Novica Blažić

A Note on Osserman Lorentzian Manifolds

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.

Osserman pseudo-Riemannian manifolds of signature (2,2)

A pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Jf?x is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of J£jhas to have a triple zero, which is the other main result. An important step in the proof is based on Walkers study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable JacobToperators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds. 2000 Mathematics subject classification: primary 53B30, 53C50.

CONFORMALLY OSSERMAN MANIFOLDS AND CONFORMALLY COMPLEX SPACE FORMS

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.

CURVATURE STRUCTURE OF SELF-DUAL 4-MANIFOLDS

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 of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds.

Four-dimensional Lie algebras with a para-hypercomplex structure

In this paper we classify 4-dimensional real Lie algebras g admitting an integrable, left invariant, para-hypercomplex structure (Cliff(1,1)-structure). The equivalence classes of compatible structures are classified. The metric of split signature (2, 2), canonically determined by the para-hypercomplex structure, is very convenient in understanding the structure of g. Moreover, these structures provide many examples of left invariant metrics of anti-self-dual metric of split signature. Conformal geometry and the curvature of the canonical metric on the corresponding Lie groups are also discussed. For example, the holonomy algebras of this canonical metrics are determined.

Foliation of a dynamically homogeneous neutral manifold

It is known that Riemannian and Lorentzian four-dimensional dynamically homogeneous manifolds are two-point homogeneous spaces. This is not true for signature (−−++) (neutral or Kleinian signature). In order to better understand their rich structure we study the geometry of nonsymmetric dynamically homogeneous spaces (types II and III): they admit autoparallel distributions and they are locally foliated by totally geodesic, flat, isotropic two-dimensional submanifolds. Moreover we characterize them locally in terms of the existence of an appropriate coordinate system (in the sense of A. G. Walker [Q. J. Math. 1, 69–79 (1950)]).

Solutions of Yang–Mills equations on generalized Hopf bundles

Trautman has constructed natural self-dual connections on the Hopf bundles over complex and quaternionic projective spaces CPn and HPn; the associated connections are SU(n+1) and Sp(n+1) invariant. Trautman wondered if these connections could be generalized to the case of the corresponding projective spaces defined by indefinite metrics. In this note, we extend the work of Trautman in two different directions. We first define self-dual connections on the Hopf bundles over the projective spaces CP(p,q) and HP(p,q) which are U(p,q+1) and Sp(p,q+1) invariant. We also define self-dual connections over the Hopf bundles associated with the para-complex and para-quarternionic projective spaces CP(p,q) and HP(p,q). Finally, the topology of these projective spaces is investigated.

Pontrjagin forms, Chern Simons classes, Codazzi transformations, and affine hypersurfaces

Abstract We show that the primary and secondary characteristic classes vanish in the context of affine differential geometry. This gives rise to obstructions to realizing a conformal class of metrics on a manifold either as the first or as the second fundamental form of an affine immersion.