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Mathematics
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Riemannian Geometry of Submanifolds: Volume II
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Riemannian Geometry of Submanifolds: Volume II

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (31 August 2022) | Viewed by 2111

Special Issue Editor

Prof. Dr. Luc Vrancken

E-Mail Website
Guest Editor
1. LAMAV, ISTV2 Université de Valenciennes, Campus du Mont Houy, CEDEX 9, 59313 Valenciennes, France
2. KU Leuven, Department of Mathematics, Celestijnenlaan 200B – Box 2400, BE-3001 Leuven, Belgium
Interests: submanifold theory; Lagrangian submanifolds; affine differentiable geometry; Kaehler and nearly Kaehler geometry
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Special Issue Information

Dear Colleagues,

Submanifold theory can be thought of as a generalization of the study of surfaces in the 3-dimensional Euclidean space. In the general theory, both the dimension of the submanifold and the codimension, which is the difference between the dimension of the ambient space and the dimension of the submanifold, can be arbitrarily high, and the ambient space does not need to be flat. A key problem in the theory is to study the relations and the interplay between intrinsic invariants, which only depend on the submanifold as a manifold itself, and extrinsic invariants, which depend on the immersion, i.e., on the shape that the submanifold takes in the ambient space.

Prof. Dr. Luc Vrancken
Guest Editor

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Keywords

  • Minimal submanifolds 
  • Pseudo-Riemannian geometry of submanifolds 
  • Affine differential geometry 
  • Lagrangian submanifolds 
  • CR submanifolds of almost complex manifolds or almost contact manifolds

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Published Papers (1 paper)

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Research

12 pages, 286 KiB  
Article
Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in S6(1)
by Miroslava Antić and Djordje Kocić
Mathematics 2022, 10(13), 2271; https://doi.org/10.3390/math10132271 - 29 Jun 2022
Cited by 14 | Viewed by 1482
Abstract
It is well known that the sphere S6(1) admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector [...] Read more.
It is well known that the sphere S6(1) admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector field ξ=−JN is said to be characteristic or the Reeb vector field. The Jacobi operator with respect to ξ is called the structure Jacobi operator, and is denoted by l=R(·,ξ)ξ, where R is the curvature tensor on M. The study of Riemannian submanifolds in different ambient spaces by means of their Jacobi operators has been highly active in recent years. In particular, many recent results deal with questions around the existence of hypersurfaces with a structure Jacobi operator that satisfies conditions related to their parallelism. In the present paper, we study the parallelism of the structure Jacobi operator of real hypersurfaces in the nearly Kähler sphere S6(1). More precisely, we prove that such real hypersurfaces do not exist. Full article
(This article belongs to the Special Issue Riemannian Geometry of Submanifolds: Volume II)
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