Article

# Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators

07/2010; DOI:abs/1007.2569
Source: arXiv

ABSTRACT We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n,n) for any n which is at least 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither satisfy the third Gray identity nor are they integrable.

0 0
·
0 Bookmarks
·
41 Views
• ##### Article:A NOTE ON OSSERMAN LORENTZIAN MANIFOLDS
[show abstract] [hide abstract]
ABSTRACT: Let p be a point of a Lorentzian manifold M. We show that if M is spacelike Osserman at p, then M has constant sectional curvature at p; similarly, if M is timelike Osserman at p, then M has constant sectional curvature at p. The reverse implications are immediate. The timelike case and 4-dimensional spacelike case were first studied in [3]; we use a different approach to this case.
Bulletin of the London Mathematical Society 02/1997; 29(02):227 - 230. · 0.54 Impact Factor
• Source
##### Article:Relating the curvature tensor and the complex Jacobi operator of an almost Hermitian manifold
[show abstract] [hide abstract]
ABSTRACT: Let J be a unitary almost complex structure on a Riemannian manifold (M,g). If x is a unit tangent vector, let P be the associated complex line spanned by x and by Jx. We show that if (M,g) is Hermitian or if (M,g) is nearly Kaehler, then either the complex Jacobi operator (JC(P)y=R(y,x)x+R(y,Jx)Jx) or the complex curvature operator (RC(P)y=R(x,Jx)y) completely determine the full curvature operator; this generalizes a well known result in the real setting to the complex setting. We also show this result fails for general almost Hermitian manifold.
12/2006;
• ##### Book:The Geometry of Walker Manifolds
01/2009; Morgan & Claypool Publishers.

Available from

### Keywords

neutral signature

non-nilpotent Jacobi operators

non-trivial Jordan normal form

semi para-complex Osserman