Semi-Riemannian manifold

A semi-Riemannian $𝐶 𝛼$ -manifold $(𝑀, 𝒜, 𝑔)$ is a $𝐶 𝛼$ -manifold $(𝑀, 𝒜)$ equipped with a special kind of tensor field $𝑔 𝑎 𝑏 \in T 02 (𝑀)$ called a metric tensor. #m/def/geo/diff This means $𝑔 𝑎 𝑏$ must satisfy

symmetry, i.e. $𝑔 𝑎 𝑏 = 𝑔 𝑏 𝑎$ ;
non-degeneracy, i.e. $𝑔 𝑎 𝑏 𝑣 𝑎 = 0$ iff $0 = 𝑣 𝑎 \in 𝔛 (𝑀)$ ;

Note that by non-degeneracy we can define the inverse metric tensor $𝑔 𝑎 𝑏$ so that $𝑔 𝑎 𝑏 𝑔 𝑏 𝑐 = 𝛿 𝑎 𝑐$ . See Raising and lowering of indices.

Further terminology

Locally the metric tensor can be brought into the form of a diagonal matrix with entries $\pm 1$ . For a connected manifold, the numbers of positive and negative entries is an invariant $(𝑠, 𝑡)$ , called the signature.

$(𝑛, 0)$ gives a Riemannian manifold
$(𝑛 - 1, 1)$ gives a Lorentzian manifold

Semi-Riemannian manifold

Further terminology

See also