Marcin Bobienski
A nonstandard (maximal) inclusion SO(3) in SO(5) associated with the irreducible representation ρ_5 of SO(3) in R^5 is considered. The topological obstructions for admitting the SO(3) structure on the frame bundle over 5-manifold are investigated. The necessary and sufficient conditions are formulated.
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For a conformal manifold, we describe a new relation between the ambient obstruction tensor of Fefferman and Graham and the holonomy of the normal conformal Cartan connection. This relation allows us to prove several results on the vanishing and the rank of the obstruction tensor, for example for conformal structures admitting twistor spinors or normal conformal Killing forms. As our main tool we introduce the notion of a conformal holonomy distribution and show that its integrability is closely related to the exceptional conformal structures in dimensions five and six that were found by Nurow…
SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle
受泡利不相容原理三元推广的启发，本文提出三阶三元反对称协变张量概念，分解其十维复空间并找到两个SO(3)不变量，在五维复埃尔米特流形上构造SO(3)不可约几何结构并研究其联络的曲率与挠率。
We consider the question if a five dimensional manifold can be embedded into a Calabi-Yau manifold of complex dimension three such that the real part of the holomorphic volume form induces a given closed 3-form on the 5-manifold. We define an open set of 3-forms in dimension five which we call strongly pseudoconvex, and show that for closed strongly pseudoconvex 3-forms the perturbative version of this embedding problem can be solved if a finite dimensional vector space of obstructions vanishes.
Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. We compute the obstruction for a vector bundle $S$ over $(M,g)$ to admit a Dirac operator whose principal symbol induces on $S$ the structure of a bundle of irreducible real Clifford modules of complex type, that is, a real spinor bundle of irreducible complex type. In order to do this, we use the theory of Lipschitz structures in signature $p-q\equiv_8 3,7$ to reformulate the problem as the obstruction problem for $(M,g)$ to admit a $\mathrm{Spin}^{o}_α$ structure with $α= -1$ if $p-q \equiv_{8} 3$ or $α= +1$ if $ p-q \equiv_{8…
We consider the nonstandard inclusion of SO(3) in SO(5) associated with a 5-dimensional irreducible representation. The tensor $Υ$ representing this reduction is found to be given by a ternary symmetric form with special properties. A 5-dimensional manifold $(M,g,Υ)$ with Riemannian metric $g$ and ternary form generated by such a tensor has a corresponding SO(3) structure, whose Gray-Hervella type classification is established using so(3)-valued connections with torsion. Structures with antisymmetric torsions, we call them the nearly integrable SO(3) structures, are studied in detail. In parti…
We extend the notions of CR GJMS operators and Q-curvature to the case of partially integrable CR structures. The total integral of the CR Q-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR manifolds equipped with an orientation of the bundle of contact forms, which is nontrivial in dimension at least five. It is shown that its variation is given by the curvature-type quantity called the CR obstruction tensor, which is introduced in the author's previous work. Moreover, we consider the linearized CR obstruction operator. Based on a scattering-theore…
We use classical (Penrose) two-component spinors to set up the differential geometry of two parabolic contact structures in five dimensions, namely $G_2$ contact geometry and Legendrean contact geometry. The key players in these two geometries are invariantly defined directional derivatives defined only in the contact directions. We explain how to define them and their usage in constructing basic invariants such as the harmonic curvature, the obstruction to being locally flat from the parabolic viewpoint. As an application, we calculate the invariant torsion of the $G_2$ contact structure on t…
For a homogeneous space $G/H$ of reductive type, we consider the tangential homogeneous space $G_θ/H_θ$. In this paper, we give obstructions to the existence of compact Clifford-Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford-Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford-Klein forms. The existence problem of compact Clifford-Klein forms for homogeneous spaces of reductive type…
We obtain the topological obstructions to existence of a bundle of irreducible real Clifford modules over a pseudo-Riemannian manifold $(M,g)$ of arbitrary dimension and signature and prove that bundles of Clifford modules are associated to so-called real Lipschitz structures. The latter give a generalization of spin structures based on certain groups which we call real Lipschitz groups. In the fiberwise-irreducible case, we classify the latter in all dimensions and signatures. As a simple application, we show that the supersymmetry generator of eleven-dimensional supergravity in "mostly plus"…
We study the equations governing rigid N=1 supersymmetry in five dimensions. If the supersymmetry spinor satisfies a reality condition, these are foliations admitting families of almost complex structures on the leaves. In other words, all these manifolds have families of almost Cauchy-Riemann (CR) structures. After deriving integrability conditions under which circumstances the almost CR structure defines a CR manifold or a transversally holomorphic foliation (THF), we discuss implications on localization. We also discuss potential global obstructions to the existence of solutions.
A Quantitative Reeb-Form Obstruction for Real Hypersurfaces of Strict Nearly Kähler Six-Manifolds
本文证明六维严格近凯勒流形的实超曲面上Reeb一次形式的外微分逐点非零，并由此得出Yano联络的Ricci张量处处不对称，排除了满足dη=0的诱导几乎切触类。
We study five dimensional geometries associated with the 5-dimensional irreducible representation of GL(2,R). These are special Weyl geometries in signature (3,2) having the structure group reduced from CO(3,2) to GL(2,R). The reduction is obtained by means of a conformal class of totally symmetric 3-tensors. Among all GL(2,R) geometries we distinguish a subclass which we term `nearly integrable GL(2,R)geometries'. These define a unique gl(2,R) connection which has totally skew symmetric torsion. This torsion splits onto the GL(2,R) irreducible components having respective dimensions 3 and 7. …
We investigate orthogonal representations of compact Lie groups from the point of view of their quotient spaces, considered as metric spaces. We study metric spaces which are simultaneously quotients of different representations and investigate properties of the corresponding representations. We obtain some structural results and apply them to study irreducible representations of small copolarity. As an important tool, we classify all irreducible representations of connected groups with cohomogeneity four or five.
Regularized determinants of the Rumin complex in irreducible unitary representations of the (2,3,5) nilpotent Lie group
研究(2,3,5) nilpotent Lie群的Rumin微分在不可约酉表示中的量子谐振子推广，计算Schrodinger表示的谱和正则化行列式，以及通用表示中的交替积即Rumin复形的解析 torsion。
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