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| {{otheruses4|a concept from differential geometry|the algebraic concept|Zariski–Riemann space}}
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| {{distinguish2|[[Riemann surface]]}}
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| In [[differential geometry]], a '''(smooth) Riemannian manifold''' or '''(smooth) Riemannian space''' (''M'',''g'') is a real [[smooth manifold]] ''M'' equipped with an [[Inner product space|inner product]] <math> g_p </math> on the [[tangent space]] <math> T_pM </math> at each point <math>p</math>
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| that varies smoothly from point to point in the sense that if ''X'' and ''Y'' are [[vector fields]] on ''M'', then
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| <math> p \mapsto g_p(X(p),Y(p))</math> is a [[smooth function]].
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| The family <math> g_p </math> of inner products is called a '''Riemannian metric (tensor)'''.
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| These terms are named after the German mathematician [[Bernhard Riemann]].
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| The study of Riemannian manifolds comprises the subject called [[Riemannian geometry]].
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| A Riemannian metric (tensor) makes it possible to define various geometric notions on a Riemannian manifold, such as [[angle]]s, lengths of [[curve]]s, [[area]]s (or [[volume]]s), [[curvature]], [[gradient]]s of functions and [[divergence]] of [[vector field]]s.
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| ==Introduction==
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| 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 [[Gaussian curvature|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 was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. [[Albert Einstein]] 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.
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| == Overview ==
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| 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 function|smooth curve]] α(''t''): [0, 1] → ''M'' has tangent vector α′(''t''<sub>0</sub>) in the tangent space T''M''(α(''t''<sub>0</sub>)) at any point ''t''<sub>0</sub> ∈ (0, 1), and each such vector has length ‖α′(''t''<sub>0</sub>)‖, where ‖·‖ denotes the [[norm (mathematics)|norm]] induced by the inner product on T''M''(α(''t''<sub>0</sub>)). The [[integral]] of these lengths gives the length of the curve α:
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| :<math>L(\alpha) = \int_0^1{\|\alpha'(t)\|\, \mathrm{d}t}.</math>
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| Smoothness of α(''t'') for ''t'' in [0, 1] guarantees that the integral ''L''(α) exists and the length of this curve is defined.
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| In many instances, in order to pass from a linear-algebraic concept to a differential-geometric one, the smoothness requirement is very important.
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| Every smooth [[submanifold]] of '''R'''<sup>''n''</sup> has an induced Riemannian metric ''g'': the [[inner product]] on each tangent space is the restriction of the inner product on '''R'''<sup>''n''</sup>. In fact, as follows from the [[Nash embedding theorem]], all Riemannian manifolds can be realized this way.
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| In particular one could ''define'' Riemannian manifold as a [[metric space]] which is [[Isometry|isometric]] to a smooth submanifold of '''R'''<sup>''n''</sup> 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]].
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| === Riemannian manifolds as metric spaces ===
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| Usually a Riemannian manifold is defined as a smooth manifold with a smooth [[Section (fiber bundle)|section]] of the [[positive-definite]] [[quadratic form]]s on the [[tangent bundle]]. Then one has to work to show that it can be turned to a [[metric space]]:
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| If γ: [''a'', ''b''] → ''M'' is a continuously differentiable [[curve]] in the Riemannian manifold ''M'', then we define its length ''L''(γ) in analogy with the example above by
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| :<math>L(\gamma) = \int_a^b \|\gamma'(t)\|\, \mathrm{d}t.</math>
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| With this definition of length, every [[connected space|connected]] Riemannian manifold ''M'' becomes a [[metric space]] (and even a [[intrinsic metric|length metric space]]) in a natural fashion: the distance ''d''(''x'', ''y'') between the points ''x'' and ''y'' of ''M'' is defined as
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| :''d''(''x'',''y'') = [[infimum|inf]]{ L(γ) : γ is a continuously differentiable curve joining ''x'' and ''y''}.
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| Even though Riemannian manifolds are usually "curved," there is still a notion of "straight line" on them: the [[geodesic]]s. These are curves which locally join their points along [[Geodesic|shortest path]]s.
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| Assuming the manifold is [[compact set|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]] '''R'''<sup>2</sup> \ {0}, the distance between the points (−1, 0) and (1, 0) is 2, but there is no geodesic realizing this distance.
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| === Properties ===
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| In Riemannian manifolds, the notions of [[geodesic]] [[complete space|completeness]], [[topological space|topological]] completeness and [[metric space|metric]] completeness are the same: that each implies the other is the content of the [[Hopf–Rinow theorem]].
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| == Riemannian metrics ==
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| Let ''M'' be a [[differentiable manifold]] of dimension ''n''. A '''Riemannian metric''' on ''M'' is a family of ([[definite bilinear form|positive definite]]) [[inner product]]s
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| :<math>g_p \colon T_pM\times T_pM\longrightarrow \mathbf R,\qquad p\in M</math>
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| such that, for all differentiable [[vector fields]] ''X'',''Y'' on ''M'',
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| :<math> p\mapsto g_p(X(p), Y(p))</math>
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| defines a [[smooth function]] ''M'' → '''R'''.
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| In other words, a Riemannian metric ''g'' is a symmetric (0,2)-tensor that is positive definite (i.e. ''g''(''X'', ''X'') > 0 for all tangent vectors ''X'' ≠ 0).
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| In a system of [[local coordinates]] on the manifold ''M'' given by ''n'' real-valued functions ''x''<sup>1</sup>,''x''<sup>2</sup>, …, ''x''<sup>''n''</sup>, the vector fields
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| :<math>\left\{\frac{\partial}{\partial x^1},\dotsc, \frac{\partial}{\partial x^n}\right\}</math>
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| give a [[basis of a vector space|basis]] of [[Tangent space|tangent vector]]s at each point of ''M''. Relative to this coordinate system, the components of the metric tensor are, at each point ''p'',
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| :<math>g_{ij}(p):=g_p\Biggl(\left(\frac{\partial }{\partial x^i}\right)_p,\left(\frac{\partial }{\partial x^j}\right)_p\Biggr).</math>
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| Equivalently, the [[metric tensor]] can be written in terms of the [[dual basis]] {d''x''<sup>1</sup>, …, d''x''<sup>''n''</sup>} of the cotangent bundle as
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| :<math> g=\sum_{i,j}g_{ij}\mathrm d x^i\otimes \mathrm d x^j.</math>
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|
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| Endowed with this metric, the [[differentiable manifold]] (''M'', ''g'') is a '''Riemannian manifold'''.
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| === Examples ===
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| * With <math>\frac{\partial }{\partial x^i}</math> identified with ''e<sub>i</sub>'' = (0, …, 1, …, 0), the standard metric over an [[open set|open subset]] ''U'' ⊂ '''R'''<sup>''n''</sup> is defined by
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| ::<math>g^{\mathrm{can}}_p \colon T_pU\times T_pU\longrightarrow \mathbf R,\qquad \left(\sum_ia_i\frac{\partial}{\partial x^i},\sum_jb_j\frac{\partial}{\partial x^j}\right)\longmapsto \sum_i a_ib_i.</math>
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| :Then ''g'' is a Riemannian metric, and
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| ::<math>g^{\mathrm{can}}_{ij}=\langle e_i,e_j\rangle = \delta_{ij}.</math>
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| :Equipped with this metric, '''R'''<sup>''n''</sup> is called '''[[Euclidean space]]''' of dimension ''n'' and ''g''<sub>ij</sub><sup>can</sup> is called the (canonical) '''[[Euclidean metric]]'''.
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| * Let (''M'',''g'') be a Riemannian manifold and ''N'' ⊂ ''M'' be a [[submanifold]] of ''M''. Then the restriction of ''g'' to vectors tangent along ''N'' defines a Riemannian metric over ''N''.
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| * More generally, let ''f'': ''M''<sup>''n''</sup>→''N''<sup>''n''+''k''</sup> be an [[immersion (mathematics)|immersion]]. Then, if ''N'' has a Riemannian metric, ''f'' induces a Riemannian metric on ''M'' via [[pullback (differential geometry)|pullback]]:
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| ::<math>g^M_p \colon T_pM\times T_pM\longrightarrow \mathbf R,</math>
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| ::<math>(u,v)\longmapsto g^M_p(u,v):=g^N_{f(p)}(T_pf(u), T_pf(v)).</math>
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| :This is then a metric; the positive definiteness follows on the injectivity of the differential of an immersion.
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| * Let (''M'', ''g''<sup>''M''</sup>) be a Riemannian manifold, ''h'':''M''<sup>''n''+''k''</sup>→''N''<sup>''k''</sup> be a differentiable map and ''q''∈''N'' be a [[regular value]] of ''h'' (the [[pushforward (differential)|differential]] ''dh''(''p'') is surjective for all ''p''∈''h''<sup>−1</sup>(''q'')). Then ''h''<sup>−1</sup>(''q'')⊂''M'' is a submanifold of ''M'' of dimension ''n''. Thus ''h''<sup>−1</sup>(''q'') carries the Riemannian metric induced by inclusion.
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| * In particular, consider the following map :
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| ::<math>h\colon \mathbf R^n\longrightarrow \mathbf R,\qquad (x^1, \dotsc, x^n)\longmapsto \sum_{i=1}^n(x^i)^2-1.</math>
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| :Then, ''0'' is a regular value of ''h'' and
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| ::<math>h^{-1}(0)= \left \{x\in\mathbf R^n\vert \sum_{i=1}^n(x^i)^2=1 \right \}= \mathbf{S}^{n-1}</math>
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| :is the unit sphere '''S'''<sup>''n'' − 1</sup> ⊂ '''R'''<sup>''n''</sup>. The metric induced from '''R'''<sup>''n''</sup> on '''S'''<sup>''n'' − 1</sup> is called the '''canonical metric''' of '''S'''<sup>''n'' − 1</sup>.
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| * Let ''M''<sub>1</sub> and ''M''<sub>2</sub> be two Riemannian manifolds and consider the cartesian product ''M''<sub>1</sub> × ''M''<sub>2</sub> with the product structure. Furthermore, let π<sub>1</sub>: ''M''<sub>1</sub> × ''M''<sub>2</sub> → ''M''<sub>1</sub> and π<sub>2</sub>: ''M''<sub>1</sub> × ''M''<sub>2</sub> → ''M''<sub>2</sub> be the natural projections. For (''p,q'') ∈ ''M''<sub>1</sub> × ''M''<sub>2</sub>, a Riemannian metric on ''M''<sub>1</sub> × ''M''<sub>2</sub> can be introduced as follows :
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| ::<math>g^{M_1\times M_2}_{(p,q)}\colon T_{(p,q)}(M_1\times M_2)\times T_{(p,q)}(M_1\times M_2) \longrightarrow \mathbf R,</math>
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| ::<math>(u,v)\longmapsto g^{M_1}_p(T_{(p,q)}\pi_1(u), T_{(p,q)}\pi_1(v))+g^{M_2}_q(T_{(p,q)}\pi_2(u), T_{(p,q)}\pi_2(v)).</math>
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| :The identification
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| ::<math>T_{(p,q)}(M_1\times M_2) \cong T_pM_1\oplus T_qM_2</math>
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| :allows us to conclude that this defines a metric on the product space.
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| :The torus '''S'''<sup>1</sup> × … × '''S'''<sup>1</sup> = '''T'''<sup>''n''</sup> possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from '''R'''<sup>2</sup> on the circle '''S'''<sup>1</sup> ⊂ '''R'''<sup>2</sup> and then taking the product metric. The torus '''T'''<sup>''n''</sup> endowed with this metric is called the [[flat torus]].
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| * Let ''g''<sub>0</sub>, ''g''<sub>1</sub> be two metrics on ''M''. Then,
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| ::<math>\tilde g:=\lambda g_0 + (1-\lambda)g_1,\qquad \lambda\in [0,1],</math>
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| :is also a metric on ''M''.
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| === The pullback metric ===
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| If ''f'':''M''→''N'' is a differentiable map and (''N'',''g<sup>N</sup>'') a Riemannian manifold, then the [[pullback (differential geometry)|pullback]] of ''g''<sup>N</sup> along ''f'' is a quadratic form on the tangent space of ''M''. The pullback is the quadratic form ''f''*''g<sup>N</sup>'' on ''TM'' defined for ''v'', ''w'' ∈ ''T''<sub>p</sub>''M'' by
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| :<math> (f^*g^N)(v,w) = g^N(df(v),df(w))\,.</math>
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| where ''df(v)'' is the [[pushforward (differential)|pushforward]] of ''v'' by ''f''.
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| The quadratic form ''f''*''g<sup>N</sup>'' is in general only a semi definite form because ''df'' can have a kernel. If ''f'' is a [[diffeomorphism]], or more generally an [[immersion (mathematics)|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 [[covering space]] inherits a metric from covering a Riemannian manifold.
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| === Existence of a metric ===
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| Every [[paracompact]] differentiable manifold admits a Riemannian metric. To prove this result, let ''M'' be a manifold and {(''U''<sub>α</sub>, φ(''U''<sub>α</sub>))|α ∈ ''I''} a [[locally finite collection|locally finite]] [[atlas (topology)|atlas]] of open subsets ''U'' of ''M'' and diffeomorphisms onto open subsets of '''R'''<sup>''n''</sup>
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| :<math>\phi \colon U_\alpha\to \phi(U_\alpha)\subseteq\mathbf{R}^n.</math>
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| Let τ<sub>α</sub> be a differentiable [[partition of unity]] subordinate to the given atlas. Then define the metric ''g'' on ''M'' by
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| :<math>g:=\sum_\beta\tau_\beta\cdot\tilde{g}_\beta,\qquad\text{with}\qquad\tilde{g}_\beta:=\tilde{\phi}_\beta^*g^{\mathrm{can}}.</math>
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| where ''g''<sup>can</sup> is the Euclidean metric. This is readily seen to be a metric on ''M''.
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| === Isometries ===
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| Let (''M'', ''g<sup>M</sup>'') and (''N'', ''g<sup>N</sup>'') be two Riemannian manifolds, and ''f'': ''M'' → ''N'' be a diffeomorphism. Then, ''f'' is called an '''isometry''', if
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| :<math> g^M = f^* g^N\,,</math>
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| or pointwise
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| :<math>g^M_p(u,v) = g^N_{f(p)}(df(u), df(v))\qquad \forall p\in M, \forall u,v\in T_pM.</math>
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| Moreover, a differentiable mapping ''f'': ''M'' → ''N'' is called a '''local isometry''' at ''p'' ∈ ''M'' if there is a neighbourhood ''U'' ⊂ ''M'', ''p'' ∈ ''U'', such that ''f'': ''U'' → ''f(U)'' is a diffeomorphism satisfying the previous relation.
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| == Riemannian manifolds as metric spaces ==
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| A [[connected space|connected]] Riemannian manifold carries the structure of a [[metric space]] whose distance function is the arclength of a minimizing [[geodesic]].
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| Specifically, let (''M'',''g'') be a connected Riemannian manifold. Let ''c'': [''a,b''] → ''M'' be a parametrized curve in ''M'', which is differentiable with velocity vector ''c''′. The length of ''c'' is defined as
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| :<math>L_a^b(c) := \int_a^b \sqrt{g(c'(t),c'(t))}\,\mathrm d t = \int_a^b\|c'(t)\|\,\mathrm d t.</math>
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| By [[change of variables]], the arclength is independent of the chosen parametrization. In particular, a curve [''a,b''] → ''M'' can be parametrized by its arc length. A curve is parametrized by arclength if and only if <math>\|c'(t)\|=1</math> for all <math>t\in[a,b]</math>.
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| The distance function ''d'' : ''M''×''M'' → [0,∞) is defined by
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| :<math> d(p,q) = \inf L(\gamma)</math>
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| where the [[infimum]] extends over all differentiable curves γ 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 [[normal coordinates|normal coordinate system]], which also allows one to show that the topology induced by ''d'' is the same as the original topology on ''M''.
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| === Diameter ===
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| The '''diameter''' of a Riemannian manifold ''M'' is defined by
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| :<math>\mathrm{diam}(M):=\sup_{p,q\in M} d(p,q)\in \mathbf R_{\geq 0}\cup\{+\infty\}.</math>
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| The diameter is invariant under global isometries. Furthermore, the [[Heine–Borel theorem|Heine–Borel property]] holds for (finite-dimensional) Riemannian manifolds: ''M'' is [[compact space|compact]] if and only if it is [[complete metric space|complete]] and has finite diameter.
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| === Geodesic completeness ===
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| A Riemannian manifold ''M'' is '''geodesically complete''' if for all ''p'' ∈ ''M'', the [[Exponential_map#Riemannian_geometry|exponential map]] <math>\exp_p</math> is defined for all <math>v\in T_pM</math>, i.e. if any geodesic <math>\gamma(t)</math> starting from ''p'' is defined for all values of the parameter ''t'' ∈ '''R'''. The [[Hopf-Rinow theorem]] asserts that ''M'' is geodesically complete if and only if it is [[complete metric space|complete as a metric space]].
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| 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.
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| == See also ==
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| * [[Riemannian geometry]]
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| * [[Finsler manifold]]
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| * [[sub-Riemannian manifold]]
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| * [[pseudo-Riemannian manifold]]
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| * [[Metric tensor]]
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| * [[Hermitian manifold]]
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| * [[Space (mathematics)]]
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| == References ==
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| * {{Citation | last1=Jost | first1=Jürgen | title=Riemannian Geometry and Geometric Analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=5th | isbn=978-3-540-77340-5 | year=2008}}
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| * {{Citation | last1=do Carmo | first1=Manfredo | title=Riemannian geometry | publisher=Birkhäuser | location=Basel, Boston, Berlin | isbn=978-0-8176-3490-2 | year=1992}} [http://www.amazon.fr/Riemannian-Geometry-Manfredo-P-Carmo/dp/0817634908/ref=sr_1_1?ie=UTF8&s=english-books&qid=1201537059&sr=8-1]
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| ==External links==
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| *{{springer|id=R/r082180|title=Riemannian metric|author=L.A. Sidorov}}
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| [[Category:Riemannian manifolds|*]]
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