Pseudo-Riemannian manifold

In <a href="/facts/Mathematical_physics/sLKg8T1G">mathematical physics</a>, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> with a <a href="/facts/Metric_tensor/tbqek1gJ">metric tensor</a> that is everywhere <a href="/facts/Non-degenerate_bilinear_form/9TLGBvSH">nondegenerate</a>. This is a generalization of a <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> in which the requirement of <a href="/facts/Positive-definite_bilinear_form/utlBuVpN">positive-definiteness</a> is relaxed. 
Every <a href="/facts/Tangent_space/crvpbSeG">tangent space</a> of a pseudo-Riemannian manifold is a <a href="/facts/Pseudo-Euclidean_vector_space/UW6sbGeK">pseudo-Euclidean vector space</a>.
A special case used in <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> is a four-dimensional Lorentzian manifold for modeling <a href="/facts/Spacetime/uRIN1x4b">spacetime</a>, where tangent vectors can be classified as <a href="/facts/Causal_structure/GVTdz7r9">timelike, null, and spacelike</a>.

Pseudo-Riemannian manifold open-in-new

Pseudo-Riemannian manifold