# Cut on hypersurfaces and angular defects

I like very much the elementary property that if one cuts a geodesic triangle onto a sphere (one can use 3 plans that contain $0$). The cut surface of the sphere is given by the sum of the angles of this triangle minus $\pi$. More generaly, we have the Gauss Bonnet Theorem saying that on any surface this is equal to the integral of the Curvature of the cut surface.

Because Gauss Bonnet is valid on higher dimensional manifold, I wonder if we can state a similar "elementary" result on higher dimension. For exemple if one cut $\mathbb{S}^4$ using 5 hyperplan, can we obtained the volume of the cut with a formula using only the angles between the hyperplans?

My other motivation is that this summer I tried to explain (and almost prove) the Gauss Bonnet Theorem to 15 years old student with the anglular defects https://en.wikipedia.org/wiki/Angular_defect#Descartes.27_theorem . I would like to know if it is also possible to defined (such that a 15 years old boy can understand) an "angular defect" on higher dimensional polyhedron and such that we have a similar theorem : "the sum of the (high dimensional) angular defects are equal to the volume of the hypersphere"

• There is a proof of Gauss-Bonnet in higher dimensions by Allendoerefer-Weil that might be more along the lines that you want, see McMullen's notes on 'cone manifolds' : math.harvard.edu/~ctm/papers/home/text/papers/gb/gb.pdf Sep 21, 2018 at 9:04