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Arguably, the most important type of pseudo-Riemannian manifold is a Lorentzian manifold. Lorentzian manifolds occur in the General Theory Of Relativity as models of curved 4-dimensional Spacetime . Just as Riemannian manifolds may be thought of as being locally modeled on Euclidean Space , Lorentzian manifolds are locally modeled on Minkowski Space .


FORMAL DEFINITION


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 '''.

Every nondegenerate, symmetric, Bilinear Form on a
Vector Space can be assigned a Signature (p,q). Here p and q denote the number of positive and negative Eigenvalue s of the form. The signature of a pseudo-Riemannian manifold is just the signature of the metric on any given tangent space (one should insist that the signature is the same on every Connected Component ). Note that p + q = n is the dimension of the manifold. A Riemannian metric has signature (n,0).


LORENTZIAN MANIFOLDS


Pseudo-Riemannian metrics of signature (p,1) (or sometimes (1,q), see Sign Convention ) are called Lorentzian metrics. A manifold equipped with a Lorentzian metric is naturally called a '''Lorentzian manifold'''. After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important because of their physical applications to the theory of General Relativity . A principal assumption of general relativity is that Spacetime can be modeled as a Lorentzian manifold of signature (3,1).


PROPERTIES OF PSEUDO-RIEMANNIAN MANIFOLDS


Just as Euclidean Space R''n'' can be thought of as the model Riemannian Manifold , Minkowski Space R''p'',1 with the flat Minkowski Metric is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (p,q) is R''p'',''q'' with the metric
:g = dx_1^2 + \cdots + dx_p^2 - dx_{p+1}^2 - \cdots - dx_{p+q}^2

Some basic theorems of Riemannian geometry can be generalized to the pseudo-Riemannian case. In particular, the Fundamental Theorem Of Riemannian Geometry is true of pseudo-Riemannian manifolds as well. This allows one to speak of the Levi-Civita Connection on a pseudo-Riemannian manifold along with the associated Curvature Tensor . On the other hand, there are many theorems in Riemannian geometry which do not hold in the generalized case. For example, it is ''not'' true that every smooth manifold admits a pseudo-Riemannian metric of a given signature; there are certain Topological obstructions.