A question about sum of angles in a non-positive curvature Riemannian manifold

Question

Suppose on a non-positive curvature Riemannian manifold,we have a geodesic triangle $\triangle abc$ ,and counterpart edges donates $\alpha,\beta,\gamma$.

If now I get

$$ a^2 \geqq b^2+c^2-2bc cos\alpha $$

How can I induce that $\alpha+\beta+\gamma \leqq \pi$?

I will appreciate your help!

Remember that $\cos$ is a decreasing function on $[0,\pi ]$. — Jeppe Stig Nielsen, Dec 19 '14 at 13:52
Strictly speaking, your first inequality holds only if the manifold is complete and simply-connected. — Moishe Kohan, Dec 19 '14 at 16:11

score 1 · Answer 1 · answered Dec 19 '14 at 13:54

1

Take a Euclidean triangle with sides $a,b,c$ (same as the hyperbolic triangle) and angles $\alpha',\beta',\gamma'$. Comparing your inequality with $$a^2=b^2+c^2-2bc\cos\alpha'$$yields$$\alpha\leq\alpha'.$$

answered Dec 19 '14 at 13:54

Amitai Yuval

19,308

A question about sum of angles in a non-positive curvature Riemannian manifold

1 Answers1