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My question is an exercise in Peter Petersen "Riemannian Geometry" Chapter 5 #10

Let $N \subset M$ be a submanifold of Riemannian manifold $(M,g)$.

(a) The distance from N to $x \in M$ is defined as $d(x,N) = \inf\{ d(x,y)\ |\ p \in N\}$. A unit speed curve $\sigma : [a,b] \to M$ with $\sigma(a) \in N,\sigma(a)$ and $l(\sigma) = d(x,N)$ is called a segment from $x$ to $N$. Show that $\sigma$ is also a segment from $N$ to any $\sigma(t),t. Show that $\sigma'(a)$ is perpendicular to $N$.

(b) Show that if $N$ is a closed subspace of $M$ and $(M,g)$ is complete, then any point in $M$ can be joined to $N$ by segments.

(d) Show that $d(\dot \ ,N)$ is smooth on a neighborhood of $N$ and that the integral curves for its gradient are the geodesics that perpendicular to $N$.

Please give me a answer as complete as possible,. Thank you very much!

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    A hint for the part b can be: you can use the first variation formula for the energy function (see for instance DoCarmo or either Petersen). The variation to be considered is that wich have "final vector" tangent to the submanifold. The condition on minimal distance will be useful because a minimal of the energy makes the derivative of energy zero, and hence you can get "zero inner-products" in order to prove orthogonality.2012-04-21

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