This question is a cross post from MathOverflow. I have requested the migration of the question, but unfortunately it is not possible after two months of posting.
I was reading about geometry in metric spaces from different books, two of them are: (1) A course in metric geometry by Y. Burago, D. Burago and S. Ivanov; and (2) Metric spaces of non-positive curvature by M. Bridson and A. Häfliger. Both develop the Alexandrov's approach to curvature, which uses comparison triangles with the constant curvature model spaces.
For a normed space $X$, the following statements are equivalent:
- $X$ has curvature $\leq\kappa$ in Alexandrov's sense, for some real number $\kappa$.
- $X$ has curvature $\leq 0$ in Alexandrov's sense.
- The norm on $X$ is induced by an inner product.
So it seems to me that Alexandrov's approach is not very informative in the normed case. On the other hand, a geodesic space has non-positive curvature in the Busemann's sense if its metric is convex, in general this is a weaker notion than Alexandrov's, and in the normed case the following statements are equivalent:
- $X$ has non-positive curvature in the Busemann's sense.
- $X$ is uniquely geodesic, that is, every pair of points is joined by a unique geodesic (the linear segment between them).
- $X$ is strinctly convex, that is, the ball in $X$ is strictly convex which means that for every pair of different vectors $v$ and $w$ of norm equal to $1$ we have that $tv+(1-t)w$ has norm strictly less than $1$ for every $t$ in $(0,1)$.
So it seems to me that this weaker notion is the appropriate notion for non-positive curvature in the normed case and I think also for finsler manifolds. I have never studied finsler geometry, but I am very interested in studying metric geometry from this approach. And I do not know where I should start.
My question is: What is a good introductory book about finsler manifolds from the metric geometry point of view? What is a good introductory book for the Busemann's approach? If there was not an introductory book available, a reference to an advanced one along with references that cover the necessary background would be very welcome.
Thanks in advance!