Curvature Properties of Weyl Geometries

Gilkey, Peter Β.; Nikčević, Stana; Simon, Udo

doi:10.1007/s00025-011-0111-3

Cited by 3 publications

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“…Weyl [34] used these geometries in an attempt to unify gravity with electromagnetism -although this approach failed for physical reasons, the resulting geometries are still of importance and there is a vast literature on the subject. See, for example, [1,10,14,15,20,21]; note that the indefinite signature setting is of particular importance [3,11,20,22] as is the complex setting [18,19,23]. The field is a vast one and we only cite a few representative recent examples.…”

Section: Introductionmentioning

confidence: 99%

Homogeneous 4-Dimensional Kähler–Weyl Structures

et al. 2013

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Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kähler-Weyl structure; this structure is locally conformally Kähler if and only if the alternating Ricci tensor ρa vanishes. The tensor ρa takes values in a certain representation space. In this paper, we show that any algebraic possibility Ξ in this representation space can in fact be geometrically realized by a left-invariant Kähler-Weyl structure on a 4-dimensional Lie group in either the pseudo-Hermitian or the para-Hermitian setting. MSC 2010: 53A15, 53C15, 15A72.

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Section: Introductionmentioning

confidence: 99%

Homogeneous 4-Dimensional Kähler–Weyl Structures

et al. 2013

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Applications of Affine and Weyl Geometry

Garcı́a-Rı́o

Gilkey

Nikčević

et al. 2013

Synthesis Lectures on Mathematics &Amp; Statistics

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Applications of Affine and Weyl Geometry

Garcı́a-Rı́o¹,

Gilkey²,

Nikčević³

et al. 2013

Synthesis Lectures on Mathematics and Statistics

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Pseudo-Riemannian geometry is, to a large extent, the study of the Levi-Civita connection, which is the unique torsion-free connection compatible with the metric structure. There are, however, other affine connections which arise in different contexts, such as conformal geometry, contact structures, Weyl structures, and almost Hermitian geometry. In this book, we reverse this point of view and instead associate an auxiliary pseudo-Riemannian structure of neutral signature to certain affine connections and use this correspondence to study both geometries. We examine Walker structures, Riemannian extensions, and Kähler-Weyl geometry from this viewpoint.This book is intended to be accessible to mathematicians who are not expert in the subject and to students with a basic grounding in differential geometry. Consequently, the first chapter contains a comprehensive introduction to the basic results and definitions we shall need-proofs are included of many of these results to make it as self-contained as possible. Para-complex geometry plays an important role throughout the book and consequently is treated carefully in various chapters, as is the representation theory underlying various results. It is a feature of this book that, rather than as regarding para-complex geometry as an adjunct to complex geometry, instead, we shall often introduce the para-complex concepts first and only later pass to the complex setting. The second and third chapters are devoted to the study of various kinds of Riemannian extensions that associate to an affine structure on a manifold a corresponding metric of neutral signature on its cotangent bundle. These play a role in various questions involving the spectral geometry of the curvature operator and homogeneous connections on surfaces. The fourth chapter deals with Kähler-Weyl geometry, which lies, in a certain sense, midway between affine geometry and Kähler geometry. Another feature of the book is that we have tried wherever possible to find the original references in the subject for possible historical interest. Thus, we have cited the seminal papers of Levi-Civita, Ricci, Schouten, and Weyl, to name but a few exemplars. We have also given different proofs of various results than those that are given in the literature, to take advantage of the unified treatment of the area given herein.

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Curvature Properties of Weyl Geometries

Abstract: We examine relations between geometry and the associated curvature decompositions in Weyl Geometry.Mathematics Subject Classification (2010). 53B05, 15A72, 53A15, 53B10, 53C07, 53C25.

Cited by 3 publications

References 9 publications

Homogeneous 4-Dimensional Kähler–Weyl Structures

Homogeneous 4-Dimensional Kähler–Weyl Structures

Applications of Affine and Weyl Geometry

Applications of Affine and Weyl Geometry

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