# Conformally Osserman manifolds and self-duality in Riemannian geometry

@article{Blai2005ConformallyOM, title={Conformally Osserman manifolds and self-duality in Riemannian geometry}, author={Novica Bla{\vz}i{\'c} and Peter Gilkey}, journal={arXiv: Differential Geometry}, year={2005} }

We study the spectral geometry of the conformal Jacobi operator on a 4-dimensional Riemannian manifold (M,g). We show that (M,g) is conformally Osserman if and only if (M,g) is self-dual or anti self-dual. Equivalently, this means that the curvature tensor of (M,g) is given by a quaternionic structure, at least pointwise.

## 19 Citations

### Conformally Osserman four-dimensional manifolds whose conformal Jacobi operators have complex eigenvalues

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2006

Conformal Osserman four-dimensional manifolds are studied with special attention to the construction of new examples showing that the algebraic structure of any such curvature tensor can be realized…

### CURVATURE STRUCTURE OF SELF-DUAL 4-MANIFOLDS

- Mathematics
- 2008

We show the existence of a modified Cliff(1,1)-structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds…

### Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces

- Mathematics
- 2012

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit…

### Osserman and Conformally Osserman Manifolds with Warped and Twisted Product Structure

- Mathematics
- 2008

Abstract.We characterize Osserman and conformally Osserman Riemannian manifolds with the local structure of a warped product. By means of this approach we analyze the twisted product structure and…

### Weyl–Schouten Theorem for symmetric spaces

- Mathematics
- 2013

Let M0 be a symmetric space of dimension n > 5 whose de Rham decomposition contains no factors of constant curvature and let W0 be the Weyl tensor of M0 at some point. We prove that a Riemannian…

### Complex Osserman Kähler manifolds in dimension four

- Mathematics
- 2013

Abstract. Let be a 4-dimensional almost-Hermitian manifold which satisfies the Kähler identity. We show that is complex Osserman if and only if has constant holomorphic sectional curvature. We also…

### Complex Osserman Kaehler Manifolds

- Mathematics
- 2010

Let H be a 4 dimensional almost Hermitian manifold which satisfies the Kaehler identity. Then H is complex Osserman if and only if H has constant holomorphic sectional curvature. We also classify in…

### Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces

- MathematicsAnnali di Matematica Pura ed Applicata
- 2011

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit…

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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…

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Abstract.Let Mn be a Riemannian manifold and R its curvature tensor. For a point p ∈ Mn and a unit vector X ∈ TpMn, the Jacobi operator is defined by RX=R(X,·)X. The manifold Mn is called pointwise…