# Riemannian manifold

A is a covariant, type $(0,2)$ tensor field $g\in\Gamma(\operatorname{T}^{*}M\otimes\operatorname{T}^{*}M)$ such that at each point $p\in M$, the bilinear form   $g_{p}:\operatorname{T}_{p}M\times\operatorname{T}_{p}M\to\mathbb{R}$ is symmetric   and positive definite  . Here $T^{*}M$ is the cotangent bundle  of $M$ (defined as a sheaf), $\Gamma$ is the set of global sections of $T^{*}M\otimes T^{*}M$, and $g_{p}$ is the value of the function $g$ at the point $p\in M$.

Let $(x^{1},\ldots,x^{n})$ be a system of local coordinates on an open subset $U\subset M$, let $dx^{i},\;i=1,\ldots,n$ be the corresponding coframe of 1-forms, and let $\displaystyle\frac{\partial}{\partial x^{i}},\;i=1,\ldots,n$ be the corresponding dual frame of vector fields. Using the local coordinates, the metric tensor has the unique expression

 $g=\sum_{i,j=1}^{n}g_{ij}\,dx^{i}\otimes dx^{j},$
 $g_{ij}=g\left(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial x^{j}}\right)$

are smooth functions on $U$.

Once we fix the local coordinates, the functions $g_{ij}$ completely determine the Riemannian metric. Thus, at each point $p\in U$, the matrix $(g_{ij}(p))$ is symmetric, and positive definite. Indeed, it is possible to define a Riemannian structure on a manifold $M$ by specifying an atlas over $M$ together with a matrix of functions $g_{ij}$ on each coordinate chart which are symmetric and positive definite, with the proviso that the $g_{ij}$’s must be compatible with each other on overlaps.

A manifold $M$ together with a Riemannian metric tensor $g$ is called a Riemannian manifold.

Note: A Riemannian metric tensor on $M$ is not a distance metric on $M$. However, on a connected manifold every Riemannian metric tensor on $M$ induces a distance metric on $M$, given by

 $d(x,y):=\inf\left\{\int_{0}^{1}\left[g\!\!\left(\frac{dc}{dt},\frac{dc}{dt}% \right)_{\!c(t)}\right]^{1/2}dt\right\},\quad x,y\in M,$

where the infimum is taken over all rectifiable curves $c:[0,1]\to M$ with $c(0)=x$ and $c(1)=y$.

Often, it is the $g_{ij}$ that are referred to as the “Riemannian metric”. This, however, is a misnomer. Properly speaking, the $g_{ij}$ should be called local coordinate components of a metric tensor, where as “Riemannian metric” should refer to the distance function defined above. However, the practice of calling the collection of $g_{ij}$’s by the misnomer “Riemannian metric” appears to have stuck.

Remarks:

 Title Riemannian manifold Canonical name RiemannianManifold Date of creation 2013-03-22 13:02:54 Last modified on 2013-03-22 13:02:54 Owner djao (24) Last modified by djao (24) Numerical id 31 Author djao (24) Entry type Definition Classification msc 53B20 Classification msc 53B21 Synonym Riemann space and metric Related topic QuantumGeometry2 Related topic Gradient  Related topic CategoryOfRiemannianManifolds Related topic HomotopyCategory Related topic CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams Related topic QuantumAlgebraicTopologyOfCWComplexRepresentationsNewQATResultsForQuantumStateSpacesOfSpinNetworks Defines Riemannian metric Defines Riemannian structure Defines metric tensor