Pseudo-Riemannian manifold

A pseudo- Riemannian manifold or Riemannian manifold is a semi- mathematical object from the ( pseudo-) Riemannian geometry. It is a generalization of the earlier defined Riemannian manifold and was introduced by Albert Einstein in his general theory of relativity. However, the object after Bernhard Riemann, the founder of Riemannian geometry, was named. But even after Albert Einstein, a structure of a manifold was named. This Einstein manifolds are a special case of pseudo- Riemannian.

Definition

The tangent space at a point is called a differentiable manifold in the following. A pseudo- Riemannian manifold is a differentiable manifold together with a function. This function is tensorial, symmetric and non-degenerate, that is, for all tangent vectors and functions shall

In addition, depending on differentiable. So the function is a differentiable tensor field and is called pseudo- Riemannian metric or metric tensor.

Signature

Like any ordinary bilinear form can be assigned to a signature and the pseudo- Riemannian metric. This is independent of the choice of the coordinate system on the manifold due to the inertia set of Sylvester, and thus independent of the choice of the point. As with the determinant there is due " physics " numerous equivalent expressions. But there is non-degenerate, is the third entry in the signature is always zero and the determinant of is always equal to zero. Four-dimensional pseudo- Riemannian manifolds with signature ( 3,1,0 ) (or mostly ( 1,3,0 ) ) are called Lorentz manifolds. These play an important role in the general theory of relativity.

Pseudo - Riemannian geometry

In contrast to pseudo- Riemannian metrics the Riemannian metrics are positive definite, allowing greater demand for " non-degenerate " as is. Some results from the Riemannian geometry can be transferred to pseudo- Riemannian manifolds. For example, the fundamental theorem applies Riemannian geometry also for pseudo- Riemannian manifolds. So there exists a unique Levi- Civita connection for each pseudo- Riemannian manifold. However, in contrast to the Riemannian geometry can not find any differentiable manifold with a metric given signature one. Another important difference between Riemannian and pseudo- Riemannian geometry is the lack of an equivalent for the set of Hopf Rinow in the pseudo - Riemannian geometry. In general, metric completeness and geodesic completeness are not linked here. The signature of the metric is also present problems for the continuity of the distance function. May have the property of the distance function for Lorentz manifolds not being steadily above.

Definition variant

Notwithstanding the above definition differs Serge Lang semi- Riemannian pseudo - Riemannian manifolds and additionally requires for the former, that is positive semi-definite, ie for all.