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Let $M^n\subseteq {\mathbb{R}}^{n+1}$ be a hypersurface. Compute the sectional curvatures in all planes which are spanned by two eigenvectors $X_i, X_j$ of the Weingarten map. Also compute the Ricci tensor Ric($X_i, X_j$) and the scalar curvature.

Hint: Gauss equation

Thank you for your help.

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Ok some formulae to get you started:

The Gauss eqution reduces to $$0= \langle R(X_i, X_j)X_j, X_i\rangle +\langle s(X_i, X_j),s(X_j,X_j) \rangle - \langle s(X_i, X_i),s(X_j,X_j) \rangle$$ since the curvature tensor of the ambient space $\mathbb{R}^n$ vanishes. The sectional curvature of the subspace of $T_p M$ spanned by $X, Y$ is, by defintion, $$K(X, Y) = \frac{\langle R(X,Y)Y,Y\rangle}{A(X,Y)^2} $$ where $A(X, Y)$ denotes the area of the parallelogram spanned by $X$ and $Y$. Since you are asked to calculate the sectional curvatures for eigenvectors of the Weingarten map, you may assume they are of length $1$, so $A(X_i,X_j) =1$ if $i\neq j$. In addition, $$s( X_i,X_j) = II(X_i,X_j)\nu_M = -\langle d\nu_M(X_i), X_j\rangle$$ Now use the fact that the $X_i$ are eigenvectors of the Weingarten map.

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  • $\begingroup$ Thank you for your answer! $\endgroup$ – Peter Zin Jun 6 '12 at 10:12

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