Suppose $(M, g)$ is a connected Riemannian manifold, $S \subseteq M$ is a connected embedded submanifold, and $\tilde{g}$ is the induced Riemannian metric on $S$. \begin{enumerate}[label=(\alph*)] \item Prove that $d_{\tilde{g}}(p, q) \geq d_{g}(p, q)$ for $p, q \in S$. \item Prove that if $(M, g)$ is complete and $S$ is properly embedded, then $(S, \widetilde{g})$ is complete. \item Use (b) together with the Whitney embedding theorem to prove (without quoting Proposition 13.3 or \href{./problem 13.17}{Problem 13-17}) that every connected smooth manifold admits a complete Riemannian metric. \end{enumerate}

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Introduction to Smooth Manifolds

Problem 13.18 (Every connected smooth manifold admits a complete Riemannian metric)

Suppose $(M, g)$ is a connected Riemannian manifold, $S \subseteq M$ is a connected embedded submanifold, and $\tilde{g}$ is the induced Riemannian metric on $S$ .

(a): Prove that $d_{\tilde{g}} (p, q) \geq d_{g} (p, q)$ for $p, q \in S$ .
(b): Prove that if $(M, g)$ is complete and $S$ is properly embedded, then $(S, \tilde{g})$ is complete.
(c): Use (b) together with the Whitney embedding theorem to prove (without quoting Proposition 13.3 or Problem 13-17) that every connected smooth manifold admits a complete Riemannian metric.

Problem 13.18 (Every connected smooth manifold admits a complete Riemannian metric)

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