A Characterization of Totally Geodesic Hypersurfaces of S n+1 and CP n+1

1981 ◽  
Vol 81 (4) ◽  
pp. 603 ◽  
Author(s):  
Kinetsu Abe
Keyword(s):  
Totally Geodesic ◽  
Totally Geodesic Hypersurfaces

scholarly journals Parallel and totally geodesic hypersurfaces of non‐reductive homogeneous four‐manifolds

2020 ◽  
Vol 293 (9) ◽  
pp. 1707-1729
Author(s):  
Giovanni Calvaruso ◽  
Reinier Storm ◽  
Joeri Van der Veken
Keyword(s):  
Totally Geodesic ◽  
Totally Geodesic Hypersurfaces ◽  
Four Manifolds

On six-dimensional Vaisman — Gray submanifolds of the octave algebra

2019 ◽  
pp. 29-35
Author(s):  
M. Banaru
Keyword(s):  
Structural Equations ◽  
Totally Geodesic ◽  
Hermitian Manifolds ◽  
Cayley Algebra ◽  
Almost Hermitian Manifolds ◽  
Almost Contact Metric ◽  
Totally Geodesic Hypersurfaces ◽  
Metric Structures ◽  
Nearly Cosymplectic ◽  
Kählerian Manifolds

The W1 W4 class of almost Hermitian manifolds (in accordance with the Gray — Hervella classification) is usually named as the class of Vaisman — Gray manifolds. This class contains all Kählerian, nearly Kählerian and locally conformal Kählerian manifolds. As it is known, Vaisman — Gray manifolds are invariant under the conformal transformations of the metric. A criterion in the terms of the configuration tensor for an arbitrary six-dimensional submanifold of Cayley algebra to belong to the Vaisman — Gray class of almost Hermitian manifolds is established. The Cartan structural equations of the almost contact metric structures induced on oriented hypersurfaces of six-dimensional Vaisman — Gray submanifolds of the octave algebra are obtained. It is proved that totally geodesic hypersurfaces of six-dimensional Vaisman — Gray submanifolds of Cayley algebra admit nearly cosymplectic structures (or Endo structures). This result is a generalization of the previously proved fact that totally geodesic hypersurfaces of nearly Kählerian manifolds also admit nearly cosymplectic structures.

Real Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Structure Jacobi Operator

2014 ◽  
Vol 57 (4) ◽  
pp. 821-833 ◽  
Author(s):  
Imsoon Jeong ◽  
Seonhui Kim ◽  
Young Jin Suh
Keyword(s):  
Real Hypersurface ◽  
Jacobi Operator ◽  
Parallel Structure ◽  
Real Hypersurfaces ◽  
Structure Jacobi Operator ◽  
Totally Geodesic

AbstractIn this paper we give a characterization of a real hypersurface of Type (A) in complex two-plane GrassmanniansG2(ℂm+2), which means a tube over a totally geodesicG2(ℂm+1) inG2(ℂm+2), by means of the Reeb parallel structure Jacobi operator ∇εRε= 0.

Real Hypersurfaces in Complex Two-Plane Grassmannians with Vanishing Lie Derivative

2006 ◽  
Vol 49 (1) ◽  
pp. 134-143 ◽  
Author(s):  
Young Jin Suh
Keyword(s):  
Vector Field ◽  
Shape Operator ◽  
Real Hypersurfaces ◽  
Lie Derivative ◽  
Reeb Vector ◽  
Totally Geodesic ◽  
Reeb Vector Field

AbstractIn this paper we give a characterization of real hypersurfaces of type A in a complex two-plane Grassmannian G2(ℂm+2) which are tubes over totally geodesic G2(ℂm+1) in G2(ℂm+2) in terms of the vanishing Lie derivative of the shape operator A along the direction of the Reeb vector field ξ.

scholarly journals Homotopy of compact symmetric spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 221-228 ◽  
Author(s):  
John M. Burns
Keyword(s):  
Symmetric Spaces ◽  
Unified Approach ◽  
Topological Characterization ◽  
New Approach ◽  
Totally Geodesic ◽  
Compact Symmetric Space ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds ◽  
Compact Symmetric Spaces

In recent years a new approach to the study of compact symmetric spaces has been taken by Nagano and Chen [10]. This approach assigned to each pair of antipodal points on a closed geodesic a pair of totally geodesic submanifolds. In this paper we will show how these totally geodesic submanifolds can be used in conjunction with a theorem of Bott to compute homotopy in compact symmetric spaces. Some of the results are already known (see [1], [5], [11] for example) but we include them here for completeness and to illustrate this unified approach. We also exhibit a connection between the second homotopy group of a compact symmetric space and the multiplicity of the highest root. Using this in conjunction with a theorem of J. H. Cheng [6] we obtain a topological characterization of quaternionic symmetric spaces with antiquaternionic involutive isometry. The author would like to thank Prof T. Nagano for all his help and his detailed descriptions of the totally geodesic submanifolds mentioned above.

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