A note on the geometry and topology of almost even-Clifford Hermitian manifolds (original) (raw)

Almost even-Clifford hermitian manifolds with a large automorphism group

Proceedings of the American Mathematical Society, 2016

We study manifolds endowed with an (almost) even Clifford (hermitian) structure and admitting a large automorphism group. We classify them when they are simply connected and the dimension of the automorphism group is maximal, and also prove a gap theorem for the dimension of the automorphism group.

Spin Structures on Compact Homogeneous Pseudo-Riemannian Manifolds

Transformation Groups, 2018

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds F = G/H, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M = G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin.

Spin structures on almost-flat manifolds

We give a necessary and su?cient condition for almost- at manifolds with cyclic holonomy to admit a Spin structure. Using this condition we ?nd all 4-dimensional orientable almost- at manifolds with cyclic holonomy that do not admit a Spin structure.

Classification of Spin Structures on Four-Dimensional Almost-Flat Manifolds

Mathematika, 2018

Almost-flat manifolds were defined by Gromov as a natural generalization of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental group. Utilizing this, we classify the spin structures on all four-dimensional almost-flat manifolds that are not flat. Out of 127 orientable families, we show that there are exactly 15 that are non-spin, the rest are, in fact, parallelizable.

Spin structures of flat manifolds of diagonal type

Homology, Homotopy and Applications

For each integer d at least two, we construct non-spin closed oriented flat manifolds with holonomy group Z d 2 and with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have trivial Stiefel-Whitney classes.

Spin Structures on Flat Manifolds

Monatshefte für Mathematik, 2006

The aim of this paper is to present some results about spin structures on flat manifolds. We prove that any finite group can be the holonomy group of a flat spin manifold. Moreover we shall give some methods of constructing spin structures related to the holonomy representation.

Spin(9) and almost complex structures on 16-dimensional manifolds

arXiv (Cornell University), 2011

For a Spin(9)-structure on a Riemannian manifold M 16 we write explicitly the matrix ψ of its Kähler 2-forms and the canonical 8-form Φ Spin(9). We then prove that Φ Spin(9) coincides up to a constant with the fourth coefficient of the characteristic polynomial of ψ. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Φ Spin(9) and Φ 2 Spin(9) in the special case of holonomy Spin(9). Contents 1. Introduction 1 2. Preliminaries and notations 4 3. Low dimensions 5 4. The Kähler forms of a Spin(7)-structure 8 5. The Kähler forms of a Spin(9)-structure 10 6. The 8-form of a Spin(9)-structure 13 7. The main formula and its corollaries 16 References 18 List of Tables A Synoptic table of symbols 3 B Non-zero terms of the 8-form Φ Spin(9) in R 16 15

Special almost Hermitian geometry

Journal of Geometry and Physics, 2005

We study the classification of special almost hermitian manifolds in Gray and Hervella's type classes. We prove that the exterior derivatives of the Kähler form and the complex volume form contain all the information about the intrinsic torsion of the SU(n)-structure. Furthermore, we apply the obtained results to almost hyperhermitian geometry. Thus, we show that the exterior derivatives of the three Kähler forms of an almost hyperhermitian manifold are sufficient to determine the three covariant derivatives of such forms, i.e., the three mentioned exterior derivatives determine the intrinsic torsion of the Sp(n)-structure. : Primary 53C15; Secondary 53C10, 53C55, 53C26.