This question is a cross post from Math.SE. I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I also have found this related question, but in my opinion it is not a duplicate from mine.

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.

In Math.SE, user @HK Lee has suggested the paper On intrinsic geometry of surfaces in normed spaces by D. Burago and S. Ivanov. And I have found the following references, although I need the advice of the experts:

  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.

Thanks in advance!

  • 2
    $\begingroup$ Let me bring your attention to the paper arxiv.org/abs/1711.02951. It shows that for (regular) Finsler metrics, the Busemann non-positive curvature condition may be too strong to be interesting. Only Berwald metrics satisfy it. Berwald metrics are very special and closely related to Riemannian ones, for example in dimension 2 they are either Riemannian or flat. $\endgroup$ Sep 9, 2019 at 19:59

1 Answer 1


Busemann's 1955 book The Geometry of Geodesics is a great presentation of his approach. It includes almost all of the content of his other two papers mentioned in the post.

By contrast, Papadopoulos's book is only partly about Busemann's approach to metric geometry.

Busemann also wrote a 1970 book called "Recent Synthetic Differential Geometry" with more advanced follow-ups from the intervening years.

If you think "the weaker notion is the appropriate notion", you're likely to enjoy Busemann a lot.


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