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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
Source Normalized Impact per Paper (SNIP) 2018: 0.908

Mathematical Citation Quotient (MCQ) 2017: 0.62

Online
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1615-7168
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Volume 8, Issue 3

Issues

Relating the curvature tensor and the complex Jacobi operator of an almost Hermitian manifold

M. Brozos-Vázquez
  • Department of Geometry and Topology, Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela 15782, Spain. Email: ,
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ E. García-Río
  • Department of Geometry and Topology, Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela 15782, Spain. Email: ,
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ P. Gilkey
Published Online: 2008-09-11 | DOI: https://doi.org/10.1515/ADVGEOM.2008.023

Abstract

Let J be a unitary almost complex structure on a Riemannian manifold (M, g). If x is a unit tangent vector, let π := Span{x, Jx} be the associated complex line in the tangent bundle of M. The complex Jacobi operator and the complex curvature operators are defined, respectively, by and . We show that if (M, g) is Hermitian or if (M, g) is nearly Kähler, then either the complex Jacobi operator or the complex curvature operator completely determine the full curvature operator; this generalizes a well known result in the real setting to the complex setting. We also show that this result fails for general almost Hermitian manifolds.

Key words.: Complex curvature operator; complex Jacobi operator; almost Hermitian manifold; Hermitian manifold; nearly Kähler manifold

About the article

Received: 2006-11-15

Published Online: 2008-09-11

Published in Print: 2008-08-01


Citation Information: Advances in Geometry, Volume 8, Issue 3, Pages 353–365, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/ADVGEOM.2008.023.

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[1]
Eduardo García-Río, Peter Gilkey, Stana Nikčević, and Ramón Vázquez-Lorenzo
Synthesis Lectures on Mathematics and Statistics, 2013, Volume 6, Number 1, Page 1
[2]
E. Calviño-Louzao, E. García-Río, P. Gilkey, and R. Vázquez-Lorenzo
Geometriae Dedicata, 2012, Volume 156, Number 1, Page 151
[3]
Miguel Brozos-Vázquez and Peter Gilkey
Forum Mathematicum, 2013, Volume 25, Number 2

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