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Generalized osserman manifolds

Generalized osserman manifolds Abh. Math. Sem. Univ. Hamburg 68 (1998), 125-127 By P. B. GILKEY Let R(X, Y) be the curvature operator of a Riemannian manifold (M, g) of dimen- sion m. Fix a point P 6 M and let X be a unit vector in the tangent space Tp M to M at P. The Jacobi operator J(X) : Y ~+ R(Y, X)X is a self-adjoint endo- morphism of TeM. Let p be the Ricci tensor; for any orthonormal base {Xi} of TpM we have p(X, Y) := Eig(R(X, Xi)Xi, Y). Let Grp(TpM) be the Grassma- nian of p planes in TpM. If p(X, Y) = )~g(X, Y) or equivalently if Tr J(X) is constant on Grl (TpM), then we say that M is einstein; if Tr{J(X) k } is constant on Grl(TpM) for k = 1, 2, then we say that M is zweistein. If E c Grp(TpM), let J(E) := J(X1) +... + J(Xp) where {X1 ..... Xp} is an orthonormal basis for E; J(E) is independent of the basis chosen. Assume that 1 < p < m - 1. Following STANILOV and VIDEV [9], we say that M is Ossp at P if the eigenvalues of J(E) are http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

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References (8)

Publisher
Springer Journals
Copyright
Copyright © 1998 by Mathematisches Seminar der Universität Hamburg
Subject
Mathematics; Algebra; Differential Geometry; Combinatorics; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02942557
Publisher site
See Article on Publisher Site

Abstract

Abh. Math. Sem. Univ. Hamburg 68 (1998), 125-127 By P. B. GILKEY Let R(X, Y) be the curvature operator of a Riemannian manifold (M, g) of dimen- sion m. Fix a point P 6 M and let X be a unit vector in the tangent space Tp M to M at P. The Jacobi operator J(X) : Y ~+ R(Y, X)X is a self-adjoint endo- morphism of TeM. Let p be the Ricci tensor; for any orthonormal base {Xi} of TpM we have p(X, Y) := Eig(R(X, Xi)Xi, Y). Let Grp(TpM) be the Grassma- nian of p planes in TpM. If p(X, Y) = )~g(X, Y) or equivalently if Tr J(X) is constant on Grl (TpM), then we say that M is einstein; if Tr{J(X) k } is constant on Grl(TpM) for k = 1, 2, then we say that M is zweistein. If E c Grp(TpM), let J(E) := J(X1) +... + J(Xp) where {X1 ..... Xp} is an orthonormal basis for E; J(E) is independent of the basis chosen. Assume that 1 < p < m - 1. Following STANILOV and VIDEV [9], we say that M is Ossp at P if the eigenvalues of J(E) are

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Aug 29, 2008

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