In differential geometry, a pseudoRiemannian manifold (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.
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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.
Associated with each point p in an ndimensional differentiable manifold M is a tangent space (denoted ). This is an ndimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point 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 g we can express this as .
The map is symmetric and bilinear so if are tangent vectors at a point p in the manifold M then we have
for some real number a.
That g is nondegenerate means there are no nonzero such that for all .
For an ndimensional manifold the metric tensor (in a fixed coordinate system) has n eigenvalues. If the metric is nondegenerate then none of these eigenvalues are zero. The signature of the metric denotes the number of positive and negative eigenvalues, this quantity is independent of the chosen coordinate system by Sylvester's rigidity theorem and locally nondecreasing. If the metric has p positive eigenvalues and q negative eigenvalues then the metric signature is (p,q). For a nondegenerate metric p + q = n.
A pseudoRiemannian manifold is a differentiable manifold equipped with a nondegenerate, smooth, symmetric metric tensor 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.
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.
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 assumption 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).
Just as Euclidean space can be thought of as the model Riemannian manifold, Minkowski space with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudoRiemannian manifold of signature (p,q) is with the metric:
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 of a pseudoRiemannian manifold need not be a pseudoRiemannian manifold.
