Riemannian manifold

A Riemannian metric tensor is a covariant, type $(0,2)$ tensorfield$g\in \mathrm{\Gamma }({\mathrm{T}}^{*}M\otimes {\mathrm{T}}^{*}M)$ such that at each point $p\in M$,the bilinear form$g_p:{\mathrm{T}}_{p}M\times {\mathrm{T}}_{p}M\to ℝ$ is symmetric andpositive definite. Here $T^{*}M$ is the cotangent bundle of $M$ (defined as a sheaf), $\mathrm{\Gamma }$ 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 $(x^1,\mathrm{\dots },x^n)$ be a system of local coordinates on an opensubset $U\subset M$, let $d{x}^i,i=1,\mathrm{\dots },n$ be the correspondingcoframe of 1-forms, and let ${\displaystyle \frac{\partial }{\partial x^i}},i=1,\mathrm{\dots },n$be the corresponding dual frame of vector fields. Using the localcoordinates, the metric tensor has the unique expression

$$g=\sum _{i,j=1}^n{g}_{ij}d{x}^i\otimes d{x}^j,$$

where the metrictensor components

$$g_{ij}=g(\frac{\partial }{\partial x^i},\frac{\partial }{\partial x^j})$$

are smoothfunctions on $U$.

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

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

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

$$d(x,y):=inf\left\{{\int }_0^1{\left[g{(\frac{dc}{dt},\frac{dc}{dt})}_{c(t)}\right]}^{1/2}𝑑t\right\},x,y\in M,$$

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 “Riemannianmetric”. This, however, is a misnomer. Properly speaking, the$g_{ij}$ should be called local coordinate components of a metrictensor, where as “Riemannian metric” should refer to the distancefunction defined above. However, the practice of calling thecollection of $g_{ij}$’s by the misnomer “Riemannian metric” appearsto have stuck.

Remarks:

• Both the Riemannian manifold and Riemannian metric tensor are fundamental conceptsin Einstein’s General Relativity (GR) theory where the “Riemannian metric” and curvature of thephysical Riemannian space-time are changed by the presence of massive bodies and energyaccording to Einstein’s fundamental GR field equations (http://planetmath.org/EinsteinFieldEquations).

• The category of Riemannian manifolds (or ‘spaces’) provides an alternative framework for GR theoriesas well as algebraic quantum field theories (AQFTs);

• The category of ‘pseudo-Riemannian’ manifolds, deals in fact with extensions of Minkowski spaces, does notpossess the Riemannian metric defined in this entry on Riemannian manifolds, and is claimed as a usefulapproach to defining 4D-spacetimes in relativity theories.

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