> [!definition] > > Let $E$ be a self-dual [[Banach Space|Banach space]], then the set $\met(E)$ of non-singular [[Symmetric Bilinear Form|symmetric bilinear forms]] on $E$ is the **pseudo Riemannian metrics** on $E$. > > $\met(E)$ forms an [[Open Set|open]] subset of $\symbi(E)$. > > If $E$ is a [[Hilbert Space|Hilbert space]], then the set $\ri(E)$ of [[Coercive Operator|positive definite]] operators forms the **Riemannian metrics** on $E$. > [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]], $\pi$ be a $C^{p - 1}$-[[Vector Bundle|vector bundle]] modelled on $F$, and $\xi$ be a [[Sections of Vector Bundles|section of]] $\pi$. If $\xi$ is non-singular at each point, then $\xi$ is a **pseudo Riemannian metric** on $\pi$. > > If $F$ is a [[Hilbert Space|Hilbert space]] and $\xi$ is positive definite at each point, then $\xi$ is a **Riemannian metric** on $\pi$. > [!theorem] > > The set of Riemannian metrics forms a convex cone of $\mathfrak{S}(X, \pi)$, that is, for any $a, b > 0$ and $g_1, g_2 \in \ri(\pi)$, $ag_1 + bg_2 \in \ri(\pi)$. > [!theorem] > > Let $X$ be a $C^p$-manifold modelled on $E$ and $\pi$ be a $C^{p - 1}$-vector bundle modelled on $F$. Suppose that > 1. $X$ admits a [[Partition of Unity|partitions of unity]] subordinate to each [[Open Cover|open cover]]. > 2. $F$ is a [[Hilbert Space|Hilbert space]]. > > then $\pi$ admits a Riemannian metric. > > *Proof*. Let $\bracs{(U_i, \varphi_i)}$ be a [[Partition of Unity|partition of unity]] such that for each $i$, there exists a trivialisation > $ > \pi_i: \pi^{-1}(U_i) \to U_i \times F > $ > By directly stealing an [[Inner Product|inner product]] from each of these, and summing over the partition of unity yields a Riemannian metric.