Pseudo-Riemannian manifold

Definition
It is a smooth manifold $(\mathcal{M},\mathcal{O},{\mathcal{A}}_{◻})$ together with a pseudo-Riemannian metric.
$◼$

They are, in some sense, metric spaces: we can measure lengths.

A particular case are the Riemannian manifolds.

We can compute volumes in a natural way, and therefore we have a distinguished volume form in it.

They can have symmetry of a pseudo-Riemannian manifold.

Flat manifolds

A key feature is the Riemann curvature tensor. We say a pseudo-Riemannian manifold is flat (flat metric) when the Riemann curvature tensor vanishes identically or, equivalently, if it is locally isometric to Euclidean space (every point has a neighborhood that is isometric to an open subset of ${\mathbb{R}}^n$ with its Euclidean metric). See @lee2006riemannian Theorem 7.3. Also @ivey2016cartan page 52 (Theorem 2.6.12).
In other words, the Riemann normal coordinates can be extended to a neighborhood, not only a point.

Interpretation as a Cartan geometry

It can be interpreted as a Cartan geometry.