> [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]] modelled on $E$ and $g: \symbi(TX) \to X$ be a [[Sections of Vector Bundles|section of]] the [[Bundle of Symmetric Bilinear Forms|symmetric bilinear forms]] on the [[Tangent Bundle|tangent bundle]]. If $g$ is non-singular, then $(X, g)$ is a **pseudo Riemannian manifold**. > > If $E$ is a [[Hilbert Space|Hilbert space]] and $g$ is [[Riemannian Metric|Riemannian]], then $(X, g)$ is a **Riemannian manifold**.