Recently I asked: Invariant curves...
Let $\Bbb R^{1,1}$ be a semi-Riemannian manifold called Minkowski-space. Consider the invariant timelike hyperbolas of the lower branch (rectangular hyperbolas).
$\varphi:\Bbb R^{1,1}\to\Bbb R^{1,1}_+$ with $\varphi(x,y)=(e^x,e^y)$ is a diffeomorphism. The invariant timelike hyperbola of the lower branch map to $(0,1)^2.$
Let $\Bbb R^{1,1}$ be another copy of Minkowski-space. Consider the invariant spacelike hyperbolas of the lower branch.
Use a diffeomorphism $\psi:\Bbb R^{1,1}\to\Bbb R^{1,1}$ with $\psi(x,y)=(-e^{-x},e^y).$ The invariant spacelike hyperbola of the lower branch map to $(-1,0)^2.$
Translate the invariant spacelike hyperbola to $(0,1)^2.$ Now the invariant timelike "hyperbola" of the lower branch are transversal to the invariant spacelike "hyperbola" of the lower branch, in $(0,1)^2.$
In pink are the invariant spacelike hyperbola and in bluish-green are the invariant timelike hyperbola:
How do you prove this transversality?
Doesn't the region in $(0,1)^2$ form a manifold? How can one formalize this manifold?
In order to prove the transversality of the invariant spacelike and timelike "hyperbola" I need to prove that the tangent spaces of each generate the ambient space. So for submanifolds of $M$ being $S$ and $T,$ standing for invariant spacelike and timelike "hyperbola" respectively, I need to prove that:
$$\forall p\in S \cap T,~ T_pM=T_pS + T_pT $$
I believe that the tangent spaces span a space that is isometric to $\Bbb R^{1,1},$ hence the ambient manifold must be at least isometric to $\Bbb R^{1,1}$ in the bounded region $(0,1)^2.$
I'm having trouble connecting all the dots and writing down the rigorous proof though.