arXiv subjects

Stana Nikcevic

Publications and source records attributed to Stana Nikcevic.

8 recordsLinked to original sources

arXiv2013

Affine projective Osserman structures

By considering the projectivized spectrum of the Jacobi operator, we introduce the concept of projective Osserman manifold in both the affine and in the pseudo-Riemannian settings. If M is an affine projective Osserman manifold, then the modified Riemannian extension metric on the cotangent bundle is both spacelike and timelike projective Osserman. Since any rank 1 symmetric space is affine projective Osserman, this provides additional information concerning the cotangent bundle of a rank 1 Riemannian symmetric space with the modified Riemannian extension metric. We construct other examples of affine projective Osserman manifolds where the Ricci tensor is not symmetric and thus the connection is not the Levi-Civita connection of any metric. If M is an affine projective Osserman manifold of odd dimension, we use methods of algebraic topology to show the Jacobi operator has only one non-zero eigenvalue and that eigenvalue is real.

math.DG↗

arXiv2012

4-dimensional (para)-Kähler--Weyl structures

We give an elementary proof of the fact that any 4-dimensional para-Hermitian manifold admits a unique para-Kaehler--Weyl structure. We then use analytic continuation to pass from the para-complex to the complex setting and thereby show any 4-dimensional pseudo-Hermitian manifold also admits a unique Kaehler--Weyl structure.

math.DG↗

arXiv2010

Kaehler and para-Kaehler curvature Weyl manifolds

We show that the Weyl structure of an almost-Hermitian Weyl manifold of dimension at least 6 is trivial if the associated curvature operator satisfies the Kaehler identity. Similarly if the curvature of an almost para-Hermitian Weyl manifold of dimension at least 6 satisfies the para-Kaehler identity, then the Weyl structure is trivial as well.

math.DG↗

arXiv2010

Geometric realizations, curvature decompositions, and Weyl manifolds

We show any Weyl curvature model can be geometrically realized by a Weyl manifold

math.DG↗

arXiv2008

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.

math.DG↗

arXiv2008

Geometrical representations of equiaffine curvature operators

We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold.

math.DG↗

arXiv2005

Generalized plane wave manifolds

We show that generalized plane wave manifolds are complete, strongly geodesically convex, Osserman, Szabo, and Ivanov-Petrova. We show their holonomy groups are nilpotent and that all the local Weyl scalar invariants of these manifolds vanish. We construct isometry invariants on certain families of these manifolds which are not of Weyl type. Given k, we exhibit manifolds of this type which are k-curvature homogeneous but not locally homogeneous. We also construct a manifold which is weakly 1-curvature homogeneous but not 1-curvature homogeneous.

math.DG↗

arXiv2005

Complete k-curvature homogeneous pseudo-Riemannian manifolds 0-modeled on an indecomposible symmetric space

For k at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). The curvature tensor of these manifolds is modeled on that of an indecomposible symmetric space. All the local scalar Weyl curvature invariants of these manifolds vanish.

math.DG↗