Science Fair Project Encyclopedia In differential geometry, a
pseudo-Riemannian manifold is a smooth manifold equipped with a
smooth, symmetric, (0,2) tensor which is nondegenerate at each
point on the manifold. This tensor is called a pseudo-Riemannian
metric or, simply, a (pseudo-)metric tensor. The key difference
between a Riemannian metric and a pseudo-Riemannian metric is
that a pseudo-Riemannian metric need not be positive-definite,
merely nondegenerate. Since every positive-definite form is also
nondegenerate a Riemannian metric is a special case of a pseudo-
Riemannian one. Thus pseudo-Riemannian manifolds can be
considered generalizations of Riemannian manifolds. Every
nondegenerate, symmetric, bilinear form has a fixed signature
(p,q). Here p and q denote the number of positive and negative
eigenvalues of the form. The signature of a pseudo-Riemannian
manifold is just the signature of the metric (one should insist
that the signature is the same on every connected component).
Note that p + q = n is th