arXiv subjects

R. Vazquez-Lorenzo

Publications and source records attributed to R. Vazquez-Lorenzo.

10 recordsLinked to original sources

arXiv2012

Homogeneous 4-dimensional Kaehler--Weyl Structures

Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kaehler-Weyl structure; this structure is locally conformally Kaehler if and only if the alternating Ricci tensor vanishes. The alternating Ricci tensor takes values in a certain representation space. In this paper, we show that any algebraic possibility in this representation space can in fact be geometrically realized by a left-invariant Kaehler-Weyl structure on a 4-dimensional Lie group in either the Hermitian or the para-Hermitian setting.

math.DG↗

arXiv2010

Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators

We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n,n) for any n which is at least 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither satisfy the third Gray identity nor are they integrable.

math.DG↗

arXiv2010

Compact Osserman manifolds with neutral metric

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.

math.DG↗

arXiv2009

Geometric Realizations of para-Hermitian curvature models

We show that a para-Hermitian algebraic curvature model satisfies the para-Gray identity if and only if it is geometrically realizable by a para-Hermitian manifold. This requires extending the Tricerri-Vanhecke curvature decomposition to the para-Hermitian setting. Additionally, the geometric realization can be chosen to have constant scalar curvature and constant *-scalar curvature.

math.DG↗

arXiv2009

The geometry of modified Riemannian extensions

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and give necessary and sufficient conditions so that a modified Riemannian extension is Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3,3) whose Jacobi operators have non-trivial Jordan normal form and which are not nilpotent. We present new four dimensional results in Osserman geometry.

math.DG↗

arXiv2008

Osserman and conformally Osserman manifolds with warped and twisted product structure

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.

math.DG↗

arXiv2008

Four-dimensional Osserman metrics of neutral signature

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.

math.DG↗

arXiv2007

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.

math.DG↗

arXiv2007

Examples of signature (2,2) manifolds with commuting curvature operators

We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A.

math.DG↗

arXiv2006

Completeness, Ricci blowup, the Osserman and the conformal Osserman condition for Walker signature (2,2) manifolds

Walker manifolds of signature (2,2) have been used to provide examples of Osserman and of conformal Osserman manifolds of signature (2,2). We study questions of geodesic completeness and Ricci blowup in this context.

math.DG↗