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The American Mathematical Monthly is currently published by Mathematical Association of America. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
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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
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Aug 29, 2008
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