For a compact submanifold $\mathcal{M}$ embedded in Euclidean space, its sectional curvature is positive, given $x,y\in \mathcal{M}, \eta \in T_x\mathcal{M}$, where $T_x\mathcal{M}$ denote the tangent space on $\mathcal{M}$ at $x$, I want to prove $$ \|P_{M}(x+\eta) - y \| \leq \|x+\eta - y\|, $$ where $P_{\mathcal{M}}$ is the orthogonal projection on $\mathcal{M}$. I guess that the manifold with positive sectional curvature is locally similar to the convex set (such as a sphere), and the projection has 1-Lipschitz properties.
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