# Pseudo-riemannian manifolds with commuting jacobi operators

@article{BrozosVzquez2006PseudoriemannianMW, title={Pseudo-riemannian manifolds with commuting jacobi operators}, author={Miguel Brozos-V{\'a}zquez and Peter Gilkey}, journal={Rendiconti del Circolo Matematico di Palermo}, year={2006}, volume={55}, pages={163-174} }

We study the geometry of pseudo-Riemannian manifolds which are Jacobi-Tsankov, i.e. ℊ(x)ℊ(y)=ℊ(y)ℊ(x) for allx, y. We also study manifolds which are 2-step Jacobi nilpotent, i.e. ℊ(x)ℊ(y)=0 for allx, y.

## 5 Citations

### Commutative curvature operators over four-dimensional homogeneous manifolds

- Mathematics
- 2015

Four-dimensional pseudo-Riemannian homogeneous spaces whose isotropy is non-trivial with commuting curvature operators have been studied. The only example of homogeneous Einstein four-manifold which…

### Examples of signature (2, 2) manifolds with commuting curvature operators

- Mathematics
- 2007

We exhibit Walker manifolds of signature (2, 2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator and the Jacobi operator. If the Walker metric is a…

### LORENTZIAN 3-MANIFOLDS WITH COMMUTING CURVATURE OPERATORS

- Mathematics
- 2008

Three-dimensional Lorentzian manifolds with commuting curvature operators are studied. A complete description is given at the algebraic level. Consequences are obtained at the differentiable setting…

### Jacobi--Tsankov manifolds which are not 2-step nilpotent

- Mathematics
- 2006

An algebraic curvature tensor A is said to be Jacobi-Tsankov if J(x)J(y)=J(y)J(x) for all x,y. This implies J(x)J(x)=0 for all x; necessarily A=0 in the Riemannian setting. Furthermore, this implies…

### Stanilov-Tsankov-Videv Theory ?

- Mathematics
- 2007

We survey some recent results concerning Stanilov-Tsankov-Videv theory, con- formal Osserman geometry, and Walker geometry which relate algebraic properties of the curvature operator to the…

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ROBERT OSSERMAN wrote a Ph.D. thesis on Riemann surfaces under the direction of Lars V. Ahlfors at Harvard University. He gradually moved from geometric function theory to minimal surfaces,…