Do carmo problem. exe 6 section 8

Question

Calculate the mean curvature and the sectional curvature of the umbilic hypersurface of the hyperspace. please introduce a book that calculate this. or show how i can calculate this. This is a part of exe 6 in chapter 8

Answer 1

The umbilic hypersurfaces of $ H^n $, up to isometries, are given by $ H^n\cap S $, where $ S $ is an euclidean $ (n-1)- $ sphere with center belonging to $ H^n $ or belonging to $ \partial H^n $. This fact can be proved using point 2) and 3) of the exercise that you cite.

Now let $ S $ be a such euclidean sphere of radius $ r $ and euclidean center $ p=(p_1,\ldots p_n) \in H^n \cup \partial H^n $. Then, as an hypersurface on the euclidean space, it is an umbilic hypersurface of principal curvature $ \lambda= \frac{1}{r} $.

Using formula given in 2) you can easily calculate (with $ \mu=\frac{1}{x_{n}^{2}} $ and $ \lambda= \frac{1}{r} $ ) the principal curvature of $ S $ as an hypersurface of $ H^n $. It is given by $ \overline{\lambda}=\frac{p_n}{r} $.

Therefore the mean curvature is $ H = \overline{\lambda} $ and the sectional curvature follows immediately by Gauss equation.