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A pseudo-Riemannian manifold is a manifold  together with a non degenerate, symmetric section  of
 (2-covariant tensor bundle over  ).
Unlike with a Riemannian manifold,  is not positive definite. That is, there exist vectors
 such that
 .
A well known result from linear algebra permits us to make a change of basis such that in the new base  is represented by a diagonal matrix with  or  elements in the diagonal. If there are
 ,  elements in the diagonal and  ,  , the tensor is said to have signature 
The signature will be invariant in every connected component of  , but usually the restriction that it be a global invariant is added to the definition of a pseudo-Riemannian manifold.
Unlike a Riemannian metric, some manifolds do not admit a pseudo-Riemannian metric.
Pseudo-Riemannian manifolds are crucial in Physics and in particular in General Relativity where space-time is modeled as a 4-pseudo Riemannian manifold with signature (1,3)1.
Intuitively pseudo-Riemannian manifolds are generalizations of Minkowski's space just as a Riemannian manifold is a generalization of a vector space with a positive definite metric.
Footnotes
- 1
- also referred to as
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