I am learning general relativity, and the geometry is very hard for me.
The problem:
Use the Gauss equation to show that the hyperboloid, given by equation $x^2 + y^2 + z^2 - t^2 = -1$ in Minkowski space with metric $-dt^2+dx^2+dy^2+dz^2$ has constant sectional curvature $-1$.
My progress:
Let $g_{ab} = \mathrm{diag}(1,1,1,-1)$ be the metric of the Minkowski space, and $h_{ab}$ be the (induced) metric of the 3-submanifold. We can get it with the formula $$ h_{ab} = g_{ab} + n_a n_b, $$ where $n$ is a unit normal vector of the 3-submanifold. In our case, $$n^x = x, n^y = y, n^z = z, n^t = t,\\ n_x = x, n_y = y, n_z = z, n_t = -t.$$ With this, I can write the 4x4 matrix $h_{ab}$.
However, I don't know how to use this 4x4 metric to calculate the curvatures. In my opinion, I need a 3x3 metric to do so.