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 :$g\backslash left(\backslash frac,\backslash frac\backslash right):U\backslash to\backslash mathbb$ 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 $g\_p:T\_pM\backslash times\; T\_pM\backslash to\backslash mathbb,$ along with which comes a norm $|\backslash cdot|\_p:T\_pM\backslash to\backslash mathbb$ defined by $|v|\_p=\backslash sqrt.$ The smooth manifold $M$ endowed with this metric $g$ is a Riemannian manifold, denoted $(M,g)$. When given a system of smooth local coordinates on $M,$ given by $n$ real-valued functions $(x^1,\backslash ldots,x^n):U\backslash to\backslash mathbb^n,$ the vectors :$\backslash left\backslash$ form a basis of the vector space $T\_pM,$ for any $p\backslash in\; U.$ Relative to this basis, one can define metric tensor "components" at each point $p$ by :$g\_|\_p:=g\_p\backslash left(\backslash left.\backslash frac\backslash \_p,\backslash left.\backslash frac\backslash \_p\backslash right).$ One could consider these as $n^2$ individual functions $g\_:U\backslash to\backslash mathbb$ or as a single $n\backslash times\; n$ 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 :$g=\backslash sum\_g\_\; \backslash ,\; \backslash mathrm\; d\; x^i\backslash otimes\; \backslash mathrm\; d\; x^j.$


Isometries

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


Regularity of a Riemannian metric

One says that the Riemannian metric $g$ is continuous if $g\_:U\backslash to\backslash mathbb$ are continuous when given any smooth coordinate chart $(U,x).$ 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 $(M,g),$ there is a (usually large) number $N$ and an embedding $F:M\backslash to\backslash mathbb^N$ such that the pullback by $F$ of the standard Riemannian metric on $\backslash mathbb^n$ 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 $x^1,\backslash ldots,x^n$ denote the standard coordinates on $\backslash mathbb^n.$ Then define $g^\_p:\; T\_p\backslash mathbb^n\backslash times\; T\_p\backslash mathbb^n\backslash to\backslash mathbb$ by : $\backslash left(\backslash sum\_ia\_i\backslash frac,\backslash sum\_jb\_j\backslash frac\backslash right)\backslash longmapsto\; \backslash sum\_i\; a\_ib\_i.$ Phrased differently: relative to the standard coordinates, the local representation $g\_:U\backslash to\backslash mathbb$ is given by the constant value $\backslash delta\_.$ This is clearly a Riemannian metric, and is called the standard Riemannian structure on $\backslash mathbb^n.$ 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 $(M,g)$ be a Riemannian manifold and let $N\backslash subset\; M$ 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 $S^=\backslash ,$ 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 $\backslash mathbb^3$ has a natural Riemannian metric. The graph of a smooth function $f:\backslash mathbb^3\backslash to\backslash mathbb$ is an embedded submanifold, and so has a natural Riemannian metric as well.


Immersions

Let $(M,g)$ be a Riemannian manifold and let $f:\backslash Sigma\backslash to\; M$ be a differentiable map. Then one may consider the pullback of $g$ via $f$, which is a symmetric 2-tensor on $\backslash Sigma$ defined by : $(f^\backslash ast\; g)\_p(v,w)=g\_\backslash big(df\_p(v),df\_p(w)\backslash big),$ where $df\_p(v)$ is the pushforward of $v$ by $f.$ In this setting, generally $f^\backslash ast\; g$ will not be a Riemannian metric on $\backslash Sigma,$ since it is not positive-definite. For instance, if $f$ is constant, then $f^\backslash ast\; g$ is zero. In fact, $f^\backslash ast\; g$ is a Riemannian metric if and only if $f$ is an immersion, meaning that the linear map $df\_p:T\_p\backslash Sigma\backslash to\; T\_M$ is injective for each $p\backslash in\backslash Sigma.$ * An important example occurs when $(M,g)$ is not simply-connected, so that there is a covering map $\backslash widetilde\backslash to\; M.$ 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 $(M,g)$ and $(N,h)$ be two Riemannian manifolds, and consider the cartesian product $M\backslash times\; N$ with the usual product smooth structure. The Riemannian metrics $g$ and $h$ naturally put a Riemannian metric $\backslash widetilde$ on $M\backslash times\; N,$ which can be described in a few ways. * Considering the decomposition $T\_(M\backslash times\; N)\backslash cong\; T\_pM\backslash oplus\; T\_qN,$ one may define : $\backslash widetilde\_(u\backslash oplus\; x,v\backslash oplus\; y)=g\_p(u,v)+h\_q(x,y).$ * Let $(U,x)$ be a smooth coordinate chart on $M$ and let $(V,y)$ be a smooth coordinate chart on $N.$ Then $(U\backslash times\; V,(x,y))$ is a smooth coordinate chart on $M\backslash times\; N.$ For convenience let $\backslash operatorname\_^+$ denote the collection of positive-definite symmetric $n\backslash times\; n$ real matrices. Denote the coordinate representation of $g$ relative to $(U,x)$ by $g\_U:U\backslash to\backslash operatorname\_^+$ and denote the coordinate representation of $h$ relative to $(V,y)$ by $h\_V:V\backslash to\backslash operatorname\_^+.$ Then the local coordinate representation of $\backslash widetilde$ relative to $(U\backslash times\; V,(x,y))$ is $\backslash widetilde\_:U\backslash times\; V\backslash to\backslash operatorname\_^+$ given by :: A standard example is to consider the n-torus $T^n,$ define as the n-fold product $S^1\backslash times\backslash cdots\backslash times\; S^1.$ If one gives each copy of $S^1$ its standard Riemannian metric, considering $S^1\backslash subset\backslash mathbb^2$ 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 $\backslash lambda\backslash in,1$ :$\backslash tilde\; g:=\backslash lambda\; g\_0\; +\; (1-\backslash lambda)g\_1$ 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 $\backslash gamma:,bto\; M$ is differentiable, then it assigns to each $t\backslash in(a,b)$ a vector $\backslash gamma\text{'}(t)$ in the vector space $T\_M,$ the size of which can be measured by the norm $|\backslash cdot|\_.$ So $t\backslash mapsto|\backslash gamma\text{'}(t)|\_$ defines a nonnegative function on the interval $(a,b).$ 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 $\backslash gamma$ to be continuously differentiable, so that the function to be integrated is nonnegative and continuous, and hence the length of $\backslash gamma,$ : $L(\backslash gamma)=\backslash int\_a^b|\backslash gamma\text{'}(t)|\_\backslash ,dt,$ 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 $d\_g:M\backslash times\; M\backslash to[0,\backslash infty)$ by :$d\_g(p,q)\; =\; \backslash inf\; \backslash .$ It is mostly straightforward to check the well-definedness of the function $d\_g,$ its symmetry property $d\_g(p,q)=d\_g(q,p),$ its reflexivity property $d\_g(p,p)=0,$ and the triangle inequality $d\_g(p,q)+d\_g(q,r)\backslash geq\; d\_g(p,r),$ 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 $p\backslash neq\; q$ ensures $d\_g(p,q)>0,$ 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 $(M,d\_g)$ 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 $d\_g:M\backslash times\; M\backslash to\backslash mathbb$ 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 $(M,g)$ is an ellipsoid.


Geodesics

As in the previous section, let $(M,g)$ be a connected and continuous Riemannian manifold; consider the associated metric space $(M,d\_g).$ Relative to this metric space structure, one says that a path $c:[a,b]\backslash to\; M$ is a unit-speed [[geodesic if for every $t\_0\backslash in[a,b]$ there exists an interval $J\backslash subset[a,b]$ which contains $t\_0$ and such that : $d\_g(c(s),c(t))=|s-t|\backslash qquad\backslash forall\; s,t\backslash in\; J.$ 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 $|c\text{'}(t)|\_=1$ for all $t,$ then one automatically has $d\_g(c(s),c(t))\backslash leq\; |s-t|$ 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 $c(s)$ and $c(t)$ 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 $(M,g)$ may force even the most innocuous geodesics to bend back and intersect themselves. * Consider the case that $(M,g)$ is the circle $S^1$ with its standard Riemannian metric, and $c:\backslash mathbb\backslash to\; S^1$ is given by $t\backslash mapsto(\backslash cos\; t,\backslash sin\; t).$ 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 $(M,g)$ 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 $(M,g)$ is $\backslash mathbb^2$ with its standard Riemannian metric. Then a unit-speed line such as $t\backslash mapsto\; (2^t,2^t)$ 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 $(M,g)$ be a connected and continuous Riemannian manifold. The Hopf-Rinow theorem, in this setting, says that (Gromov 1999) * if the metric space $(M,d\_g)$ is complete (i.e. every $d\_g$-Cauchy sequence converges) then ** every closed and bounded subset of $M$ is compact. ** given any $p,q\backslash in\; M$ there is a unit-speed geodesic $c:,bto\; M$ from $p$ to $q$ such that $d\_g(c(s),c(t))=|s-t|$ for all $s,t\backslash in,b$ 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 $\backslash overline,$ to a sequence of piecewise continuously-differentiable unit-speed curves from $p$ to $q$ whose lengths approximate $d\_g(p,q).$ The resulting subsequential limit is the desired geodesic. The assumed completeness of $(M,d\_g)$ is important. For example, consider the case that $(M,g)$ is the punctured plane $\backslash mathbb^2\backslash smallsetminus\backslash$ with its standard Riemannian metric, and one takes $p=(1,0)$ and $q=(-1,0).$ There is no unit-speed geodesic from one to the other.


The diameter

Let $(M,g)$ be a connected and continuous Riemannian manifold. As with any metric space, one can define the diameter of $(M,d\_g)$ to be : $\backslash operatorname(M,d\_g)=\backslash sup\backslash .$ The Hopf-Rinow theorem shows that if $(M,d\_g)$ is complete and has finite diameter, then it is compact. Conversely, if $(M,d\_g)$ is compact, then the function $d\_g:M\backslash times\; M\backslash to\backslash mathbb$ has a maximum, since it is a continuous function on a compact metric space. This proves the following statement: * If $(M,d\_g)$ 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. 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 : $M=\backslash Big\backslash$ with the uniform metric : $d(f,g)=\backslash sup\_|f(x)-g(x)|.$ So, although all of the terms in the above corollary of the Hopf-Rinow theorem involve only the metric space structure of $(M,g),$ 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.


See also

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


References

* * * *

External links

* {{Authority control *