Planar Pseudo-geodesics and Totally Umbilic Submanifolds

Steen Markvorsen, Matteo Raffaelli

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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 ι : M ,→ 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 R3.
Original languageEnglish
Article number53
JournalJournal of Geometric Analysis
Volume34
Issue number2
Number of pages15
ISSN1050-6926
DOIs
Publication statusPublished - 2024

Keywords

  • Isotropic immersion
  • Normalized mean curvature vector
  • Planar curve
  • Planar geodesic immersion
  • Pseudo-geodesic
  • Totally umbilic submanifold

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