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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).

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You mention sectional curvatures but this is not enough, as the disks $D_i$ may have curved extrinsic geometry. They may even be flat and you still can't get any lower bounds without some control over the second fundamental form.

If you do have bounds on the second fundamental form, it should be easy to get estimates, because essentially you are working with curves. Take a pair of points of intersection and connect them by geodesics $\gamma_i\subset D_i$. The curve $\gamma_i$ is not a geodesic in the ambient space $M$, but its geodesic curvature as a curve in $M$ is controlled by the size of the second fundamental form of $D_i\subset M$. Assuming for simplicity that $M$ is Euclidean, you are reduced to finding a lower bound for the size of a "digon" whose angle is bounded from below, and whose sides have curvature bounded from above.

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  • $\begingroup$ I apologize for not having mentioned this explicitly in the problem statement, but I had intended for $D_1, D_2$ to have the induced metric from their embedding into $(M,g)$. What you've said is true, however, if the embeddings have nothing to do with the intrinsic geometry we place on the $D_i$. $\endgroup$ – A Blumenthal Jul 23 '13 at 20:41
  • $\begingroup$ I think you misunderstood my answer. Working with the induced metric, one can't get lower bounds without controlling the extrinsic geometry of the imbedding. Consider, for example, the graph of the function $\sin nx$. For large $n$, this will have arbitrarily close zeros, even though you can equip the graph with the metric induced from the Euclidean embedding. Note that, in the higher dimensional case, the sectional curvature of the embedded surface does not control the extrinsic geometry. $\endgroup$ – Mikhail Katz Jul 24 '13 at 6:54
  • $\begingroup$ I agree, I have misunderstood your answer. Thank you for your patience. I'm going to edit the problem statement now. $\endgroup$ – A Blumenthal Jul 24 '13 at 15:53

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