- Riemannian manifold
In

Riemannian geometry , a**Riemannian manifold**("M","g") (with**Riemannian metric**"g") is a realdifferentiable manifold "M" in which eachtangent space is equipped with an inner product "g" in a manner which varies smoothly from point to point. The metric "g" is apositive definite metric tensor . This allows one to define various notions such asangle s, lengths ofcurve s,area s (orvolume s),curvature ,gradient s of functions anddivergence ofvector field s. In other words, a Riemannian manifold is a differentiable manifold in which the tangent space at each point is a finite-dimensionalHilbert space . The terms are named after German mathematicianBernhard Riemann .**Overview**The

tangent bundle of asmooth manifold "M" assigns to each fixed point of "M" a vector space called thetangent 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 curve α("t"): [0, 1] → "M" has tangent vector α′("t"_{0}) in the tangent space T"M"("t"_{0}) at any point "t"_{0}∈ (0, 1), and each such vector has length ||α′("t"_{0})||, where ||·|| denotes the norm induced by the inner product on T"M"("t"_{0}). Theintegral of these lengths gives the length of the curve α::$L(alpha)\; =\; int\_0^1\{|alpha^\{prime\}(t)|,\; mathrm\{d\}t\}.$

Smoothness of ||α′("t")|| for "t" in [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**^{"n"}has an induced Riemannian metric "g": theinner product on each tangent space is the restriction of the inner product on**R**^{"n"}. In fact, as follows from theNash embedding theorem , all Riemannian manifolds can be realized this way.In particular one could "define" Riemannian manifold as ametric space which is isometric to a smooth submanifold of**R**^{"n"}with the inducedintrinsic 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 inRiemannian geometry .**Riemannian manifolds as metric spaces**Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of the positive-definite quadratic forms on the

tangent bundle . Then one has to work to show that it can be turned to a metric space: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:$L(gamma)\; =\; int\_a^b\; |gammaprime(t)|,\; mathrm\{d\}t.$

With this definition of length, every connected Riemannian manifold "M" becomes a

metric space (and even a length metric space) in a natural fashion: the distance "d"("x", "y") between the points "x" and "y" of "M" is defined as:"d"("x","y") = inf{ L(γ) : γ is a continuously differentiable curve joining "x" and "y"}.

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 shortest paths.Assuming the manifold is 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**^{2}{0}, the distance between the points (−1, 0) and (1, 0) is 2, but there is no geodesic realizing this distance.**Properties**In Riemannian manifolds, the notions of

geodesic completeness, topological completeness and metric completeness are the same: that each implies the other is the content of theHopf-Rinow theorem .**Riemannian metrics**Let "M" be a

second countable Hausdorffdifferentiable manifold of dimension "n". A**Riemannian metric**on "M" is a family of (positive definite )inner product s:$g\_p\; :\; T\_pM\; imes\; T\_pMlongrightarrow\; mathbb\; R,qquad\; pin\; M$

such that, for all differentiable

vector fields "X","Y" on "M",:$pmapsto\; g\_p(X(p),\; Y(p))$

defines a differentiable function "M" →

**R**. The assignment of an inner product "g"_{"p"}to each point "p" of the manifold is called a.metric tensor In a system of

local coordinates on the manifold "M" given by "n" real-valued functions "x"_{1},"x"_{2}, …, "x"_{"n"}, the vector fields:$left\{frac\{partial\}\{partial\; x\_1\},dots,\; frac\{partial\}\{partial\; x\_n\}\; ight\}$

give a basis of tangent vectors at each point of "M". Relative to this coordinate system, the components of the metric tensor are, at each point "p",

:$g\_\{ij\}(p):=g\_pBiggl(left(frac\{partial\; \}\{partial\; x\_i\}\; ight)\_p,left(frac\{partial\; \}\{partial\; x\_j\}\; ight)\_pBiggr).$

Equivalently, the metric tensor can be written in terms of the

dual basis {d"x"_{1}, …, d"x"_{"n"}} of thecotangent bundle as:$g=sum\_\{i,j\}g\_\{ij\}mathrm\; d\; x\_iotimes\; mathrm\; d\; x\_j.$ Endowed with this metric, the differentiable manifold ("M","g") is a

**Riemannian manifold**.**Examples*** With $frac\{partial\; \}\{partial\; x\_i\}$ identified with $e\_i=(0,dots,\; 1,dots,\; 0)$, the standard metric over an open subset $Usubsetmathbb\; R^n$ is defined by

::$g^\{mathrm\{can\_p\; :\; T\_pU\; imes\; T\_pUlongrightarrow\; mathbb\; R,qquad\; left(sum\_ia\_ifrac\{partial\}\{partial\; x\_i\},sum\_jb\_jfrac\{partial\}\{partial\; x\_j\}\; ight)longmapsto\; sum\_i\; a\_ib\_i.$

:Then "g" is a Riemannian metric, and

::$g^\{mathrm\{can\_\{ij\}=langle\; e\_i,e\_j\; angle\; =\; delta\_\{ij\}.$

:Equipped with this metric,

**R**^{n}is calledof dimension "n" and "g"Euclidean space _{ij}^{can}is called the.Euclidean metric

* Let ("M","g") be a Riemannian manifold and $Nsubset\; M$ be asubmanifold of "M". Then the restriction of "g" to vectors tangent along "N" defines a Riemannian metric over "N".

* More generally, let "f":"M"^{n}→"N"^{n+k}be an immersion. Then, if "N" has a Riemannian metric, "f" induces a Riemannian metric on "M" via pullback:::$g^M\_p\; :\; T\_pM\; imes\; T\_pMlongrightarrow\; mathbb\; R,qquad\; (u,v)longmapsto\; g^M\_p(u,v):=g^N\_\{f(p)\}(T\_pf(u),\; T\_pf(v)).$

:This is then a metric; the positive definiteness follows of the injectivity of the differential of an immersion.

* Let ("M","g"^{M}) be a Riemannian manifold, "h":"M"^{n+k}→"N"^{k}be a differentiable application and "q"∈"N" be aregular value of "h" (the differential "dh"("p") is surjective for all "p"∈"h"^{-1}("q")). Then "N"="h"^{-1}("q")⊂"M" is a submanifold of "M" of dimension "n". Thus "N" carries the Riemannian metric induced by inclusion.* In particular, consider the following application :

::$h:\; mathbb\; R^nlongrightarrow\; mathbb\; R,qquad\; (x\_1,\; dots,\; x\_n)longmapsto\; sum\_\{i=1\}^nx\_i^2-1.$

:Then, "0" is a regular value of "h" and

::$h^\{-1\}(0)=\{xinmathbb\; R^nvert\; sum\_\{i=1\}^nx\_i^2=1\}=S^\{n-1\}$

:is the unit sphere $S^\{n-1\}subset\; mathbb\; R^n$. The metric induced from $mathbb\; R^n$ on $S^\{n-1\}$ is called the

**canonical metric**of $S^\{n-1\}$.

* Let $M\_1$ and $M\_2$ be two Riemannian manifolds and consider the cartesian product $M\_1\; imes\; M\_2$ with the product structure. Furthermore, let $pi\_1:M\_1\; imes\; M\_2\; ightarrow\; M\_1$ and $pi\_2:M\_1\; imes\; M\_2\; ightarrow\; M\_2$ be the natural projections. For $(p,q)in\; M\_1\; imes\; M\_2$, a Riemannian metric on $M\_1\; imes\; M\_2$ can be introduced as follows :::$g^\{M\_1\; imes\; M\_2\}\_\{(p,q)\}:T\_\{(p,q)\}(M\_1\; imes\; M\_2)\; imes\; T\_\{(p,q)\}(M\_1\; imes\; M\_2)\; longrightarrow\; mathbb\; R,qquad\; (u,v)longmapsto\; g^\{M\_1\}\_p(T\_\{(p,q)\}pi\_1(u),\; T\_\{(p,q)\}pi\_1(v))+g^\{M\_2\}\_q(T\_\{(p,q)\}pi\_2(u),\; T\_\{(p,q)\}pi\_2(v)).$

:The identification

::$T\_\{(p,q)\}(M\_1\; imes\; M\_2)\; cong\; T\_pM\_1oplus\; T\_qM\_2$

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

:The torus $S^1\; imesdots\; imes\; S^1=T^n$ possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from $mathbb\; R^2$ on the circle $S^1subset\; mathbb\; R^2$ and then taking the product metric. The torus $T^n$ endowed with this metric is called the

flat torus .

* Let $g\_0,g\_1$ be two metrics on $M$. Then,::$ilde\; g:=lambda\; g\_0\; +\; (1-lambda)g\_1,qquad\; lambdain\; [0,1]\; ,$

:is also a metric on "M".

**The pullback metric**If "f":"M"→"N" is a diffeomorphism and ("N","g"

^{N}) be a Riemannian manifold, then the pullback of "g"^{N}along "f" is a Riemannian metric on "M". The pullback is the metric "f"*"g"^{N}on "M" defined for "v", "w" ∈ "T"_{p}"M" by:$(f^*g^N)(v,w)\; =\; g^N(df(v),df(w)),.$

**Existence of a metric**Every paracompact differentiable manifold admits a Riemannian metric. To prove this result, let "M" be a manifold and {("U"

_{α}, φ("U"_{α}))|α∈"I"} alocally finite atlas of open subsets "U" of "M" and diffeomorphisms onto open subsets of**R**^{n}:$phi\; :\; U\_alpha\; o\; phi(U\_alpha)subseteqmathbb\{R\}^n.$

Let τ

_{α}be a differentiablepartition of unity subordinate to the given atlas. Then define the metric "g" on "M" by:$g:=sum\_eta\; au\_etacdot\; ilde\{g\}\_eta,qquad\; ext\{with\}qquad\; ilde\{g\}\_eta:=\; ilde\{phi\}\_eta^*g^\{mathrm\{can.$

where "g"

^{can}is the Euclidean metric. This is readily seen to be a metric on "M".**Isometries**Let $(M,\; g^M)$ and $(N,\; g^N)$ be two Riemannian manifolds, and $f:M\; ightarrow\; N$ be a diffeomorphism. Then, "f" is called an

**isometry**, if:$g^M\_p(u,v)\; =\; g^N\_\{f(p)\}(T\_pf(u),\; T\_pf(v))qquad\; forall\; pin\; M,\; forall\; u,vin\; T\_pM.$

Moreover, a differentiable mapping $f:M\; ightarrow\; N$ is called a

**local isometry**at $pin\; M$ if there is a neighbourhood $Usubset\; M$, $U\; i\; p$, such that $f:U\; ightarrow\; f(U)$ is a diffeomorphism satisfying the previous relation.**Riemannian manifolds as metric spaces**A connected Riemannian manifold carries the structure of a

metric space whose distance function is the arclength of a minimizinggeodesic .Specifically, let ("M","g") be a connected Riemannian manifold. Let $c:\; [a,b]\; ightarrow\; M$ be a parametrized curve in "M", which is differentiable with velocity vector "c"′. The length of "c" is defined as:$L\_a^b(c)\; :=\; int\_a^b\; sqrt\{g(c\text{'}(t),c\text{'}(t))\},mathrm\; d\; t\; =\; int\_a^b|c\text{'}(t)|,mathrm\; d\; t$

By

change of variables , the arclength is independent of the chosen parametrization. In particular, a curve $[a,b]\; ightarrow\; M$ can be parametrized by its arc length. A curve is parametrized by arclength if and only if $|c\text{'}(t)|=1$ for all $tin\; [a,b]$.The distance function "d" : "M"×"M" → [0,∞) is defined by:$d(p,q)\; =\; inf\; L(gamma)$where the

infimum extends over all differentiable curves γ beginning at "p"∈"M" and ending at "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 "d"("p","q")=0 implies that "p"="q". For this property, one can use a 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:$mathrm\{diam\}(M):=sup\_\{p,qin\; M\}\; d(p,q)in\; mathbb\; R\_\{geq\; 0\}cup\{+infty\}.$

The diameter is invariant under global isometries. Furthermore, the Heine-Borel property holds for (finite-dimensional) Riemannian manifolds: "M" is compact if and only if it is complete and has finite diameter.

**Geodesic completeness**A Riemannian manifold "M

**is**geodesically complete"' if for all $pin\; M$, the exponential map $exp\_p$ is defined for all $vin\; T\_pM$, i.e. if any geodesic $gamma(t)$ starting from "p" is defined for all values of the parameter $tinmathbb\; R$. TheHopf-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 a proper submanifold of any other Riemannian manifold. The converse is not true, however: there exist non-extendable manifolds which are not complete.

**See also***

Riemannian geometry

*Finsler manifold

*sub-Riemannian manifold

*pseudo-Riemannian manifold

*Metric tensor

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

**References***

* [*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*]

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Riemannian manifold**— noun in Riemannian geometry, a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, in a manner which varies smoothly from point to point … Wiktionary**Pseudo-Riemannian manifold**— In differential geometry, a pseudo Riemannian manifold (also called a semi Riemannian manifold) is a generalization of a Riemannian manifold. It is one of many things named after Bernhard Riemann. The key difference between the two is that on a… … Wikipedia**Sub-Riemannian manifold**— In mathematics, a sub Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub Riemannian manifold,you are allowed to go only along curves tangent to so called horizontal… … Wikipedia**Cut locus (Riemannian manifold)**— In Riemannian geometry, the cut locus of a point p in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from p, but it may contain additional points where the minimizing geodesic… … Wikipedia**pseudo-Riemannian manifold**— noun in differential geometry, a generalization of a Riemannian manifold Syn: semi Riemannian manifold … Wiktionary**Riemannian geometry**— Elliptic geometry is also sometimes called Riemannian geometry. Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric , i.e. with an inner product on the tangent… … Wikipedia**Manifold**— For other uses, see Manifold (disambiguation). The sphere (surface of a ball) is a two dimensional manifold since it can be represented by a collection of two dimensional maps. In mathematics (specifically in differential geometry and topology),… … Wikipedia**Riemannian submersion**— In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Let (M, g) and (N, h) be… … Wikipedia**Riemannian connection on a surface**— For the classical approach to the geometry of surfaces, see Differential geometry of surfaces. In mathematics, the Riemannian connection on a surface or Riemannian 2 manifold refers to several intrinsic geometric structures discovered by Tullio… … Wikipedia**Riemannian submanifold**— A Riemannian submanifold N of a Riemannian manifold M is a smooth manifoldequipped with the induced Riemannian metric from M . The image of an isometric immersion is a Riemannian submanifold … Wikipedia