Planar pseudo-geodesics and totally umbilic submanifolds

TL;DR

This paper characterizes totally umbilic isometric immersions between Riemannian manifolds using planar pseudo-geodesics, extending classical results and providing new insights into the geometry of such immersions.

Contribution

It introduces planar pseudo-geodesics as a new tool to characterize totally umbilic immersions with parallel normalized mean curvature vector, extending classical surface results.

Findings

Characterization of totally umbilic immersions via planar pseudo-geodesics.

Existence and uniqueness of planar pseudo-geodesics.

Extension of classical surface results to higher codimension.

Abstract

We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e., those having nonzero mean curvature vector and such that the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. Such characterization is based on a family of curves, called planar pseudo-geodesics, representing a natural extrinsic generalization of both geodesics and Riemannian circles: being planar, their Cartan development in the tangent space is planar in the ordinary sense; being pseudo-geodesics, their geodesic and normal curvatures satisfy a linear relation. We study these curves in detail and, in particular, establish their local existence and uniqueness. Moreover, in the case of codimension-one immersions, we prove the following statement: an isometric immersion $\iota \colon M \hookrightarrow Q$ is totally umbilic if and only if the extrinsic shape of every geodesic of $M$ is planar. This extends a well-known result about surfaces in $\mathbb{R}^{3}$.