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. 
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References

SHOWING 1-10 OF 18 REFERENCES

Natural curvature operators of bounded spectrum

Complete curvature homogeneous pseudo-Riemannian manifolds

Abstract. For k ≥ 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). All the…

Manifolds with commuting Jacobi operators

Abstract.We characterize Riemannian manifolds of constant sectional curvature in terms of commutation properties of their Jacobi operators.

On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection

We give a complete decomposition of the space of curvature tensors with the symmetry properties as the curvature tensor associated with a symmetric connection of Riemannian manifold. We solve the…

Osserman manifolds of dimension 8

Abstract.For a Riemannian manifold Mn with the curvature tensor R, the Jacobi operator RX is defined by RXY=R(X,Y)X. The manifold Mn is called pointwise Osserman if, for every p ∈ Mn, the eigenvalues…

New examples of Osserman metrics with nondiagonalizable Jacobi operators

Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor

Algebraic curvature tensors the skew-symmetric curvature operator the Jacobi operator controlling the eigenvalue structure.

Curvature in the eighties

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,…

On a problem of Osserman in Lorentzian geometry

Bouncing Braneworlds