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Projections from Tubular Neighborhoods

  1. Aug 24, 2012 #1
    Hello! Could anybody give me some hint with the following problem? Consider a smooth, compact embedded submanifold [itex]M = M^m\subset \mathbb{R}^n[/itex], and consider a tubular neighborhood [itex]U = E(V)\supset M[/itex], where [itex]E: (x, v) \in NM \mapsto x + v \in M[/itex] is a diffeomorphism from a open subset of the normal bundle [itex]NM[/itex] of the form [itex]V = \left\{(x, v) \in NM \: : \: \left| v \right| < \delta\right\}[/itex]. We know that [itex]r:=\pi \circ E^{-1}[/itex] is a smooth retraction, where [itex]\pi:(x, v) \in NM \mapsto x \in M[/itex] is the projection. How can I prove that if [itex]y \in U[/itex], where [itex]U[/itex] is a sufficiently small tubular neighborhood, then [itex]r(y)[/itex] realizes the minimum of the distance from the points of [itex]M[/itex]? I just proved, following Lee's hints, that if [itex]y \in \mathbb{R}^n[/itex] has a closest point [itex]x \in M[/itex], then [itex]y - x \in N_x M[/itex], but I can't realize how to use this information!
     
  2. jcsd
  3. Aug 24, 2012 #2
    Maybe, I've solved: let [itex]y = x + v[/itex], for some [itex](x, v) \in V[/itex], and consider any other point [itex]p \in M[/itex], together with a curve [itex]\gamma: (-\epsilon, \epsilon) \to M[/itex] such that [itex]\gamma(0) = x, \dot{\gamma}(0) = v[/itex] joining [itex]x, p[/itex]. Without loss of generality, [itex]y = 0[/itex]. If [itex]f(t):=\frac{1}{2}\left|\gamma(t)\right|^2[/itex], then
    [tex]\dot{f}(0) = \left<\gamma(0), \dot{\gamma}(0)\right> = \left<x, v\right> = 0[/tex]
    since [itex]y - r(y) = x + v - x = v \in N_x M[/itex]. Since [itex]f[/itex] is convex, [itex]0[/itex] is a minimum, hence the thesis for the arbitrariness of the curve [itex]\gamma[/itex]. Am I wrong?
     
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