It is the study of smooth manifolds equipped with a non-degenerate metric tensor, not necessarily positive-definite (and hence a generalisation of [riemannian-geometry]). Included in this are metric tensors with index 1, called "Lorentzian", which are used to model spacetimes in (general-relativity).
A semi-Riemannian manifold is a pair $(M,g)$ where $M$ is a smooth manifold and $g$ a section (usually assumed smooth, but continuous or perhaps even lower regularity are sometimes considered) of $T^{0,2}M$ (in other words, $g$ is a covariant two-tensor field), such that $g$ satisfies the conditions:
- $g(X,Y) = g(Y,X)$ for any vector fields $X,Y$. (Symmetric)
- $g(X,\cdot)$, as a one-form, is the zero-one form if and only if $X$ is the zero vector. (Non-degenerate)
If in addition $g(X,X) > 0$ when $X \neq 0$, we say that the geometry is Riemannian.
While much of the algebraic theory of Riemannian geometry (by which I mean the curvature identities, Gauss-Codazzi equations, and other statements that are obtained by purely algebraic manipulations of the definitions) can be reproduced identically for semi-Riemannian geometry, there are crucial differences. For example, the theorem of Hopf-Rinow on geodesic completeness is no longer true. (Roughly speaking, the reason is that in the proof of Hopf-Rinow it is used the fact that in Riemannian geometry, the set of vectors $g(X,X) = 1$ is topologically a sphere, and is compact. In semi-Riemannian geometry, the corresponding set can be non-compact.)
Much of the study of semi-Riemannian geometry focuses on clarifying which statements of Riemannian geometry have appropriate analogues.
Now, since $g$ is continuous, from Sylvester's law of inertia we have that the metric signature of $g$ is constant on every connected component of $M$. The case where the signature is (-++...+) deserves special mention, as this is the setting in which one studies general-relativity. This case is often called Lorentzian geometry.
In the Lorentzian case we have some more tools available to us compared to the general semi-Riemannian manifolds. The most important difference being that the set $\{g(X,X) < 0\}$ in $T_pM$ has two connected components. In more general semi-Riemannian settings the analogue set is connected. This property allows us to locally determine a notion of future and past, which leads to the development of causal geometry, which carries a fundamental role in the study of mathematical relativity.
