Let $H_c:=\{x \in \mathbb{R}^n_{\nu} \vert <x,x>=c \}$. I want to proof the following statements:
1) $H_c$ is a pseudo-Riemannian hypersurface $\Leftrightarrow$ $c \neq 0$.
2) $H_0 \setminus \{0\}$ is a smooth hypersurface, but not a pseudo-Riemannian hypersurface.
3) The Weingarten map of $H_c$ with $\zeta(x)=x$ is $S^{\zeta}=-Id$
What I have so far:
1) I know that for some smooth $F \in C^{\infty}(\mathbb{R}^n_{\nu})$ it holds that $F^{-1}(0)$ is a smooth manifold if $0$ is a regular value of $F$ (Though I am not sure if this is equivalent). Define $F(x):= \sum\limits_{i=1}^{\nu} -x_1^2 + \sum\limits_{j=\nu+1}^n x_j^2 -c$
Now $grad F=(-2x_1 ... -2x_{\nu} \ 2x_{\nu+1} ... 2x_n)^T$, so $F^{-1}(0)$ is a smooth manifold if $x \neq 0$ so $c \neq 0$. Is that correct?
2)Now the first part follows with the same argumentation as above, but I'm not sure how to do the second part. I need to find some $v,w\neq 0 \in T_pH_c$, s.t. $g_p(v,w)=0$.
3) I need to show that $\zeta$ is a unit normal and that $-\nabla_X \zeta=-Id$ for all $X \in \mathfrak{X}(H_c)$ right? Now the second part is obvious, because $\nabla_X \zeta = \zeta(X)=X$, is that correct? And how to show the first part?
Thanks already in advance for any answers!