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{{about|a concept from differential geometry|the algebraic concept|Zariski–Riemann space}}
{{distinguish|Riemann surface}}

In [[differential geometry]], a ('''smooth''') '''Riemannian [[manifold]]''' or ('''smooth''') '''Riemannian space''' {{nowrap|(''M'', ''g'')}} is a real, [[smooth manifold]] ''M'' equipped with an [[Inner product space|inner product]] ''g''<sub>''p''</sub> on the [[tangent space]] ''T''<sub>''p''</sub>''M'' at each point ''p'' that varies smoothly from point to point in the sense that if ''X'' and ''Y'' are differentiable [[vector fields]] on ''M'', then {{nowrap|''p'' ↦ ''g''<sub>''p''</sub>(''X''{{!}}<sub>''p''</sub>, ''Y''{{!}}<sub>''p''</sub>)}} is a [[smooth function]]. The family ''g''<sub>''p''</sub> of inner products is called a [[metric tensor|Riemannian metric (or Riemannian metric tensor)]]. These terms are [[list of things named after Bernhard Riemann|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 [[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 (formally pseudo-Riemannian manifolds) to develop his [[general theory of relativity]]. In particular, his equations for gravitation are [[Constraint (mathematics)|constraint]]s on the curvature of space-time.

== Overview ==
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 space|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]] {{nowrap|''α'' : [0, 1] → ''M''}} has tangent vector {{nowrap|''α''&prime;(''t''<sub>0</sub>)}} in the tangent space T''M''(''α''(''t''<sub>0</sub>)) at any point {{nowrap|''t''<sub>0</sub> ∈ (0, 1)}}, and each such vector has length ‖''α''&prime;(''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'':

:<math>L(\alpha) = \int_0^1 \|\alpha'(t)\|\, \mathrm{d}t.</math>

Smoothness of ''α''(''t'') for ''t'' in {{nowrap|[0, 1]}} guarantees that the integral ''L''(''α'') exists and the length of this curve is defined.

In many instances, in order to pass from a linear-algebraic concept to a differential-geometric one, the smoothness requirement is very important.

Every smooth [[submanifold]] of '''R'''<sup>''n''</sup> with a Euclidean metric has an induced Riemannian metric ''g'': the [[inner product]] on each tangent space is the [[Restriction (mathematics)|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.
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]].

=== Riemannian manifolds as metric spaces ===
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]]. This definition allows the construction of an accompanying [[metric space]]:

If {{nowrap|''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

:<math>L(\gamma) = \int_a^b \| \gamma ' (t) \| \, \mathrm{d}t.</math>

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 {{nowrap|''d''(''x'', ''y'')}} between the points ''x'' and ''y'' of ''M'' is defined as

:<math>d(x, y) = \inf \{ L(\gamma) : \gamma\text{ a continuously differentiable curve joining }x\text{ and }y \} .</math>

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.

Assuming the manifold is [[complete metric space|complete]], any two points ''x'' and ''y'' can be connected with a geodesic whose length is {{nowrap|''d''(''x'', ''y'')}}.  Without completeness, this need not be true. For example, in the [[punctured plane]] {{nowrap|'''R'''<sup>2</sup> ∖ {0},}} the distance between the points {{nowrap|(&minus;1, 0)}} and {{nowrap|(1, 0)}} is 2, but there is no geodesic realizing this distance.

=== Properties ===
In Riemannian manifolds, the notions of [[geodesic]] [[complete space|completeness]] and [[metric space]] completeness are the same: that each implies the other is the content of the [[Hopf–Rinow theorem]].

== Riemannian metrics ==
{{main|Metric tensor}}
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
:<math>g_p \colon T_pM\times T_pM\rightarrow \mathbf R,\qquad p\in M</math>

such that, for every pair of differentiable [[vector fields]] ''X'', ''Y'' on ''M'',
:<math> p\mapsto g_p(X|_p, Y|_p)</math>

defines a [[smooth function]] {{nowrap|''M'' → '''R'''}}.

We can also regard a Riemannian metric ''g'' as a symmetric {{nowrap|(0, 2)}}-tensor field that is positive-definite at every point (i.e. {{nowrap|1=''g''(''X'', ''X''){{!}}<sub>''p''</sub> := ''g''<sub>''p''</sub>(''X''{{!}}<sub>''p''</sub>, ''X''{{!}}<sub>''p''</sub>) > 0}} whenever {{nowrap|''X''{{!}}<sub>''p''</sub> ≠ 0}}).

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
:<math>\left\{\frac{\partial}{\partial x^1},\dotsc, \frac{\partial}{\partial x^n}\right\}</math>

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'',
:<math>g_{ij}|_p:=g_p\left(\left.\frac{\partial }{\partial x^i}\right|_p,\left.\frac{\partial }{\partial x^j}\right|_p\right).</math>

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
:<math> g=\sum_{i,j}g_{ij} \, \mathrm d x^i\otimes \mathrm d x^j.</math>
 
The [[differentiable manifold]] ''M'' endowed with this metric ''g'' is a '''Riemannian manifold''', denoted {{nowrap|(''M'', ''g'')}}.

=== Examples ===
* With <math>\frac{\partial }{\partial x^i}</math> identified with {{nowrap|1=''e<sub>i</sub>'' = (0, ..., 1, ..., 0)}}, the standard metric over an [[open set|open subset]] {{nowrap|''U'' ⊂ '''R'''<sup>''n''</sup>}} is defined by

::<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>

:Then ''g'' is a Riemannian metric, and
::<math>g^{\mathrm{can}}_{ij}=\langle e_i,e_j\rangle = \delta_{ij}.</math>

: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]]'''.
* Let {{nowrap|(''M'', ''g'')}} be a Riemannian manifold and {{nowrap|''N'' ⊂ ''M''}} be a [[submanifold]] of ''M''. Then the [[Restriction (mathematics)|restriction]] of ''g'' to vectors tangent along ''N'' defines a Riemannian metric over ''N''.
* More generally, {{nowrap|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]]:

::<math>g^M_p \colon T_pM\times T_pM\longrightarrow \mathbf R,</math>
::<math>(u,v)\longmapsto g^M_p(u,v):=g^N_{f(p)}(df_p(u), df_p(v)).</math>

:This is then a metric; the positive definiteness follows from the injectivity of the differential of an immersion.
* Let {{nowrap|(''M'', ''g''<sup>''M''</sup>)}} be a Riemannian manifold, {{nowrap|''h'' : ''M''<sup>''n''+''k''</sup> → ''N''<sup>''k''</sup>}} be a differentiable map and {{nowrap|''q'' ∈ ''N''}} be a [[regular value]] of ''h'' (the [[pushforward (differential)|differential]] ''dh''(''p'') is surjective for all {{nowrap|''p'' ∈ ''h''<sup>&minus;1</sup>(''q'')}}). Then {{nowrap|''h''<sup>&minus;1</sup>(''q'') ⊂ ''M''}} is a submanifold of ''M'' of dimension ''n''. Thus ''h''<sup>&minus;1</sup>(''q'') carries the Riemannian metric induced by inclusion.
* In particular, consider the following map :
::<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>

:Then 0 is a regular value of ''h'' and 
::<math>h^{-1}(0)= \left\{ x\in\mathbf R^n \,\left\vert\, \sum_{i=1}^n(x^i)^2=1 \right. \right\}= \mathbf{S}^{n-1}</math>

:is the unit sphere {{nowrap|'''S'''<sup>''n''&minus;1</sup> ⊂ '''R'''<sup>''n''</sup>}}. The metric induced from '''R'''<sup>''n''</sup> on '''S'''<sup>''n''&minus;1</sup> is called the '''canonical metric''' of '''S'''<sup>''n''&minus;1</sup>.
* Let ''M''<sub>1</sub> and ''M''<sub>2</sub> be two Riemannian manifolds and consider the cartesian product {{nowrap|''M''<sub>1</sub> × ''M''<sub>2</sub>}} with the product structure.  Furthermore, let {{nowrap|{{pi}}<sub>1</sub> : ''M''<sub>1</sub> × ''M''<sub>2</sub> → ''M''<sub>1</sub>}} and {{nowrap|{{pi}}<sub>2</sub> : ''M''<sub>1</sub> × ''M''<sub>2</sub> → ''M''<sub>2</sub>}} be the natural projections. For {{nowrap|(''p'', ''q'') ∈ ''M''<sub>1</sub> × ''M''<sub>2</sub>}}, a Riemannian metric on {{nowrap|''M''<sub>1</sub> × ''M''<sub>2</sub>}} can be introduced as follows :

::<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>
::<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>

:The identification
::<math>T_{(p,q)}(M_1\times M_2) \cong T_pM_1\oplus T_qM_2</math>

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

:The torus {{nowrap|1='''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 {{nowrap|'''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]].
* Let ''g''<sub>0</sub>, ''g''<sub>1</sub> be two metrics on ''M''. Then,
::<math>\tilde g:=\lambda g_0 + (1-\lambda)g_1,\qquad \lambda\in [0,1],</math>

:is also a metric on ''M''.

=== The induced (pullback) metric ===
If {{nowrap|''f'' : ''M'' → ''N''}} is a differentiable map and {{nowrap|(''N'', ''g<sup>N</sup>'')}} a Riemannian manifold, then the [[pullback (differential geometry)|pullback]] of ''g''<sup>''N''</sup> along ''f'' is the [[Induced_metric|induced metric]] on the manifold ''M'', in other words it is a bilinear form on the tangent bundle of ''M''.  The pullback is the form ''f''<sup>∗</sup>''g<sup>N</sup>'' on ''TM'' defined for {{nowrap|''v'', ''w'' ∈ ''T''<sub>''p''</sub>''M''}} by
:<math> (f^*g^N)(v,w) = g^N(df(v),df(w))\,,</math>

where ''df''(''v'') is the [[pushforward (differential)|pushforward]] of ''v'' by ''f''.

The form ''f''<sup>∗</sup>''g<sup>N</sup>'' is in general only semidefinite, because ''df'' can have a kernel. If ''f'' is a [[diffeomorphism]], or more generally an [[immersion (mathematics)|immersion]], then ''f''<sup>∗</sup>''g<sup>N</sup>'' is a Riemannian metric on ''M'' called 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.

=== Existence of a metric ===
Every [[paracompact]] differentiable manifold admits a Riemannian metric.  

{{Math proof|title=Proof|proof=
Let ''M'' be a differentiable manifold and {{nowrap|{(''U''<sub>''α''</sub>, ''φ''<sub>''α''</sub>) {{!}} ''α'' ∈ ''I''} }} a [[locally finite collection|locally finite]] [[atlas (topology)|atlas]] of open subsets ''U<sub>α</sub>'' of ''M'' and diffeomorphisms onto open subsets of '''R'''<sup>''n''</sup>
:<math>\varphi_\alpha \colon U_\alpha\to \varphi_\alpha(U_\alpha)\subseteq\mathbf{R}^n.</math>

Let {''τ<sub>α</sub>''}<sub>''α''∈''I''</sub> be a differentiable [[partition of unity]] <u>subordinate to</u> the given atlas.  

Then define the metric ''g'' on ''M'' by
:<math>g:=\sum_{\beta \in I}\tau_\beta\cdot\tilde{g}_\beta,\qquad\text{with}\qquad\tilde{g}_\beta:=\varphi_\beta^*g^{\mathrm{can}} \,\,\text{on}\,\, U_\alpha,</math>

where ''g''<sup>can</sup> is the Euclidean metric on '''R'''<sup>''n''</sup> and <math>\varphi_\beta^*g^{\mathrm{can}}</math> is its pullback along ''φ''<sub>''β''</sub>.  

This is readily seen to be a metric on ''M''.}}

=== Isometries ===
Let {{nowrap|(''M'', ''g<sup>M</sup>'')}} and {{nowrap|(''N'', ''g<sup>N</sup>'')}} be two Riemannian manifolds, and {{nowrap|''f'' : ''M'' → ''N''}} be a diffeomorphism. Then, ''f'' is called an '''isometry''', if
:<math> g^M = f^* g^N\,,</math>

or pointwise
:<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>

Moreover, a differentiable mapping {{nowrap|''f'' : ''M'' → ''N''}} is called a '''local isometry''' at {{nowrap|''p'' ∈ ''M''}} if there is a neighbourhood {{nowrap|''U'' ⊂ ''M''}}, {{nowrap|''p'' ∈ ''U''}}, such that {{nowrap|''f'' : ''U'' → ''f''(''U'')}} is a diffeomorphism satisfying the previous relation.

== Riemannian manifolds as metric spaces ==
A [[connected space|connected]] Riemannian manifold carries the structure of a [[metric space]] whose distance function is the arc length of a minimizing [[geodesic]]. Moreover, this metric space's natural topology agrees with the manifold's topology.<ref>{{cite book |first=John M. |last=Lee |title=Introduction to Smooth Manifolds |edition=2nd |location=New York |publisher=Springer |year=2013 |at=Theorem 13.29 |isbn=978-1-4419-9981-8 }}</ref>

Specifically, let {{nowrap|(''M'', ''g'')}} be a connected Riemannian manifold. Let {{nowrap|''c'' : [''a'', ''b''] → ''M''}} be a parametrized curve in ''M'', which is differentiable with velocity vector ''c''&prime;.  The length of ''c'' is defined as
:<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>

By [[change of variables]], the arclength is independent of the chosen parametrization.  In particular, a curve {{nowrap|[''a'', ''b''] → ''M''}} can be parametrized by its arc length.  A curve is parametrized by arclength if and only if {{nowrap|1=‖''c''′(''t'')‖ = 1}} for all {{nowrap|''t'' ∈ [''a'', ''b'']}}.

The distance function {{nowrap|''d'' : ''M'' × ''M'' → [0, ∞)}} is defined by
:<math> d(p,q) = \inf L(\gamma)</math>
where the [[infimum]] extends over all differentiable curves ''γ'' beginning at {{nowrap|''p'' ∈ ''M''}} and ending at {{nowrap|''q'' ∈ ''M''}}.

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 {{nowrap|1=''d''(''p'', ''q'') = 0}} implies that {{nowrap|1=''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''.

=== Diameter ===
The '''diameter''' of a Riemannian manifold ''M'' is defined by

:<math>\operatorname{diam}(M):=\sup_{p,q\in M} d(p,q)\in \mathbf R_{\geq 0}\cup\{+\infty\}.</math>

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.

=== Geodesic completeness ===
A Riemannian manifold ''M'' is '''geodesically complete''' if for all {{nowrap|''p'' ∈ ''M''}}, the [[Exponential map (Riemannian geometry)|exponential map]] exp<sub>''p''</sub> is defined for all {{nowrap|v ∈ ''T''<sub>''p''</sub>''M''}}, i.e. if any geodesic ''γ''(''t'') starting from ''p'' is defined for all values of the parameter {{nowrap|''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]].

If ''M'' is complete, then ''M'' is non-extendable in the sense that it is not isometric to an open proper submanifold of any other Riemannian manifold.  The converse is not true, however: there exist non-extendable manifolds which are not complete.

== See also ==
{{Div col|colwidth=25em}}
* [[Riemannian geometry]]
* [[Finsler manifold]]
* [[Sub-Riemannian manifold]]
* [[Pseudo-Riemannian manifold]]
* [[Metric tensor]]
* [[Hermitian manifold]]
* [[Space (mathematics)]]
* [[Wave maps equation]]
{{Div col end}}

== References ==
{{reflist}}

== Further reading ==
* {{Cite book | last1=Jost | first1=Jürgen | title=Riemannian Geometry and Geometric Analysis | publisher=[[Springer-Verlag]] | location=Berlin | edition=5th | isbn=978-3-540-77340-5 | year=2008}}
* {{Cite book | last1=do Carmo | first1=Manfredo |authorlink=Manfredo do Carmo | title=Riemannian geometry | publisher=Birkhäuser | location=Basel | isbn=978-0-8176-3490-2 | year=1992}}

==External links==
*{{springer|id=R/r082180|title=Riemannian metric|author=L.A. Sidorov}}

{{Authority control}}

[[Category:Riemannian manifolds|*]]