Peano property, two dimensional triangles, constant curvature

Abstract

In the present note, we discuss the connection of the convex geometric and the analytic properties of the underlying manifold. As an application of the main results we show that in Riemannian manifolds of dimension at least three, local versions the Peano property, the drop completeness, or the “thinness” of triangles are equivalent with having constant curvature, and global versions characterize Euclidean and hyperbolic spaces among Cartan–Hadamard manifolds.