TheInfoList

In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, a Riemannian manifold or Riemannian space is a
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
,
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
''M'' equipped with a positive-definite
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
''g''''p'' on the
tangent space In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...
''T''''p''''M'' at each point ''p''. A common convention is to take ''g'' to be
smooth Smooth may refer to: Mathematics * Smooth function is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ...
, which means that for any smooth
coordinate chartIn topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological spa ...
on ''M'', the ''n''2 functions :$g\left\left(\frac,\frac\right\right):U\to\mathbb$ are
smooth function is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over ...

s. In the same way, one could also consider
LipschitzLipschitz, Lipshitz, or Lipchitz is an Ashkenazi Jewish surname. The surname has many variants, including: Lifshitz (Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lipschütz), Lipshutz, Lüpschütz; ...

Riemannian metrics or
measurable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
Riemannian metrics, among many other possibilities. The family ''g''''p'' of inner products is called a Riemannian metric (or Riemannian metric tensor) after the German mathematician
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
.
Riemannian geometry#REDIRECT Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and m ...
is the study of Riemannian manifolds. A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

at an intersection, length of a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

,
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

of a surface and higher-dimensional analogues (
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

, etc.), extrinsic curvature of submanifolds, and
intrinsic curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a Surface (mathematics), surface deviates from being a plane (ge ...
of the manifold itself.

Introduction

In 1828,
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

proved his ''Theorema Egregium'' ("remarkable theorem" in Latin), establishing an important property of surfaces. Informally, the theorem says that the 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
''.
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
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 Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

used the theory of
pseudo-Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differenti ...
s (a generalization of Riemannian manifolds) to develop his
general theory of relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
. In particular, his equations for gravitation are constraints on the curvature of spacetime.

Definition

The
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

of a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
$M$ assigns to each point $p$ of $M$ a vector space $T_pM$ called the
tangent space In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...
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 In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
$M$ endowed with this metric $g$ is a Riemannian manifold, denoted $\left(M,g\right)$. When given a system of smooth
local coordinates Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples: * Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow t ...
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 In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
, the
metric tensor In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
can be written in terms of the
dual basis In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, 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 Geometric analysis is a mathematics, mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology. The use o ...
, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
of
John NashJohn Nash may refer to: Arts and entertainment *John Nash (architect) (1752–1835), Anglo-Welsh architect *John Nash Round, English architect active in the mid-19th-century Kent *"Jolly" John Nash (1828–1901), English music hall entertainer *Joh ...
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 In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward. Precomposition Precomposition with a Function (mathematics), function probabl ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 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 Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
of dimension ''n'' and ''g''''ij''can is also called the (canonical)
Euclidean metric In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
.

Embedded submanifolds

Let $\left(M,g\right)$ be a Riemannian manifold and let $N\subset M$ be an
embedded submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which pro ...
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 In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward. Precomposition Precomposition with a Function (mathematics), function probabl ...
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 The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" ope ...
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 .

Convex combinations of metrics

Let $g_0$ and $g_1$ be two Riemannian metrics on $M.$ Then, for any number :$\tilde g:=\lambda g_0 + \left(1-\lambda\right)g_1$ 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 unityIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

The metric space structure of continuous connected Riemannian manifolds

The length of piecewise continuously-differentiable curves

If 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 In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, 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 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 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
LipschitzLipschitz, Lipshitz, or Lipchitz is an Ashkenazi Jewish surname. The surname has many variants, including: Lifshitz (Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lipschütz), Lipshutz, Lüpschütz; ...

, 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 from $p$ to $q$ such that $d_g\left(c\left(s\right),c\left(t\right)\right)=, s-t,$ for all 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 This is a glossary of some terms used in the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and c ...
$\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 Hopf–Rinow theorem is a set of statements about the Geodesic manifold, geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931. Statement Let (''M'', ''g'') be a conne ...
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.

Infinite-Dimensional Manifolds

The statements and theorems above are for finite-dimensional manifolds—manifolds whose charts map to open subsets of $\R^n.$ These can be extended, to a certain degree, to infinite-dimensional manifolds; that is, manifolds that are modeled after a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an Abstra ...
; 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 $M$ is a smooth function $g : TM \times TM \to \R,$ such that for any $x \in M$ the restriction $g_x : T_xM \times T_xM \to \R$ is an inner product on $T_xM.$ * A strong Riemannian metric on $M$ is a weak Riemannian metric, such that $g_x$ induces the topology on $T_xM.$ Note that if $M$ is not a Hilbert manifold then $g$ cannot be a strong metric.

Examples

* If $\left(H, \langle \,\cdot, \cdot\, \rangle\right)$ is a
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, then for any $x \in H,$ one can identify $H$ with $T_xH.$ By setting for all $x, u, v \in H$ $g_x\left(u,v\right) = \langle u, v \rangle$ one obtains a strong Riemannian metric. * Let $\left(M, g\right)$ be a compact Riemannian manifold and denote by $\operatorname\left(M\right)$ its diffeomorphism group. It is a smooth manifold ( see here) and in fact, a
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. Its tangent bundle at the identity is the set of smooth vector fields on $M.$ Let $\mu$ be a
volume form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
on $M.$ Then one can define $G,$ the $L^2$ weak Riemannian metric, on $\operatorname\left(M\right).$ Let $f\in \operatorname\left(M\right),$ $u, v \in T_f\operatorname\left(M\right).$ Then for $x \in M, u\left(x\right) \in T_M$ and define $G_f\left(u,v\right) = \int _M g_ \left(u\left(x\right),v\left(x\right)\right) d\mu \left(x\right).$ The $L^2$ weak Riemannian metric on $\operatorname\left(M\right)$ induces vanishing geodesic distance, see Michor and Mumford (2005).

The metric space structure

Length of curves is defined in a way similar to the finite-dimensional case. The function $d_g : M \times M \to$
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. If $g$ is a weak Riemannian metric, then no notion of completeness implies the other in general.