In differential geometry, a pseudoRiemannian manifold ^{[1]}^{[2]} (also called a semiRiemannian manifold) is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudoRiemannian manifold is that on a pseudoRiemannian manifold the metric tensor need not be positivedefinite. Instead a weaker condition of nondegeneracy is imposed.
Introduction
Manifolds
In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an ndimensional Euclidean space any point can be specified by n real numbers. These are called the coordinates of the point.
An ndimensional differentiable manifold is a generalisation of ndimensional Euclidean space. In a manifold it may only be possible to define coordinates locally. This is achieved by defining coordinate patches: subsets of the manifold which can be mapped into ndimensional Euclidean space.
See Manifold, differentiable manifold, coordinate patch for more details.
Tangent spaces and metric tensors
Associated with each point $\backslash scriptstyle\; p$ in an $\backslash scriptstyle\; n$dimensional differentiable manifold $\backslash scriptstyle\; M$ is a tangent space (denoted $\backslash scriptstyle\; T\_pM$). This is an $\backslash scriptstyle\; n$dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point $\backslash scriptstyle\; p$.
A metric tensor is a nondegenerate, smooth, symmetric, bilinear map which assigns a real number to pairs of tangent vectors at each tangent space of the manifold. Denoting the metric tensor by $\backslash scriptstyle\; g$ we can express this as
 $g:\; T\_pM\; \backslash times\; T\_pM\; \backslash to\; \backslash mathbb\{R\}$.
The map is symmetric and bilinear so if $\backslash scriptstyle\; X,Y,Z\; \backslash in\; T\_pM$ are tangent vectors at a point $\backslash scriptstyle\; p$ to the manifold $\backslash scriptstyle\; M$ then we have
 $\backslash ,g(X,Y)\; =\; g(Y,X)$
 $\backslash ,g(aX\; +\; Y,\; Z)\; =\; a\; g(X,Z)\; +\; g(Y,Z)$
for any real number $\backslash scriptstyle\; a\backslash in\backslash mathbb\{R\}$.
That $\backslash scriptstyle\; g$ is nondegenerate means there are no nonzero $X\; \backslash in\; T\_pM$ such that $\backslash ,g(X,Y)\; =\; 0$ for all $Y\; \backslash in\; T\_pM$.
Metric signatures
Given a metric tensor g on an ndimensional real manifold, the quadratic form q(x) = g(x,x) associated with the metric tensor applied to each vector of any orthogonal basis produces n real values. By Sylvester's rigidity theorem, the number of each positive, negative and zero values produced in this manner are invariants of the metric tensor, independent of the choice of orthogonal basis. The signature (p,q,r) of the metric tensor gives these numbers, shown in the same order. For a nondegenerate metric tensor r = 0 and the signature may be denoted (p,q), where p + q = n.
Definition
A pseudoRiemannian manifold $\backslash ,(M,g)$ is a differentiable manifold $\backslash ,M$ equipped with a nondegenerate, smooth, symmetric metric tensor $\backslash ,g$ which, unlike a Riemannian metric, need not be positivedefinite, but must be nondegenerate. Such a metric is called a pseudoRiemannian metric and its values can be positive, negative or zero.
The signature of a pseudoRiemannian metric is (p, q) where both p and q are nonnegative.
Lorentzian manifold
A Lorentzian manifold is an important special case of a pseudoRiemannian manifold in which the signature of the metric is (1, n−1) (or sometimes (n−1, 1), see sign convention). Such metrics are called Lorentzian metrics. They are named after the physicist Hendrik Lorentz.
Applications in physics
After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudoRiemannian manifolds. They are important because of their physical applications to the theory of general relativity.
A principal basis of general relativity is that spacetime can be modeled as a 4dimensional Lorentzian manifold of signature (3, 1) or, equivalently, (1, 3). Unlike Riemannian manifolds with positivedefinite metrics, a signature of (p, 1) or (1, q) allows tangent vectors to be classified into timelike, null or spacelike (see Causal structure).
Properties of pseudoRiemannian manifolds
Just as Euclidean space $\backslash mathbb\{R\}^n$ can be thought of as the model Riemannian manifold, Minkowski space $\backslash mathbb\{R\}^\{n1,1\}$ with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudoRiemannian manifold of signature (p, q) is $\backslash mathbb\{R\}^\{p,q\}$ with the metric
 $g\; =\; dx\_1^2\; +\; \backslash cdots\; +\; dx\_p^2\; \; dx\_\{p+1\}^2\; \; \backslash cdots\; \; dx\_\{p+q\}^2$
Some basic theorems of Riemannian geometry can be generalized to the pseudoRiemannian case. In particular, the fundamental theorem of Riemannian geometry is true of pseudoRiemannian manifolds as well. This allows one to speak of the LeviCivita connection on a pseudoRiemannian 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 pseudoRiemannian metric of a given signature; there are certain topological obstructions. Furthermore, a submanifold does not always inherit the structure of a pseudoRiemannian manifold; for example, the metric tensor becomes zero on any lightlike curve. The Clifton–Pohl torus provides an example of a pseudoRiemannian manifold that is compact but not complete, a combination of properties that the Hopf–Rinow theorem disallows for Riemannian manifolds.^{[3]}
See also
Notes
References



 G. Vrănceanu & R. Roşca (1976) Introduction to Relativity and PseudoRiemannian Geometry, Bucarest: Editura Academiei Republicii Socialiste România.
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