A semi-Riemannian metric 
on a smooth manifold 
is a smooth symmetric 
tensor
field whose associated bilinear form is nondegenerate.
Its index is constant
on each connected component and may be zero.
A semi-Riemannian manifold is a smooth
manifold equipped with such a semi-Riemannian metric.
The terms semi-Riemannian and pseudo-Riemannian are commonly used synonymously, as in pseudo-Riemannian manifolds. With this convention, Riemannian manifolds are the index-zero case. Some authors reserve pseudo-Riemannian for metrics which are not positive definite.
At each point, an appropriate vector basis puts 
in the normal form
 |
where 
,
..., 
are the dual covectors and 
. The metric signature
is 
,
and 
is allowed. This is a pointwise vector basis statement
and does not assert the existence of flat coordinates
on a neighborhood.
Lorentzian manifolds are examples with one negative direction under the mostly-positive sign convention.
