Let $(M,g)$ be a connected Riemannian manifold, and let $x$ and $y$ be two points of $M$.
The Riemannian distance between $x$ and $y$ is defined as the infimum of lenghts of all piecewise smooth curves connecting $x$ and $y$, so by definition the distance can be approached by taking the lenghts of some sequence of piecewise smooth curves. But I want to know (and I am not sure if this is true) if the curves of that sequence can be chosen to be geodesics. If I word it differently, does there exist a sequence of piecewise smooth geodesic curves such that their lengths converge to the distance between $x$ and $y$?
Best Answer
Proof by picture
Proof by unnecessary complicated sequence of symbols
Let $x$ and $y$ be two distinct points in the connected Riemannian manifold $(M,g)$. Let $\gamma\colon [0,L]\to M$ be any piecewise-$\mathcal{C}^1$ path joining $x$ and $y$, parametrised by arc length. Let us prove that there exists a piecewise-geodesic path joining $x$ and $y$ with length less than that of $\gamma$.
For any $p \in \gamma\left([0,L]\right)$, there exists $\varepsilon_p>0$ such that the ball $B(p,\varepsilon_p)$ is a normal neighbourhood of $p$. By compactness of $\gamma\left([0,L]\right)$, there exists $\varepsilon > 0$ such that for any $p \in \gamma\left([0,L]\right)$, $B(p,\varepsilon)$ is a normal neighbourhood of $p$.
Choose $N>0$ large enough so that $\frac{L}{N} < \varepsilon$. Consider the sequence of points $\{x_i\}_{i=0,\ldots, N}$ defined by $x_i = \gamma\left(i\frac{L}{N}\right)$. For $i\in \{0,\ldots,N-1\}$, we have $d(x_i,x_{i+1}) \leqslant \frac{L}{N}$, since $\gamma$ is parametrised by arc length. Hence, $x_{i+1}$ belongs to the normal neighbourhood $B(x_i,\varepsilon)$ of $x_i$. Let $c_i$ be the unique geodesic segment joining $x_i$ to $x_{i+1}$. It has length less than $\frac{L}{N}$.
Define $c$ to be the concatenation of $c_0,\ldots,c_{N-1}$. Then $c$ is a piecewise-geodesic joining $x$ to $y$, of length less than $L$, which is the length of $\gamma$.
As a corollary, one can find a piecewise-geodesic joining $x$ to $y$ of length less than $d(x,y) + \delta$, for any $\delta > 0$.