I came across the following problem while reading some literature in Dynamical Systems.
Say I have an ambient Riemannian manifold $(M,g)$ and a pair of transverse embedded disks $D_1, D_2$ of complementary dimension.
With some conditions, the common result in this situation is that the set $D_1 \cap D_2$ is finite. For my purposes, though, what I'd like is to have a lower bound on the distance between any two points of $D_1, D_2$.
Obviously I need a notion of distance, hence the Riemannian metric $g$ on the ambient space $M$. However, it's also clear that I need some way of controlling the 'robustness' of the transversality of $D_1, D_2$, and also some way of controlling how wild the embeddings of $D_1, D_2$ are.
Question 1: Given a uniform bound on the minimal angle $\theta_z$ between $T_z D_1$ and $T_z D_2$ in the metric $g$, where $z \in D_1 \cap D_2$, and some control over the second fundamental forms of $D_1, D_2$, is there a way to bound from below the minimal distance between two distinct points of $D_1 \cap D_2$?
Clarification: The situation in my head is where $D_1$ is practically flat and $D_2$ can be very curved. What I want is a lower bound on either of the $D_1$ or $D_2$ distances between elements of $D_1 \cap D_2$.
Question 2: Is there a name for this kind of problem?
I feel like the methods used in the solution to this problem will be simple, but the overall estimate will likely be somewhat more tedious than an exercise. I would be very happy to be pointed to a geometry reference that carries out an example of this kind of computation.
EDIT: One needs control over the extrinsic geometry of the embedding, and not just the intrinsic curvatures of the disks $D_i$ (see user72694's answer below).