---- > [!definition] Definition. ([[metric tensor]]) > Let $M$ be a [[smooth manifold]]. A **metric tensor (field)**, or simply **metric**, on $M$ is smooth [[vector bundle|section]] $g$ of $\mathbb{S}^{2} T^{*}M$, the second [[symmetric power]] [[tensor product of vector bundles|of the]] [[cotangent bundle]], that is [[nondegenerate bilinear form|nondegenerate]] at each point. > $M$ is called a **pseudo-Riemannian manifold**. If the [[bilinear map|bilinear form]] $g_{p}$ has [[signature of a bilinear form|signature]] $(-,+,\dots,+)$ or $(+,-,\dots,-)$ for all $p \in M$, is called **Lorentzian**. The signature-$(+,\dots,+)$ case is the [[Riemannian manifold|Riemannian]] one. Many results of "Riemannian geometry" in fact hold (with suitable modifications) for "pseudo-Riemannian geometry". ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```