Total curvature of curves in Riemannian manifolds
The global properties of the curvature of curves embedded in \({\mathbb{R}}^2\) or in \({\mathbb{R}}^3\) have been studied in many papers. In the present paper, these properties are studied for curves embedded in arbitrary Riemannian manifolds. The integral curvature of a curve is approached using geodesic polygons inscribed in the curve. For non-positive sectional curvature of the manifold, the integral curvature of a curve is the least upper bound of the total curvatures of geodesic polygons inscribed in the curve. Examples are given such that this property fails for positive sectional curvature. For oriented surfaces, the curvature can be equipped with the sign and the above result holds for the signed total curvature. At the end, it is proved that the least upper bound for signed total curvature among simple inscribed polygons for curves closing a simple domain on surfaces of positive curvature is necessarily \(2\pi\).
scientific article; zbMATH DE number 5687509
scientific article