I need help to find isometries of a submanifold of a semi-Riemannian manifold. To be crystal clear, let me start with what I mean by an isometry.
$\textbf{Definition:}$ Let $(M,g)$ be a semi-Riemannian manifold.
i) $F:M\longrightarrow M$ is called an $\textbf{isometry}$ iff $F$ is an diffeomorphism and $F^\star g=g$ where $F^\star g$ is the $\textit{pullback}$ metric.
On the other hand,
ii) If $S\subseteq M $ is a submanifold (embedded or immersed), then $\mathcal{i}^\star g$ is a metric on $S$, where $\mathcal{i}:S\hookrightarrow M$ is the $\textit{inclusion}$ map and $\mathcal{i}^\star g$ is the $\textit{pullback}$ of the metric $g$ by $F$.
Now, consider the manifold $(S,\mathcal{i}^\star g)$, then an isometry of $S$ is a diffeomorphism $F:S\longrightarrow S$ such that $F^\star (\mathcal{i}^\star g)=\mathcal{i}^\star g$. The question is the following,
$\textbf{Question:}$ Is there any relation between isometries of the ambient manifold $M$ and isometries of the submanifold $S$ with induced metric?
The question arises when I try to find isometries of the pseudo-sphere. That is, consider the following semi-Riemannian manifold,
$(\mathbf{R}^{k+l},g)$ where $\mathbf{R}^{k+l}$ has Euclidean topology and standard smooth structure but the metric $g$ is defined as $g=\eta_{\mu\nu}dx^\mu dx^\nu$ where for any $\mu,\nu \in I_{k+l}, \eta_{\mu\nu}$ is the component in the $\mu$'th row and $\nu$'th coloumn of the matrix $\eta=diag(\underbrace{+,..,+}_{\text{k many}},\underbrace{-,..,-}_{\text{l many}})$.
Then, one can show that any isometry $F: \mathbf{R}^{k+l} \longrightarrow \mathbf{R}^{k+l}$ can be written as \begin{equation} F^\nu(x)= \Lambda^\nu_\lambda x^\lambda + a^\nu \ \text{for all} \ \nu \in I_{k+l}, x \in \mathbf{R}^{k+l} \end{equation} where $\Lambda^\nu_\lambda$'s satisfy \begin{equation} \eta_{\mu\nu}=\Lambda^\alpha_\mu \Lambda^\beta_\nu \eta_{\alpha\beta}. \end{equation}
So far, I was able to prove every claim. Now the tough part comes,
Define the subset which will be called $\textbf{pseudosphere}$ as follows, \begin{equation} \mathbf{S}^{(k-1,l)}=\{x \in \mathbf{R}^{k+l} : \eta_{\mu \nu} x^\mu x^\nu = 1 \} \subseteq \mathbf{R}^{k+l}. \end{equation} One can then show that it is a submanifold of $\mathbf{R}^{k+l}$. Thus, the metric $\mathbf{R}^{k+l}$ induces a metric on $\mathbf{S}^{(k-1,l)}$ and as you may guess, we ask the isometries of $\mathbf{S}^{(k-1,l)}$.
Obviously, if $a^\nu=0$ for any $\nu$, then the isometry $F^\nu(x)= \Lambda^\nu_\lambda x^\lambda$ of $\mathbf{R}^{k+l}$ can be restricted to an isometry of $\mathbf{S}^{(k-1,l)}$. That is, $F|_{\mathbf{S}^{(k-1,l)}}$ gives an isometry of $\mathbf{S}^{(k-1,l)}$ if $a^\nu=0$. However, These are just some isometries of $\mathbf{S}^{(k-1,l)}$. How can we be sure that there is no more isometries but those achieved by restricting isometries of $\mathbf{R}^{k+l}$?
$\text{Note:}$ I need a way that does not require to explicitely write the induced metric on $\mathbf{S}^{(k-1,l)}$ in a coordinate chart.
Thanks in advance for your help.