Planar subspaces are intrinsically CAT(0)

Abstract

Let $M^2_{\kappa}$ be the complete, simply connected, Riemannian 2-manifold of constant curvature $\kappa \leq 0$. Let $E$ be a closed, simply connected subspace of $M^2_{\kappa}$ with the property that every pair of points in $E$ is connected by a rectifiable path in $E$. We show that under the induced path metric, $E$ is a complete CAT($\kappa$) space. We also show that the natural notions of angle coming from the intrinsic and extrinsic metrics coincide for all simple geodesic triangles.