Let $C$ be the cone of $R^{n+1}$ defined by $$ C=\lbrace x=(x_1,..,x_{n+1})|\quad \sum_1^n x_i^2 >x_{n+1}^2 \rbrace.$$
I am interested in the set $\mathcal{C}$ of $n\times n$ matrices $M$ which preserve the cone $$ M.C \subset C $$
I am trying to figure out what are the properties of this set, and I search lecture notes / references describing it. Can anyone help?
Thank you.
The matrices preserving the (relativity) quadratic form :
$$q(x)=x_1^2+x_2^2+\cdots+x_n^2-x_{n+1}^2$$
and in particular preserving its sign are the matrices verifying:
$$MQM^T=Q$$
where $$Q=diag(\underbrace{1,1...1}_{n \ \text{terms}},-1)$$
These matrices constitute a classical group named $O(n,1)$.
For more see here.
Remark: In the case $n=1$, the group $0(1,1)$ is constituted of matrices of the form:
$$R_a=\begin{pmatrix}\cosh(a)&\sinh(a)\\ \sinh(a)&\cosh(a)\end{pmatrix}$$