Angles on a riemannian manifold

Question

How can you compute the sum of the angles on a non-euclidean surface, on the surface of a $S^2$ sphere for example ?

I know that (for an elliptic geometry) if the triangle is small enough, the sum would be $180$ degrees, and grow with the size of the triangle, so there might be an integral over the triangle involved, and the metric tensor would be of some use at one point...

So if you could give the outline and directions, it will be a good start to know which notions to check and learn, even if I do not understand the answer immediately.

Thank you for your help,

JD

Check out http://en.wikipedia.org/wiki/Angular_deficiency and http://en.wikipedia.org/wiki/Gauss-Bonnet_theorem. — joriki, Jul 16 '11 at 07:00
@ks0830: Welcome to the site. I've converted your answer to a comment (it sounded like this is what you intended). Also, if you register your account, you will gain the ability to delete your own answers. — Zev Chonoles, Jul 16 '11 at 07:23
Note that it's not true in elliptic geometry that "if the triangle is small enough, the sum would be $180$ degrees." On a surface with positive Gaussian curvature, the angle sum of a geodesic triangle is always strictly greater than $180^\circ$, as you can see from the formula quoted from my book in TheGeekGreek's answer. What is true, though, is that as the area of the triangle gets smaller, the angle sum asymptotically approaches $180^\circ$. — Jack Lee, Dec 27 '16 at 23:43

score 2 · Accepted Answer · answered Dec 27 '16 at 23:24

A good way to start would be having a look into the book Riemannian Manifolds - An Introduction to Curvature by John M. Lee. As an example there is an exercise where you prove that for a geodesic triangle on a orientable Riemannian $2$-manifold $(M,g)$ we have the formula $$\varphi_1 + \varphi_2 + \varphi_3 = \pi + KA$$ where $\varphi_i$ denotes an interior angle of the triangle, $K$ is the Gaussian curvature of $g$ and $A$ is the area of the region bounded by the triangle. This is a direct application of the Gauss-Bonnet formula which also leads to the Gauss-Bonnet theorem.

Angles on a riemannian manifold

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