Holonomy and contact geometry

On a Riemannian manifold, any parallel form is preserved by the flow of any Killing vector field with constant magnitude. As a consequence, on a 2n+1-dimensional K-contact manifold, there are no nontrivial parallel forms except of degrees 0 and 2n+1. Flat contact metrics on 3-manifolds are characterized by reducible holonomy.     

Holonomy of supermanifolds

Holonomy groups and holonomy algebras for connections on locally free sheaves over supermanifolds are introduced. A one-to-one correspondence between parallel sections and holonomy-invariant vectors, and a one-to-one correspondence between parallel locally direct subsheaves and holonomy-invariant vector supersubspaces are obtained. As the special case, the holonomy of linear connections on supermanifolds is studied. Examples of parallel geometric structures on supermanifolds and the corresponding holonomies are given. For Riemannian supermanifolds an analog of the Wu theorem is proved. Berger superalgebras are defined and their examples are given. Supported from the Basic Research Center no. LC505 (Eduard Čech Center for Algebra and Geometry) of Ministry of Education, Youth and Sport of Czech Republic.     

Modal characterisation theorems over special classes of frames

We investigate model theoretic characterisations of the expressive power of modal logics in terms of bisimulation invariance. The paradigmatic result of this kind is van Benthem’s theorem, which says that a first-order formula is invariant under bisimulation if, and only if, it is equivalent to a formula of basic modal logic. The present investigation primarily concerns ramifications for specific classes of structures. We study in particular model classes defined through conditions on the underlying frames, with a focus on frame classes that play a major role in modal correspondence theory and often correspond to typical application domains of modal logics. Classical model theoretic arguments do not apply to many of the most interesting classes-for instance, rooted frames, finite rooted frames, finite transitive frames, well-founded transitive frames, finite equivalence frames-as these are not elementary. Instead we develop and extend the game-based analysis (first-order Ehrenfeucht-Fraïssé versus bisimulation games) over such classes and provide bisimulation preserving model constructions within these classes. Over most of the classes considered, we obtain finite model theory analogues of the classically expected characterisations, with new proofs also for the classical setting. The class of transitive frames is a notable exception, with a marked difference between the classical and the finite model theory of bisimulation invariant first-order properties. Over the class of all finite transitive frames in particular, we find that monadic second-order logic is no more expressive than first-order as far as bisimulation invariant properties are concerned — though both are more expressive here than basic modal logic. We obtain ramifications of the de Jongh-Sambin theorem and a new and specific analogue of the Janin-Walukiewicz characterisation of bisimulation invariant monadic second-order for finite transitive frames.     

Multiplier theorems for special Hermite expansions on Cn

The weak type (1,1) estimate for special Hermite expansions on Cn is proved by using the Calderón-Zygmund decomposition. Then the multiplier theorem in Lp(1相似文献   

Riemannian, Symplectic and Weak Holonomy

Much of the early work of Alfred Gray was concerned with the investigation of Riemannian manifolds with special holonomy, one of the most vivid fields of Riemannian geometry in the 1960s and the following decades. It is the purpose of the present article to give a brief summary and an appreciation of Gray's contributions in this area on the one hand, and on the other hand to describe some of the more recent developments in the theory of non-Riemannian or,more specifically, symplectic holonomy groups. Namely, we show that the Merkulov twistor space of a connection on a symplectic manifold M whose holonomy group is irreducible and properly contained in Sp(V) consists of maximal totally geodesic Lagrangian submanifolds of M.     

Multiplier theorems for special Hermite expansions on C n

The weak type (1,1) estimate for special Hermite expansions on Cⁿ is proved by using the Calder/'on-Zygmund decomposition. Then the multiplier theorem in L^p(lpα) is obtained. The special Mermite expansions in twisted Hardy space are also considered. As an application, the multipliers for a certain kind of Laguerre expansions are given in L^p space.     

Holonomy and Parallel Transport for Abelian Gerbes

In this paper, we establish a one-to-one correspondence between U(1)-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with group U(1) on a simply connected manifold M is a group morphism from the thin second homotopy group to U(1), satisfying a smoothness condition, where a homotopy between maps from [0,1]² to M is thin when its derivative is of rank 2. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two special Lie groupoids, which we call Lie 2-groups. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case.     

Holonomy transformations for singular foliations

In order to understand the linearization problem around a leaf of a singular foliation, we extend the familiar holonomy map from the case of regular foliations to the case of singular foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations cannot be attached to classes of paths in the foliation, but rather to elements of the holonomy groupoid of the singular foliation.     

Lie Algebroids, Holonomy and Characteristic Classes

We extend the notion of connection in order to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of a covariant connection. It allows us to define holonomy of the orbit foliation of a Lie algebroid and prove a Stability Theorem. We also introduce secondary or exotic characteristic classes, thus providing invariants which generalize the modular class of a Lie algebroid.     

Geometric Structures on Orbifolds and Holonomy Representations

An orbifold is a topological space modeled on quotient spaces of a finite group actions. We can define the universal cover of an orbifold and the fundamental group as the deck transformation group. Let G be a Lie group acting on a space X. We show that the space of isotopy-equivalence classes of (G, X)-structures on a compact orbifold is locally homeomorphic to the space of representations of the orbifold fundamental group of to G following the work of Thurston, Morgan, and Lok. This implies that the deformation space of (G, X)-structures on is locally homeomorphic to the character variety of representations of the orbifold fundamental group to G when restricted to the region of proper conjugation action by G.     

Complex Projective Structures with Schottky Holonomy

Let S be a closed orientable surface of genus at least two. Let ?? be a Schottky group whose rank is equal to the genus of S, and ?? be the domain of discontinuity of ??. Pick an arbitrary epimorphism : ${{\rho : \pi_{1}(S) \rightarrow \Gamma}}$ . Then ??/?? is a surface homeomorphic to S carrying a (complex) projective structure with holonomy ??. We show that every projective structure with holonomy ?? is obtained by (2??)grafting ??/?? along a multiloop on S.     

Holonomy groups of four-dimensional pseudo-Riemann spaces

Four-dimensional pseudo-Riemann spacesV ₁ with a metric having the signature (3, 1) are investigated. Subgroups of the Lorentz group are described which can be holonomy groups of the pseudo-Riemann spacesV: a) with zero Ricci curvature and b) symmetric. The reducibility of the above class of spaces is determined as a function of the holonomy group.Translated from Matematicheskie Zametki, Vol. 9, No.1, pp. 59–66, January, 1971.In conclusion, the author wishes to thank his scientific director D. V. Alekseevskii for his help in this work.     

Best proximity point theorems: unriddling a special nonlinear programming problem

This paper addresses the non-linear programming problem of globally minimizing the real valued function x?d(x,Sx) where S is a generalized proximal contraction in the setting of a metric space. Eventually, one can obtain optimal approximate solutions to some fixed-point equations in the event that they have no solution.     

Holonomy on poisson manifolds and the modular class

(1) It is shown that ifc is real-valued measurable then the Maharam type of (c, P(c),σ) is 2 ^c . This answers a question of D. Fremlin [Fr, (P2f)]. (2) A different construction of a model with a real-valued measurable cardinal is given from that of R. Solovay [So]. This answers a question of D. Fremlin [Fr, (P1)]. (3) The forcing with aκ-complete ideal over a setX, |X| ≥κ cannot be isomorphic to Random × Cohen or Cohen × Random. The result forX=κ was proved in [Gi-Sh1] but, as was pointed out to us by M. Burke, the application of it in [Gi-Sh2] requires dealing with anyX. The application is: ifA _n is a set of reals forn<ω then for some pairwise disjointB _n (forn<ω) we haveB _n ⊆A _n but they have the same outer Lebesgue measure. Partially supported by the Israeli Basic Research Fund. Publ. Number 582.     

Jackson theorems for generalized polynomials with special applications to trigonometric and hyperbolic functions

Jackson theorems for polynomials are transformed into Jackson theorems for more general function classes by way of special operators. In particular, Jackson-Timan and inverse theorems are shown for classes of trigonometric and hyperbolic functions.     

Holonomy group scheme of an integral curve

Let Y be a projective variety over a field k (of arbitrary characteristic). Assume that the normalization X of Y is such that is normal, being the algebraic closure of k. We define a notion of strong semistability for vector bundles on Y. We show that a vector bundle on Y is strongly semistable if and only if its pull back to X is strongly semistable and hence it is a tensor category. In case , we show that strongly semistable vector bundles on Y form a neutral Tannakian category. We define the holonomy group scheme of Y to be the Tannakian group scheme for this category. For a strongly semistable principal G‐bundle , we construct a holonomy group scheme. We show that if Y is an integral complex nodal curve, then the holonomy group of a strongly semistable vector bundle on Y is the Zariski closure of the (topological) fundamental group of Y.