Let $(\mathbb{R}^{3+1},\eta)$ be the Lorentzian spacetime with units such that $\eta_{00}=-1$ and coordinates $(t,x,y,z)$ ($\eta=-\rm{d}t^{\otimes 2}+\rm{d}x^{\otimes2}+\rm{d}y^{\otimes2}+\rm{d}z^{\otimes2}$). Consider the hypersurface $t^2=x^2+y^2+z^2+1$ with $t\geqslant 0$. Compute the pullback metric on this submanifold (that is the induced metric on the manifold).
My thoughts
I need to consider the inclusion map, compute its derivatives w.r.t its components and use the ambient metric to deduce the induced metric on the hypersurface. There is only one possibility for the inclusion map, but I do not know how to parametrize such a manifold. How can we parametrize it in order to do calculus thereafter ?