Frame vs vector basis in differential geometry

Question

Let $E \to M$ a finite dimensional vector bundle. I faced a couple of times that a vector basis of the fiber $E_x$ over $x \in M$ was often called a 'frame'. Are there any differences between the the notation of a 'frame' and a 'vector basis'? Is a 'frame' just a terminology more conventionally used in differential geometry but in truth synonymous to vector space?

Answer 1

A local frame is a choice of a basis of $E_x$ for each point $x$ in an open set $U$ of $M$, such that this choice of basis varies continuously with $x$. A global frame is a local frame with all of $M$ as the domain.

However, sometimes people may use the word 'frame' to simply mean a basis of a fiber.