Quadratic form on Vector Bundle

Question

A quadratic form of a vector space $V$ over a field $\mathbb{F}$ is a bilinear symmetric map $V\times V \rightarrow F$.

How does one define a quadratic form over a vector bundle.

Answer 1

A quadratic form on a vector bundle $E\to M$ is a quadratic form on every fiber. In other words, it is a symmetric tensor $T$ of type $(2,0)$. As always, this tensor needs to satisfy some regularity condition which depends on the category you work with. For example, if it is the smooth category, the tensor needs to be smooth. It means that if $X,Y$ are smooth vector fields on $M$, then the function $T(X,Y):M\to\mathbb{R}$ is smooth.