In this wikipedia page https://en.wikipedia.org/wiki/Sectional_curvature I've found a Theorem by Toponogov which characterizes Riemannian spaces of non positive curvature:

Thm 1 If a complete Riemannian manifold is of non positive curvature, then for all sufficiently small geodesic triangles $xyz$, given $m$ the midpoint of the side $xy$ it is verified

$d(z,m)^2\le \frac 1 2 d(z,x)^2+\frac 1 2 d(z,y)^2-\frac 1 4 d(x,y)^2$

where $d$ is the distance induced by the Riemannian metric.

A few lines below it is stated that a simple consequence of this theorem is the following:

Thm 2 A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function $f_p(x):=d(p,x)^2$ is 1-convex (which I think should mean that, for every geodesic $\gamma(t)$, the function $f_p(\gamma(t))-t^2$ is convex).

I have two questions:

1) Is the inverse implication of Thm 1 true (if the inequality is verified for every triangle then the space has non positive curvature)?

2) Do you know how to derive Thm 2 from Thm 1 (in the wikipedia page it is written that it's a simple consequence..)?

Also can you give me a good reference on these Toponogov's theorems?


One useful reference is Theorem 1A.6 (page 173) of

M.Bridson, A.Haefliger "Metric spaces of nonpositive curvature". They prove that a Riemannian is (locally) CAT(k) in the sense of Alexadrov iff sectional curvature is $\le k$ (in the Riemannian sense).

Incidentally, this book also provides a good answer to your Question 2. At some point I tried to find a good reference for this result and ended up concluding that the best one is Bridson-Haefliger.

Equivalence of the inequality in Theorem 1 and Alexandrov's CAT(0) condition, I think, is also in Bridson and Haefliger, take a look. This will give a positive answer to your Question 1. Convexity of the distance function in (globally) CAT(0) spaces id definitely in Bridson and Haefliger. To see that convexity implies CAT(0) note that it implies the inequality in Theorem 1.

Incidentally, I do not think Theorem 1 is due to Toponogov, I am nearly sure it is due to Alexadrov (but maybe it was proven by Busemann or Rauch earlier):

A. D. Alexandrow, Über eine Verallgemeinerung der Riemannschen Geometrie. (German) Schr. Forschungsinst. Math. 1 (1957), 33–84.

Toponogov proved some powerful "complementary results" about curvature bounded below.


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