О двух структурных тензорах acm-структуры

A contact form on an odd-dimensional manifold M of dimension 2n + 1 is a one-form λ such that the (2n + 1)-form Ω given by $$\displaystyle \Omega = \lambda \wedge (d\lambda )^n $$ defines a volume form on M. We observe that any manifold admitting a contact form is necessarily orientable, and that a contact form defines a natural orientation.

We present a parabolic deformation of one-forms on compact, orientable, odd-dimensional manifolds.The flow produces contact forms from the class of initial conditions called "conductive confoliations".We give applications of these techniques to new constructions of contact forms on products of contact manifolds with surfaces.In particular, we produce contact forms on the product of any three dimensional manifold with any surface.

An almost para-cosymplectic manifold is by definition an odd-dimensional differentiable manifold endowed with an almost paracontact stmcture with hyperbolic metric for which the structure forms are closed. The local structure of an almost para-cosymplectic manifold is described. We also treat some special subclasses of this class of manifolds: para-cosymplectic, weakly para-cosymplectic and almost para-cosymplectic with para-Kahlerian leaves. Necessary and sufficient conditions for an almost para-cosymplectic manifold to be para-cosymplectic are found. Necessary and sufficient conditions for an almost para-cosymplectic manifold with para-Kahlerian leaves to be weakly para-cosymplectic are also established. We construct examples of weakly para-cosymplectic manifolds, which are not paracosymplectic. It is proved that in dimensions $\geq 5$ an almost paracosymplectic manifold cannot be of non-zero constant sectional curvature. Main curvature identities which are fulfilled by any almost para-cosymplectic manifold are found.

Given an odd-dimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the \(L^2\) inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain right-invariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an “Euler–Arnold” equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a “quasi-Lipschitz” estimate on the stream function, which leads to a Beale–Kato–Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler–Arnold equations are the Camassa–Holm equation (when the manifold is one-dimensional) and the quasi-geostrophic equation in geophysics.

This paper contains a detailed study of the behavior of the first exotic contact structure of J. Gonzalo and F. Varela on S3 along a remarkable vector field v in its kernel found by Martino (Adv Nonlinear Stud, 2011). All contact forms are assumed to verify the condition that reads as follows: d(θ α)(v,.) is a contact form with the same orientation than α. α is the first exotic contact form of Gonzalo and Varela (Third Schnepfenried Geometry Conference, vol 1, Asterisque no 107–108, pp 163–168. Société Mathématique de France, Paris, 1983). We also prove in this paper that the contact homology (via dual Legendrian curves) is non-zero for a sequence of indexes tending to infinity for the contact forms θα of the first exotic contact structure of J. Gonzalo and F. Varela on S3, under the assumption that they can be connected to the first contact form of this contact structure through a path along which a special pseudo-gradient which we build in this paper is assumed to verify a Fredholm condition (see the Sect. 1 and Bahri in Morse relations and Fredholm deformations of v-convex contact forms, 2014 for the definition of this notion). We do not know whether this assumption is verified for the pseudo-gradient which we use here.

Cortical firing rates frequently display elaborate and heterogeneous temporal structure. One often wishes to compute quantitative summaries of such structure—a basic example is the frequency spectrum—and compare with model-based predictions. The advent of large-scale population recordings affords the opportunity to do so in new ways, with the hope of distinguishing between potential explanations for why responses vary with time. We introduce a method that assesses a basic but previously unexplored form of population-level structure: when data contain responses across multiple neurons, conditions, and times, they are naturally expressed as a third-order tensor. We examined tensor structure for multiple datasets from primary visual cortex (V1) and primary motor cortex (M1). All V1 datasets were ‘simplest’ (there were relatively few degrees of freedom) along the neuron mode, while all M1 datasets were simplest along the condition mode. These differences could not be inferred from surface-level response features. Formal considerations suggest why tensor structure might differ across modes. For idealized linear models, structure is simplest across the neuron mode when responses reflect external variables, and simplest across the condition mode when responses reflect population dynamics. This same pattern was present for existing models that seek to explain motor cortex responses. Critically, only dynamical models displayed tensor structure that agreed with the empirical M1 data. These results illustrate that tensor structure is a basic feature of the data. For M1 the tensor structure was compatible with only a subset of existing models.

Instantons and hypercontact structures

In this note we define a global, R-valued invariant of a compact, strictly pseudoconvex 3-dimensional CR-manifold M whose holomorphic tangent bundle is trivial. The invariant arises as the evaluation of a deRham cohomology class on the fundamental class of the manifold. To construct the relevant form, we start with the CR structure bundle Y over M (see [Ch-Mo], whose notation we follow). The form is a secondary characteristic form of this structure. By fixing a contact form and coframe, i.e., a section of Y, we obtain a form on M. Surprisingly, this form is well-defined up to an exact term, and thus its cohomology class is well-defined in H 3 (M, R). Our motivation for studying this invariant was its analogy with the R/Z secondary characteristic number associated by Chern and Simons to the conforreal class of a Riemannian 3-manifold N, which provides an obstruction to the conformal immersion of N in R 4. Though several formal analogies to the conformal case are valid for our invariant, this one does not hold up: specifically, in w below, we calculate examples which show that the CR invariant can take on any positive real value for hypersurfaces embedded in C 2. It is also clear that the invariant is neither a homotopy nor concordance invariant, but depends in an elusive way on the CR structure. Our inspiration came from the seminal papers of Chern and Moser and Chern and Simons. The idea of looking at secondary characteristic forms of higher order geometric structures in general appears in [Ko-Oc], though with a different intention. In w 2 we will quickly review the definition of a CR structure, the construction of y and its reduction to a pseudo-hermitian structure ~ ld Webster [-We]. In w 3 we define the invariant and prove that it is, in fact, R-valued, and not R/Z-valued as in the Riemannian case. We also prove that if the invariant is stationary as a function of the CR structure, then M is locally CR equivalent to the standard three sphere in C 2, paralleling a result of Chern and Simons. As noted already, w 4 is devoted to the calculation of several examples.

Elastic electron-deuteron scattering at high momentum transfer is investigated within the Bethe-Salpeter approach. The relativistic covariant Graz II separable kernel of nucleon-nucleon interactions is used to analyze the deuteron structure functions, form factors, and tensor of polarization components. The modern data for the electromagnetic structure of nucleons from the double polarization experiments, as well as some other models of the nucleon form factors, are considered.

Let M be an n-dimensional (n = 2m + 1, m ≦ 1) real differentiable manifold. if on M there exist a tensor field , a contravariant vector field ξi and a convariant vector field ηi such that then M is said to have an almost contact structure with the structure tensors (φ,ξ, η) [1], [2]. Further, if a positive definite Riemannian metric g satisfies the conditions then g is called an associated Riemannian metric to the almost contact structure and M is then said to have an almost contact metric structure. On the other hand, M is said to have a contact structure [2], [4] if there exists a 1-form η over M such that η ∧ (dη)m ≠ 0 everywhere over M where dη means the exterior derivation of η and the symbol ∧ means the exterior multiplication. In this case M is said to be a contact manifold with contact form η. It is known [2, Th. 3,1] that if η = ηidxi is a 1-form defining a contact structure, then there exists a positive definite Riemannian metric in gij such that and define an almost contact metric structure with and ηi where the symbol ∂i standing for ∂/∂xi.

In this paper we generalize the definition of symplectic connection to the contact case. It turns out that any odd-dimensional manifold equipped with a contact form admits contact connections and that any Sasakian structure induces a canonical contact connection. Furthermore (as in the symplectic case), any contact connection induces an almost CR structure on the contact twistor space which is integrable if and only if the curvature of the connection is of Ricci-type.

Hawking's chronology protection conjecture: singularity structure of the quantum stress-energy tensor

Almost contact metric (аст-)structures induced on oriented hypersurfaces of a Kählerian manifold are considered in the case when these аст- structures are of cosymplectic type, i. e. the contact form of these structures is closed. As it is known, the Kenmotsu structure is the most important non-trivial example of an almost contact metric structure of cosymplectic type. The Cartan structural equations of the almost contact metric structure of cosymplectic type on a hypersurface of a Kählerian manifold are obtained. It is proved that an almost contact metric structure of cosymplectic type on a hypersurface of a Kählerian manifold of dimension at least six cannot be a Kenmotsu structure. Moreover, it follows that oriented hypersurfaces of a Kählerian manifold of dimension at least six do not admit non-trivial almost contact metric structures of cosymplectic type that belong to any well studied class of аст-structures. The present results generalize some results on almost contact metric structures on hypersurfaces of an almost Hermitian manifold obtained earlier by V. F. Kirichenko, L. V. Stepanova, A. Abu-Saleem, M. B. Banaru and others.