Lecture 41 Inner products

A C^r vector bundle is a C^r family of vector spaces. But we can consider inner product spaces instead of just vector spaces since this could be handy in applications. Indeed, we define an orthogonal vector bundle, which is roughly speaking, a C^r family of inner product spaces.

We see that every smooth vector bundle admits a structure of orthogonal vector bundle and use such property to prove that every short exact sequence splits.

We introduce the geometric normal bundle and the algebraic normal bundle associated to a submanifold N of a manifold M.

Finally, we state a theorem without proof: every orthogonal vector bundle has a vector bundle atlas whose transition functions take values in the orthogonal group.

Lecture notes: