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How can I prove that there are no compact semi-Riemannian hypersurfaces in semi-euclidean space $\mathbb{R}_v^n$ of index $v$ with $0??.

Thanks for any help!!

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Suppose $M \subseteq \mathbb{R}_v^n$ is a compact semi-Riemannian manifold.

Let $\{e_1, \ldots, e_n\}$ be the canonical basis for $\mathbb{R}_v^n$ such that $e_1, \ldots, e_v$ are timelike and $e_{v+1}, \ldots, e_n$ are spacelike.

The functions $f, g : M \rightarrow \mathbb{R}$ given by $f(x) = x_1$ and $g(x) = x_n$, where $x = \sum x_ie_i$, must have critical points $p_f, p_g \in M$, respectively, by compactness. By construction, $T_{p_f}M$ has index $v-1$, but $T_{p_g}M$ has index $v$.