- #1
Goklayeh
- 17
- 0
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!