# Reference request: Introduction to Finsler manifolds from the metric geometry point of view (possibly from the Busemann's approach)

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:

1. $$X$$ has curvature $$\leq\kappa$$ in Alexandrov's sense, for some real number $$\kappa$$.
2. $$X$$ has curvature $$\leq 0$$ in Alexandrov's sense.
3. 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:

1. $$X$$ has non-positive curvature in the Busemann's sense.
2. $$X$$ is uniquely geodesic, that is, every pair of points is joined by a unique geodesic (the linear segment between them).
3. $$X$$ 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.

I suggest the paper "On intrinsic geometry of surfaces in normed spaces - Burago and Ivanov" which deals second variation on Finsler surface, isometric embedding of Finsler surface and a geodesic line on Finsler surface.

• I am reading it and seems extremely interesting, thanks! I am also looking for a textbook – Dante Grevino Mar 6 '19 at 16:48

I have found the following references:

1. An introductory textbook by A. Papadopoulos about the Busemann's approach: Metric Spaces, convexity and non-positive curvature.

2. A textbook by H. Busemann: The geometry of geodesics

3. Two interesting papers by H. Busemann: The geometry of finsler spaces and Spaces with non-positive curvature.

EDIT:

1. User @Matt F. in MathOverflow recommend The Geometry of Geodesics (1955), which includes almost all of the content of the other two papers mentioned above, and another Busemann's book, Recent Synthetic Differential Geometry (1970), with more advanced follow-ups from the intervening years.