Manifolds

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 spaces and metric tensors

Associated with each point ${\displaystyle p}$ in an ${\displaystyle n}$-dimensional differentiable manifold ${\displaystyle M}$ is a tangent space (denoted ${\displaystyle T_{p}M}$). This is an ${\displaystyle n}$-dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point ${\displaystyle p}$.

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 ${\displaystyle g}$ we can express this as

${\displaystyle g:T_{p}M\times T_{p}M\to \mathbb {R} .}$

The map is symmetric and bilinear so if ${\displaystyle X,Y,Z\in T_{p}M}$ are tangent vectors at a point ${\displaystyle p}$ to the manifold ${\displaystyle M}$ then we have

• ${\displaystyle \,g(X,Y)=g(Y,X)}$
• ${\displaystyle \,g(aX+Y,Z)=ag(X,Z)+g(Y,Z)}$

for any real number ${\displaystyle a\in \mathbb {R} }$.

That ${\displaystyle g}$ is non-degenerate means there is no non-zero ${\displaystyle X\in T_{p}M}$ such that ${\displaystyle \,g(X,Y)=0}$ for all ${\displaystyle Y\in T_{p}M}$.

Metric signatures

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. By Sylvester's law of inertia, 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. A non-degenerate metric tensor has r = 0 and the signature may be denoted (p, q), where p + q = n.

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After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important in applications of general relativity.

A principal premise of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold of signature (3, 1) or, equivalently, (1, 3). Unlike Riemannian manifolds with positive-definite metrics, an indefinite signature allows tangent vectors to be classified into timelike, null or spacelike. With a signature of (p, 1) or (1, q), the manifold is also locally (and possibly globally) time-orientable (see Causal structure).