Planar subspaces are intrinsically CAT(0)

Abstract

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