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result(s) for
"Nikcevic, Stana"
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Geometric realizations of curvature
A central area of study in Differential Geometry is the examination of the relationship between the purely algebraic properties of the Riemann curvature tensor and the underlying geometric properties of the manifold. In this book, the findings of numerous investigations in this field of study are reviewed and presented in a clear, coherent form, including the latest developments and proofs. Even though many authors have worked in this area in recent years, many fundamental questions still remain unanswered. Many studies begin by first working purely algebraically and then later progressing onto the geometric setting and it has been found that many questions in differential geometry can be phrased as problems involving the geometric realization of curvature. Curvature decompositions are central to all investigations in this area. The authors present numerous results including the Singer–Thorpe decomposition, the Bokan decomposition, the Nikcevic decomposition, the Tricerri–Vanhecke decomposition, the Gray–Hervella decomposition and the De Smedt decomposition. They then proceed to draw appropriate geometric conclusions from these decompositions.
eBook
Stanilov-Tsankov-Videv Theory
We survey some recent results concerning Stanilov-Tsankov-Videv theory, conformal Osserman geometry, and Walker geometry which relate algebraic properties of the curvature operator to the underlying geometry of the manifold.
Journal Article
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.
Paper
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.
Paper
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.
Paper
Geometric realizations, curvature decompositions, and Weyl manifolds
We show any Weyl curvature model can be geometrically realized by a Weyl manifold
Paper
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.
Paper
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.
Paper
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.
Paper