A Riemannian metric tensor is a covariant, type tensor field such that at each point , the bilinear form is symmetric and positive definite. Here is the cotangent bundle of (defined as a sheaf), is the set of global sections of , and is the value of the function at the point .
Let be a system of local coordinates on an open subset , let be the corresponding coframe of 1-forms, and let be the corresponding dual frame of vector fields. Using the local coordinates, the metric tensor has the unique expression
where the metric tensor components
are smooth functions on .
Once we fix the local coordinates, the functions completely determine the Riemannian metric. Thus, at each point , the matrix is symmetric, and positive definite. Indeed, it is possible to define a Riemannian structure on a manifold by specifying an atlas over together with a matrix of functions on each coordinate chart which are symmetric and positive definite, with the proviso that the ’s must be compatible with each other on overlaps.
A manifold together with a Riemannian metric tensor is called a Riemannian manifold.
Note: A Riemannian metric tensor on is not a distance metric on . However, on a connected manifold every Riemannian metric tensor on induces a distance metric on , given by
where the infimum is taken over all rectifiable curves with and .
Often, it is the that are referred to as the “Riemannian metric”. This, however, is a misnomer. Properly speaking, the 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 ’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 curvature 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 spaces, 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.
|Date of creation||2013-03-22 13:02:54|
|Last modified on||2013-03-22 13:02:54|
|Last modified by||djao (24)|
|Synonym||Riemann space and metric|