It is perhaps the simplest example of a pseudo-Riemannian manifold.

Pseudo-Riemannian manifolds of signature (3, 1) are important in general relativity.

Some results are also known for pseudo-Riemannian manifolds.

They also have applications for embedded hypersurfaces of pseudo-Riemannian manifolds.

In theoretical physics, spacetime is modeled by a pseudo-Riemannian manifold.

The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.

Smooth manifolds built from such spaces are called pseudo-Riemannian manifolds.

Further generalization to curved spacetimes form pseudo-Riemannian manifolds, such as in general relativity.

Every 2-dimensional pseudo-Riemannian manifold is conformally flat.

More formally, let (M, g) be a pseudo-Riemannian manifold.

The Riemannian volume form on a pseudo-Riemannian manifold.

Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian manifolds.

The postulates of special relativity can be expressed very succinctly using the mathematical language of pseudo-Riemannian manifolds.

In particular, the fundamental theorem of Riemannian geometry is true of pseudo-Riemannian manifolds as well.

The Einstein tensor is a rank 2 tensor defined over pseudo-Riemannian manifolds.

The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds.

A 3-dimensional pseudo-Riemannian manifold is conformally flat if and only if the Cotton tensor vanishes.

Oriented Riemannian manifolds and pseudo-Riemannian manifolds have an associated canonical volume form.

This allows one to speak of the Levi-Civita connection on a pseudo-Riemannian manifold along with the associated curvature tensor.

Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean space described by an isotropic quadratic form.

When the Hodge star is extended to pseudo-Riemannian manifolds, then the above inner product is understood to be the metric in diagonal form.

In general relativity the light is assumed to propagate in the vacuum along null geodesic in a pseudo-Riemannian manifold.

Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds.

The spacetime underlying Einstein's field equations, which mathematically describe gravitation, is a real 4-dimensional "Pseudo-Riemannian manifold".

Furthermore, a submanifold does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor becomes zero on any light-like curve.

Conformal maps can be defined between domains in higher-dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold.

According to the fundamental theorem of Riemannian geometry, there is a unique connection on any semi-Riemannian manifold that is compatible with the metric and torsion-free.

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite.

Hermann Weyl showed that this tensor measures the deviation of a semi-Riemannian manifold from conformal flatness; if it vanishes, the manifold is (locally) conformally equivalent to a flat manifold.

Causal notions are important in general relativity to the extent that the existence of an arrow of time demands that the universe's semi-Riemannian manifold be orientable, so that "future" and "past" are globally definable quantities.

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold.