The generalized stokes integral theorem for a covariant pseudotensor field

Ориентируемые континуумы играют важную роль в микрополярной теории упругости, все реализации которой возможны только в рамках псевдотензорного формализма и представления об ориентируемом многообразии. Особенно это касается теории микрополярных гемитропных упругих сред. В настоящей работе рассматриваются различные формулировки интегральной теоремы Стокса для асимметричного ковариантного пседотензорного поля, заданного веса. Тем самым достигается распространение известной интегральной формулы Стокса на случай псевдотензоров. Последнее обстоятельство позволяет использовать, указанное обобщение для микрополярных континуумов. Исследование существенно опирается на класс специальных координатных систем. Oriented continua play an important role in the micropolar theory of elasticity, all realizations of which are possible only within the framework of the pseudotensor formalism and the orientable manifold concept. This especially concerns the theory of micropolar hemitropic elastic media. In this paper, we consider various formulations of the Stokes integral theorem for an asymmetric covariant pseudotensor field of a given weight. This extends the well-known Stokes integral formula to the case of pseudotensors. The latter circumstance makes it possible to use the manifistated generalization for micropolar continua. The study relies heavily on the class of special coordinate systems.

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.

Polynomially convex embeddings of odd-dimensional closed manifolds

It is shown that any smooth closed orientable manifold of dimension 2 ⁢ k + 1 {2k+1} , k ≥ 2 {k\geq 2} , admits a smooth polynomially convex embedding into ℂ 3 ⁢ k {\mathbb{C}^{3k}} . This improves by 1 the previously known lower bound of 3 ⁢ k + 1 {3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when k = 1 {k=1} ). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.

The size of some vanishing and critical sets

We prove that the vanishing sets of all top forms on a non-orientable manifold are at least 1-dimensional in the general case and at most $1$-codimen\-sional in the compact case. We apply these facts to show that the critical sets of some differentiable maps are at least 1-dimensional in the general case and at most 1-codimensional when the source manifold is compact.

Manifolds with Odd Euler Characteristic and Higher Orientability

It is well known that odd-dimensional manifolds have Euler characteristic zero. Furthermore, orientable manifolds have an even Euler characteristic unless the dimension is a multiple of $4$. We prove here a generalisation of these statements: a $k$-orientable manifold (or more generally Poincaré complex) has even Euler characteristic unless the dimension is a multiple of $2^{k+1}$, where we call a manifold $k$-orientable if the i-th Stiefel–Whitney class vanishes for all $0<i< 2^k$ ($k\geq 0$). More generally, we show that for a $k$-orientable manifold the Wu classes $v_l$ vanish for all $l$ that are not a multiple of $2^k$. For $k=0,1,2,3$, $k$-orientable manifolds with odd Euler characteristic exist in all dimensions $2^{k+1}m$, but whether there exists a 4-orientable manifold with an odd Euler characteristic is an open question.

Study on De Rham Cohomology Algebra of Manifolds

In the present paper some aspects of exterior derivative, graded algebra, cohomology algebra, de Rham cohomology algebra, singular homology, cohomology class are studied. Graded subspace, smooth map, a singular P- - simplex in a manifold M, oriented n- manifold M, the space of P- cycles and P- boundaries, Pth singular homology and homology class are treated in our paper. A theorem 3.03 is established which is related to orientable manifold. Dhaka Univ. J. Sci. 64(2): 109-113, 2016 (July)

The number of framings of a knot in a 3-manifold

In view of the self-linking invariant, the number |K| of framed knots in S3 with given underlying knot K is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that |K| is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of S1 × S2; the knot K need not be zero-homologous and the manifold is not required to be compact. We show that when M is orientable, the number |K| is infinite unless K intersects a nonseparating sphere at exactly one point, in which case |K| = 2; the existence of a nonseparating sphere implies that M contains a connected sum factor of S1 × S2. For knots in nonorientable manifolds we show that if |K| is finite, then K is disorienting, or there is an orientation-preserving isotopy of the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded S2 or ℝP2 at exactly one point, or it intersects some embedded S2 at exactly two points in such a way that a closed curve consisting of an arc in K between the intersection points and an arc in S2 is disorienting.

A Spectral Sequence for the Hochschild Cohomology of a Coconnective DGA

A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence presented in [7] for the computation of the loop homology of a closed orientable manifold. Using this spectral sequence we identify a class of spaces for which the Hochschild cohomology of their mod-$p$ cochain algebra is Noetherian. This implies, among other things, that for such a space the derived category of mod-$p$ chains on its loop-space carries a theory of support varieties.

Transitivity of codimension-one Anosov actions of ℝk on closed manifolds

We consider Anosov actions of ℝk, k≥2, on a closed connected orientable manifold M, of codimension one, i.e. such that the unstable foliation associated to some element of ℝk has dimension one. We prove that if the ambient manifold has dimension greater than k+2, then the action is topologically transitive. This generalizes a result of Verjovsky for codimension-one Anosov flows.

Enumeration of orientable coverings of a non-orientable manifold

International audience In this paper we solve the known V.A. Liskovets problem (1996) on the enumeration of orientable coverings over a non-orientable manifold with an arbitrary finitely generated fundamental group. As an application we obtain general formulas for the number of chiral and reflexible coverings over the manifold. As a further application, we count the chiral and reflexible maps and hypermaps on a closed orientable surface by the number of edges. Also, by this method the number of self-dual and Petri-dual maps can be determined. This will be done in forthcoming papers by authors.