This question may be coming from a bit of ignorance, so if its obvious, I apologize. (My Riemannian geometry background up to this point is only what I needed to study general relativity).
Is there a way to describe two different (or distinct) Riemannian metrics "interacting" with each other. For example, lets say I have a metric $h_{\alpha\beta}$ and a completely different one, $g_{\alpha\beta}$, and I want to put them together, almost in a patch-work manner, to describe them interacting (like I have two $S^2$ one with radius $a$ and another with radius $r$ and I am pushing them together to make them one).
I am not even sure if this question makes sense, or if it is trivial (it may very well be), but any help is appreciated. (I may just be missing some math-vocabulary).