Planar subspaces are intrinsically CAT(0)

TL;DR

This paper proves that certain planar subspaces in non-positive curvature manifolds are intrinsically CAT(0), with coinciding intrinsic and extrinsic angles, extending the understanding of geometric properties of such spaces.

Contribution

It establishes that closed, simply connected subspaces with rectifiable paths in non-positive curvature manifolds are complete CAT(0) spaces, linking intrinsic and extrinsic angle notions.

Findings

Subspaces are complete CAT(0) spaces under the induced path metric.

Intrinsic and extrinsic angles coincide for all simple geodesic triangles.

Extension of CAT(0) space properties to specific planar subspaces.

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.