Trying to determine the following:
If the invariant curves of two sub-manifolds are transversal, does this imply that the sub-manifolds are transversal?
An example I have of invariant curves, are invariant hyperbola. In special relativity $\Delta s^2 = \Delta x^2 - c \Delta t^2$ is an invariant quantity, which is in the form of a hyperbola.
My attempt: If the invariant curves of two sub-manifolds are transversal then the tangent spaces of the sub-manifolds are transversal, implying that the sub-manifolds are transversal. I also need help with the correct notation.
I have "Intro to Manifolds" by Tu, and "Semi-Riemannian Geometry" by O'Neill but I didn't find the answer in them.
Please leave a comment. Thanks.