Riemannian manifold

A Riemannian metricMathworldPlanetmath tensor is a covariant, type (0,2) tensor field gΓ(T*MT*M) such that at each point pM, the bilinear formMathworldPlanetmathPlanetmath gp:TpM×TpM is symmetricMathworldPlanetmathPlanetmath and positive definitePlanetmathPlanetmath. Here T*M is the cotangent bundleMathworldPlanetmath of M (defined as a sheaf), Γ is the set of global sections of T*MT*M, and gp is the value of the function g at the point pM.

Let (x1,,xn) be a system of local coordinates on an open subset UM, let dxi,i=1,,n be the corresponding coframe of 1-forms, and let xi,i=1,,n be the corresponding dual frame of vector fields. Using the local coordinates, the metric tensor has the unique expression


where the metric tensor componentsPlanetmathPlanetmath


are smooth functions on U.

Once we fix the local coordinates, the functions gij completely determine the Riemannian metric. Thus, at each point pU, the matrix (gij(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 gij on each coordinate chart which are symmetric and positive definite, with the proviso that the gij’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


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

Often, it is the gij that are referred to as the “Riemannian metric”. This, however, is a misnomer. Properly speaking, the gij 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 gij’s by the misnomer “Riemannian metric” appears to have stuck.


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

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

  • The category of ‘pseudo-Riemannian’ manifolds, deals in fact with extensions of Minkowski spacesMathworldPlanetmath, does not possess the Riemannian metric defined in this entry on Riemannian manifolds, and is claimed as a useful approach 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 GradientMathworldPlanetmath
Related topic CategoryOfRiemannianManifolds
Related topic HomotopyCategory
Related topic CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams
Related topic QuantumAlgebraicTopologyOfCWComplexRepresentationsNewQATResultsForQuantumStateSpacesOfSpinNetworks
Defines Riemannian metric
Defines Riemannian structure
Defines metric tensor