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CHARACTERIZATIONS OF RICCI FLAT METRICS AND LAGRANGIAN SUBMANIFOLDS IN TERMS OF THE VARIATIONAL PROBLEM

Part of: Manifolds Symplectic geometry, contact geometry Global differential geometry

Published online by Cambridge University Press:  17 December 2014

TETSUYA TANIGUCHI  and
SEIICHI UDAGAWA
TETSUYA TANIGUCHI
Affiliation:
Department of Mathematics, School of General Education, Kitasato University, Sagamihara, Kanagawa 228-8555, Japan e-mail: tetsuya@kitasato-u.ac.jp
SEIICHI UDAGAWA
Affiliation:
Division of Mathematics, School of Medicine, Nihon University, Itabashi, Tokyo 173-0032, Japan e-mail: udagawa.seiichi@nihon-u.ac.jp
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Abstract

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Given the pair (P, η) of (0,2) tensors, where η defines a volume element, we consider a new variational problem varying η only. We then have Einstein metrics and slant immersions as critical points of the 1st variation. We may characterize Ricci flat metrics and Lagrangian submanifolds as stable critical points of our variational problem.

MSC classification

Primary: 53D12: Lagrangian submanifolds; Maslov index 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 49Q99: None of the above, but in this section
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 643 - 651
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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