A _Lorentzian manifold_ $(M, g)$ is a smooth manifold $M$ equipped with a metric tensor $g$ of signature $(-+++)$ (or $(+---)$, depending on convention). This signature distinguishes one time dimension from the spatial dimensions. Concretely: 1. **Manifold:** A _manifold_ is a topological space that is locally homeomorphic to $\mathbb{R}^n$ and is equipped with a structure that allows for the definition of smooth (infinitely differentiable) functions and maps. More precisely, a manifold $M$ of dimension $n$ is a space where every point $p \in M$ has an open neighborhood homeomorphic to an open set in $\mathbb{R}^n$, and these homeomorphisms (called charts) are required to be smoothly compatible on overlaps. 2. **Lorentzian Metric:** On a Lorentzian manifold, the metric tensor $g$ is a $(0,2)$-tensor field that at each point $p \in M$ defines an inner product on the tangent space $T_p M$ with one “time-like” dimension and three “space-like” dimensions. More formally, $g_p: T_pM \times T_pM \to \mathbb{R}$ is non-degenerate and has exactly one negative eigenvalue and the rest positive, or vice versa, depending on the chosen convention. Such a structure is the setting for modern formulations of classical field theories, including general relativity, where spacetime is modeled as a 4-dimensional Lorentzian manifold.