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Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the coordinates of each point of $M$ in the ambient space) $M$ to $M'$, and consider the "image" $\gamma'$ (of $\gamma$) in $M'$. It seems very intuitive to me that there is a geodesic $\beta$ in $M'$ very "near" $\gamma'$. However, this question is vague (I hope the idea is clear) and I am unsure as to how to formulate the notions of "nearness" so that this "statement" will allow a proof. Furthermore, it also seems intuitively true that if $\gamma$ is a periodic geodesic, we can arrange that $\beta$ is also periodic.

So, my questions are:

1) How are such "perturbation problems" usually formulated? More specifically, how can we formulate the appropriate notion of "nearness" in this particular problem?

2) How can we prove the statement that there exists a geodesic near the perturbed curve in the new manifold?

Thanks in advance for any suggestion.

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  • $\begingroup$ Usually one uses the tubular neighbourhood theorem to define a topology on a space of manifolds, like your situation. This turns it into a Frechet or Hilbert manifold depending on your precise setup. For (2), one uses the smooth dependence on initial conditions for the geodesic flow. $\endgroup$
    – Ryan Budney
    Commented Oct 15, 2014 at 3:17
  • $\begingroup$ @RyanBudney Thanks a lot for your comment. I kind of understand your statement about the geodesic flow (and I will look it up further). But could you tell me where I can learn how the topology on the space of manifolds is defined and how it becomes a Frechet/Hilbert manifold? (I am aware of the statement of the tubular neighborhood theorem) $\endgroup$
    – Rookie
    Commented Oct 15, 2014 at 4:08
  • $\begingroup$ Start with Hirsch's Differential Topology text. It's covered quite well there. $\endgroup$
    – Ryan Budney
    Commented Oct 15, 2014 at 14:11

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