metric. Then a unit-speed line such as $t\mapsto \left(2^t,2^t\right)$ is a geodesic but the curve $c$ from the first example above is not. Note that unit-speed geodesics, as defined here, are by necessity continuous, and in fact Lipschitz, but they are not necessarily differentiable or piecewise differentiable.

The Hopf-Rinow theorem

As above, let $\left(M,g\right)$ be a connected and continuous Riemannian manifold. The Hopf-Rinow theorem, in this setting, says that (Gromov 1999) * if the metric space $\left(M,d_g\right)$ is complete (i.e. every $d_g$-Cauchy sequence converges) then ** every closed and bounded subset of $M$ is compact. ** given any $p,q\in M$ there is a unit-speed geodesic $c:,bto M$ from $p$ to $q$ such that $d_g\left(c\left(s\right),c\left(t\right)\right)=|s-t|$ for all $s,t\in,b$ The essence of the proof is that once the first half is established, one may directly apply the Arzelà-Ascoli theorem, in the context of the compact metric space $\overline,$ to a sequence of piecewise continuously-differentiable unit-speed curves from $p$ to $q$ whose lengths approximate $d_g\left(p,q\right).$ The resulting subsequential limit is the desired geodesic. The assumed completeness of $\left(M,d_g\right)$ is important. For example, consider the case that $\left(M,g\right)$ is the punctured plane $\mathbb^2\smallsetminus\$ with its standard Riemannian metric, and one takes $p=\left(1,0\right)$ and $q=\left(-1,0\right).$ There is no unit-speed geodesic from one to the other.

The diameter

Let $\left(M,g\right)$ be a connected and continuous Riemannian manifold. As with any metric space, one can define the diameter of $\left(M,d_g\right)$ to be : $\operatorname\left(M,d_g\right)=\sup\.$ The Hopf-Rinow theorem shows that if $\left(M,d_g\right)$ is complete and has finite diameter, then it is compact. Conversely, if $\left(M,d_g\right)$ is compact, then the function $d_g:M\times M\to\mathbb$ has a maximum, since it is a continuous function on a compact metric space. This proves the following statement: * If $\left(M,d_g\right)$ 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. Note that, more generally, and with the same one-line proof, every compact metric space has finite diameter. However the following statement is ''false'': "If a metric space is complete and has finite diameter, then it is compact." For an example of a complete and non-compact metric space of finite diameter, consider : $M=\Big\$ with the uniform metric : $d\left(f,g\right)=\sup_|f\left(x\right)-g\left(x\right)|.$ So, although all of the terms in the above corollary of the Hopf-Rinow theorem involve only the metric space structure of $\left(M,g\right),$ it is important that the metric is induced from a Riemannian structure.

Riemannian metrics

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 ''γ''(''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 that are not complete.

* Riemannian geometry * Finsler manifold * Sub-Riemannian manifold * Pseudo-Riemannian manifold * Metric tensor * Hermitian manifold * Space (mathematics) * Wave maps equation

References

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