We have the ambient space $\mathbb{R}^{n+1}$ with the metric $ds^2 = -dx_0^2+dx_1^2+...+dx_n^2$ and the submanifold $M = \{ x\in\mathbb{R}^{n+1} | -x_0^2+x_1^2+...+x_n^2=1\}$.
I would like to compute the Christoffel Symbold of $M$ with the induced metric. For this I'd like to compute the induced metric $h$, i.e. have the matrix $(h_{ij})$. I could reparametrize the above to $-t^2+x_1^2+...+x_n^2=1$ yielding $t^2=x_1^2+...+x_n^2-1$ or $t=\varepsilon(x_1^2+...+x_n^2-1)^{1/2} =:f(x)$ with $\varepsilon = \pm1$.
My idea was to compute $dt^2$ but I am a little confused. I know $\partial t/\partial x_k = 2\varepsilon x_k / f(x)$. I am not sure how to proceed.
The induced metric is given by $$ h_{ab}=\frac{\partial x^\mu}{\partial y^a}\frac{\partial x^\nu}{\partial y^b}g_{\mu\nu}, $$ where $x^\mu$ is the coordinates on the ambient space with metric $g_{mu\nu}$ and $y^a$ are coordinates on the submanifold with metric $h_{ab}$.
We choose coordinates such that $g_{\mu\nu}=\eta_{\mu\nu}=$diag$(-1,1\dots,1)$, and $y^a=x^a$ for $a=1,\dots,n$, we then have that $x^0=\sqrt{x_1^2+\dots+x_n^2-1}$ and $$ \frac{\partial x^\mu}{\partial y^a}=\begin{cases}\frac{x_a}{\sqrt{x_1^2+\dots+x_n^2-1}}\text{ if }\mu=0\\ \delta^{\mu}_a\text{ otherwise.}\end{cases} $$ We now plug in our formula to get \begin{equation}\begin{aligned} h_{ab}dy^ady^b&=-\sum_{a,b=1}^ndy^ady^b\frac{y_ay_b}{y_1^2+\dots+y_n^2-1}+\sum_{a=1}^ndy^ady^a\left(1+\frac{y^ay^a}{y_1^2+\dots+y_n^2-1}\right)\\ &=\sum_{a=1}^ndy^ady^a-2\sum_{a>b}dy^ady^b\frac{y_ay_b}{y_1^2+\dots+y_n^2-1}. \end{aligned}\end{equation}