CAT(K) Finsler manifolds.

I was wondering if the following is true (and common knowledge):

Let $(M,F)$ be a Finsler manifold. Let d be the induced distance by the norm in the usual sense. That is, $d(x,y)=\inf${lenghts of all piece-wise smooth curves…}. We consider then $(M,d)$ as a metric space. My questions, which are probably quite easy, are these:

1) Suppose that $d$ is a righteous metric, in the sense that is also symmetric. Does that imply that $(M,F)$ was actually a Riemannian manifold for starters? If not could you please show me a counter example?

2) If the answer to 1) is negative then suppose $(M,d)$ happens to also be a $CAT(\kappa)$ space. does it follow that M is necessarily a Riemannian manifold? If not, what about $\kappa =0$?

Thanks a lot in advance for any help clearing this out. Of curse, any reference is gratefully welcomed.

Solutions Collecting From Web of "CAT(K) Finsler manifolds."

  1. The answer is no. The plane with taxicab distance $\|x\|=|x_1|+|x_2|$ is a counterexample.

  2. A $CAT(0)$ Finsler space is Riemannian. Indeed, CAT(0) condition is scale-invariant, therefore passes to tangent spaces. And the only CAT(0) normed spaces are those with an inner product norm. One reference is: Ptolemaic spaces and CAT(0) by Buckley, Falk, and Wraith; Glasgow Math. J. 51 (2009) 301–314.

  3. A $CAT(\kappa)$ Finsler space is also Riemannian, but I couldn’t locate a proof. The claim appears on MathOverflow and at the beginning of this paper where it’s said to be well-known.