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One says that a smooth manifold M is a pseudo-Riemannian manifold of signature (p,q) if the tangent bundle TM is equipped with a smooth non-degenerate symmetric inner product g of signature (p,q). Similarly one says that M is an affine manifold if TM is equipped with a torsion free connection. One says g is Osserman if the eigenvalues of the Jacobi operator are constant on the pseudo-sphere bundles of unit timelike and spacelike vectors. We extend this concept from the pseudo-Riemannian to the affine setting to define the notion of a projectively Osserman manifold. This notion is the focus of the paper. We establish some basic results concerning projectively Osserman manifolds and exhibit examples of this structure which arise in several different geometrical contexts.
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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 obtain, as a consequence, that the only Osserman manifolds which can be written as a twisted product are those of constant curvature.
It is shown that if a compact four-dimensional manifold with metric of neutral signature is Jordan-Osserman, then it is either of constant sectional curvature or Ricci flat.
We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.
In this paper, a characteristic condition of the projectively flat Kropina metric is given. By it, we prove that a Kropina metric $F=alpha^2/beta$ with constant curvature $K$ and $|beta|_{alpha}=1$ is projectively flat if and only if $F$ is locally Minkowskian.
In the algebraic context, we show that null Osserman, spacelike Osserman, and timelike Osserman are equivalent conditions for a model of signature (2,2). We also classify the null Jordan Osserman models of signature (2,2). In the geometric context, we show that a pseudo-Riemannian manifold of signature (2,2) is null Jordan Osserman if and only if either it has constant sectional curvature or it is locally a complex space form.
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