Distintion between Finsler metrics

Question

The main feature in Finsler manifolds is that the metric is not neccesary given by an inner product in the tangent apaces to every point. Are there theorems that tell you exactly when the Finsler metric is actually given by a inner product in every tangent space?

Answer 1

Knowing that a Finsler metric is nothing other than a (continuously varying) norm on the tangent space $T_p M$ at each point $p \in M$, your question requires asking, for each $p$, how to tell when the given norm on $T_p M$ is induced by an inner product. This happens if and only if the given norm on $T_p M$ satisfies a condition known as the "parallelogram law", and there are several answers on this site explaining that connection, for instance here.

Then returning to the given Finsler metric on $M$, and assuming that the parallelogram law does indeed hold on $T_p M$ for each $p \in M$, there is still be some work to do in order to show that the resulting inner products vary continuously as $p$ varies. You can see at that link that there is an actual formula for the inner product $\langle u,v \rangle$, expressed as a certain (clearly continuous) function of the values of the norms $\|u\|$, $\|v\|$ and $\|u+v\|$, for given $u,v \in T_p M$. So that settles continuity issues.

So far this does not address further issues regarding whether the inner product varies smoothly. But I guess you could just require, in addition to the parallelogram law, that value of the Finsler norm $\|u\|$ should vary smoothly as a function of $u \in TM$.