The notion of pseudo-Riemannian metric is a slight variant of that of Riemannian metric.
Where a Riemannian metric is governed by a positive-definite bilinear form, a Pseudo-Riemannian metric is governed by an indefinite bilinear form.
A pseudo-Riemannian “metric” is a nondegenerate quadratic form on a real vector space . A Riemannian metric is a positive-definite quadratic form on a real vector space. The data of such a quadratic form may be equivalently given by a nondegenerate symmetric bilinear pairing on .
A pseudo-Riemannian metric
can always be diagonalized: there exists a basis such that
where the pair is called the signature of the form . Pseudo-Riemannian metrics on are classified by their signatures; thus we have a standard metric of signature where is the standard basis of .
More generally, there is a notion of pseudo-Riemannian manifold (of type , which is an -dimensional manifold equipped with a global section
of the bundle of symmetric bilinear forms over , such that each is a nondegenerate form on the tangent space .
Certain theorems of Riemannian geometry carry over to the more general pseudo-Riemannian setting; for example, pseudo-Riemannian manifolds admit Levi-Civita connections, or in other words a unique notion of covariant differentiation of vector fields
In that case, one may define a notion of geodesic in pseudo-Riemannian manifolds , and we have a notion of “distance squared” between the endpoints along any geodesic path (which might be a negative number of course). The term “pseudo-Riemannian metric” may refer to such distances in general pseudo-Riemannian manifolds. (I guess.)
A typical example of pseudo-Riemannian manifold is a Lorentzian manifold, where the metric is of type . This is particularly so in the case , where such manifolds are the mathematical backdrop for studying general relativity and cosmological models.
|local model space||global geometry||differential cohomology||first order formulation of gravity|
|general||Klein geometry||Cartan geometry||Cartan connection|
|examples||Euclidean geometry||Riemannian geometry||affine connection||Euclidean gravity|
|Lorentzian geometry||pseudo-Riemannian geometry||spin connection||Einstein gravity|
|general||Klein 2-geometry||Cartan 2-geometry|
|higher Klein geometry||higher Cartan geometry||higher Cartan connection|
|examples||extended super Minkowski spacetime||extended supergeometry||higher supergravity: type II, heterotic, 11d|