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Concept
Pseudo-Riemannian manifold
Summary
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.
Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space.
A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.
Manifold and Differentiable manifold
In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an n-dimensional Euclidean space any point can be specified by n real numbers. These are called the coordinates of the point.
An n-dimensional differentiable manifold is a generalisation of n-dimensional 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 n-dimensional Euclidean space.
See Manifold, Differentiable manifold, Coordinate patch for more details.
Tangent space and Metric tensor
Associated with each point in an -dimensional differentiable manifold is a tangent space (denoted ). This is an -dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point .
A metric tensor is a non-degenerate, smooth, symmetric, bilinear map that assigns a real number to pairs of tangent vectors at each tangent space of the manifold. Denoting the metric tensor by we can express this as
The map is symmetric and bilinear so if are tangent vectors at a point to the manifold then we have
for any real number .
That is non-degenerate means there is no non-zero such that for all .
Metric signature
Given a metric tensor g on an n-dimensional 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.
Official source
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