I know that Riemannian manifolds of constant curvature are locally isomorphic to the spherical, flat or hyperbolic space depending on the sign of the sectional curvature.
Reading Hawking and Ellis' book "The Large Scale Structure of Space-Time", in the section about de Sitter and anti-de Sitter spaces, there's written this sentence: "The space of constant curvature with $R = 0$ is Minkowski space-time. The space for $R>0$ is de Sitter space-time". Later in the text, there's also "The space of constant curvature with $R < 0$ is called anti-de Sitter
space". $R$ is the scalar curvature.
My question is: is this an extension of the property I mentioned at the beginning? That is, Lorentian manifolds of constant curvature are locally isomorphic to de Sitter, flat or anti-de Sitter spaces? I have read also Wald' general relativity book and referred to Lee's "Manifolds and Differential Geometry" (which was a bit too technical for me, so maybe I missed something), but I couldn't find an answer to this question.
Finally, where can I explore this topic further?
Best Answer
My favorite reference for semi-Riemannian geometry is
O’Neill, Barrett, Semi-Riemannian geometry. With applications to relativity, Pure and Applied Mathematics, 103. New York-London etc.: Academic Press. xiii, 468 p. (1983). ZBL0531.53051.
It is written by a mathematician and, hence, with mathematical precision. The relevant result is on p. 223, Corollary 15:
Suppose that $M, \bar{M}$ are semi-Riemannian manifolds of the dimension, same index (i.e. signature of the metric) and the same constant curvature. Then $M, \bar{M}$ are locally isometric, i.e. any two points of $M, \bar{M}$ have isometric neighborhoods.
This answers your question affirmatively.