Manifold Bundles Research Articles

Examples of Integrable Systems with Dissipation on the Tangent Bundles of Three-Dimensional Manifolds

In this paper, we prove the integrability of certain classes of dynamical systems on the tangent bundles of four-dimensional manifolds (systems with four degrees of freedom). The force field considered possessed so-called variable dissipation; they are generalizations of fields studied earlier. This paper continues earlier works of the author devoted to systems on the tangent bundles of two- and three-dimensional manifolds.

Examples of Integrable Systems with Dissipation on the Tangent Bundles of Four-Dimensional Manifolds

Examples of Integrable Systems with Dissipation on the Tangent Bundles of Four-Dimensional Manifolds

New Cases of Homogeneous Integrable Systems with Dissipation on Tangent Bundles of Two-Dimensional Manifolds

The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of two-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.

Deformations of the tangent bundle of projective manifolds

We investigate when the tangent bundle of a projective manifold has a nontrivial first-order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.

Characteristic numbers of manifold bundles over surfaces with highly connected fibers

We study smooth bundles over surfaces with highly connected almost parallelizable fiber $M$ of even dimension, providing necessary conditions for a manifold to be bordant to the total space of such a bundle and showing that, in most cases, these conditions are also sufficient. Using this, we determine the characteristic numbers realized by total spaces of bundles of this type, deduce divisibility constraints on their signatures and $\hat{A}$-genera, and compute the second integral cohomology of ${\rm BDiff}^+(M)$ up to torsion in terms of generalized Miller--Morita--Mumford classes. We also prove analogous results for topological bundles over surfaces with fiber $M$ and discuss the resulting obstructions to smoothing them.

Topological manifold bundles and the 𝐴-theory assembly map

We give a new proof of an index theorem for fiber bundles of compact topological manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized A A -theory characteristic of such a fiber bundle factors canonically through the assembly map of A A -theory. Furthermore our main result shows a refinement of this statement by providing such a factorization for an extended A A -theory characteristic, defined on the parametrized topological cobordism category. The proof uses a convenient framework for bivariant theories and recent results of Gomez-Lopez and Kupers on the homotopy type of the topological cobordism category. We conjecture that this lift of the extended A A -theory characteristic becomes highly connected as the manifold dimension increases.

New Cases of Integrable Odd-Order Systems with Dissipation

This paper shows the integrability of certain classes of odd-order dynamical systems that are homogeneous with respect to some of the variables and in which a system on the tangent bundle of smooth manifolds is distinguished. In this case, the force fields have dissipation of different signs and generalize previously considered cases.

Integrable Seventh and Ninth Order Dynamic Systems with Dissipation

We establish the integrability for a class of homogeneous (with respect to a part of variables) dynamic systems of odd (seventh and ninth) order, admitting separation of a system on the tangent bundle of smooth manifolds. Bibliography: 13 titles.

New Cases of Integrable Ninth-Order Systems with Dissipation

Integrability is shown for some classes of fifth-order dynamic systems homogeneous in variables, in which a system on the tangent bundle of two-dimensional manifolds is separated. Here, the force fields possess the dissipation of alternating signs and generalize the fields considered earlier.

Integrable Dynamical Systems with Dissipation on Tangent Bundles of 2D and 3D Manifolds

In many problems of dynamics, one has to deal with mechanical systems whose configurational spaces are two- or three-dimensional manifolds. For such a system, the phase space naturally coincides with the tangent bundle of the corresponding manifold. Thus, the problem of a flow past a (four-dimensional) pendulum on a (generalized) spherical hinge leads to a system on the tangent bundle of a two- or threedimensional sphere whose metric has a particular structure induced by an additional symmetry group. In such cases, dynamical systems have variable dissipation, and their complete list of first integrals consists of transcendental functions in the form of finite combinations of elementary functions. Another class of problems pertains to a point moving on a two- or three-dimensional surface with the metric induced by the encompassing Euclidean space. In this paper, we establish the integrability of some classes of dynamical systems on tangent bundles of two- and three-dimensional manifolds, in particular, systems involving fields of forces with variable dissipation and of a more general type than those considered previously.

On characteristic classes of exotic manifold bundles

Given a closed simply connected manifold M of dimension 2nge 6, we compare the ring of characteristic classes of smooth oriented bundles with fibre M to the analogous ring resulting from replacing M by the connected sum Msharp Sigma with an exotic sphere Sigma . We show that, after inverting the order of Sigma in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of Sigma is necessary.

New cases of integrable fifth-order systems with dissipation

Integrability is shown for some classes of fifth-order dynamic systems homogeneous in variables, in which a system on the tangent bundle of two-dimensional manifolds is separated. Here, the force fields possess the dissipation of alternating signs and generalize the fields considered earlier.

Cohomology of torus manifold bundles

Abstract Let X be a 2n-dimensional torus manifold with a locally standard T ≅ (S 1) n action whose orbit space is a homology polytope. Smooth complete complex toric varieties and quasitoric manifolds are examples of torus manifolds. Consider a principal T-bundle p : E → B and let π : E(X) → B be the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as a H *(B)-algebra and the topological K-ring of E(X) as a K *(B)-algebra with generators and relations. These generalize the results in [17] and [19] when the base B = pt. These also extend the results in [20], obtained in the case of a smooth projective toric variety, to any smooth complete toric variety.

Integrable Third and Fifth Order Dynamical Systems with Dissipation

We establish the integrability of some homogeneous (with respect to a part of variables) dynamical systems of odd (third and fifth) order. In the class of such systems, we extract a system on the tangent bundle of smooth manifolds. The force field is separated into the internal (conservative) force field and the external force field with alternating dissipation. The external force field is introduced by using a certain unimodular transformation and generalizes the cases studied by the author earlier.

Robust Global Structure From Motion Pipeline With Parallax on Manifold Bundle Adjustment and Initialization

In this letter, we present a novel global structure from motion (SfM) pipeline that is particularly effective in dealing with low-parallax scenes and camera motion collinear with the features that represent the environment structure. It is therefore particularly suitable in Urban SLAM, in which frequent road-facing motion poses many challenges to conventional SLAM algorithms. Our pipeline includes a recently explored bundle adjustment ( BA ) method that exploits a feature parameterization using Parallax angle between on-Manifold observation rays (PMBA). It is demonstrated that this BA stage has a consistently stable optimization configuration for features with any parallax and therefore low-parallax features can stay in reconstruction without pre-filtering. To allow practical usage of PMBA, we provide a compatible initialization stage in the SfM to initialize all camera poses simultaneously, exhibiting friendliness to collinear motion. This is achieved by simplifying PMBA into a hybrid graph problem of high connectivity yet small node set size, solved using a robust linear programming technique. Using simulations and a series of publicly available real datasets including “KITTI” and “Bundle Adjustment in the Large,” we demonstrate the robustness of the position initialization stage in handling collinear motion and outlier matches, superior convergence performance of the BA stage in the presence of low-parallax features, and effectiveness of our pipeline to handle many sequential or out-of-order urban scenes.

On topological and measurable dynamics of unipotent frame flows for hyperbolic manifolds

We study the dynamics of unipotent flows on frame bundles of hyperbolic manifolds of infinite volume. We prove that they are topologi-cally transitive, and that the natural invariant measure, the so-called " Burger-Roblin measure ", is ergodic, as soon as the geodesic flow admits a finite measure of maximal entropy, and this entropy is strictly greater than the codi-mension of the unipotent flow inside the maximal unipotent flow. The latter result generalises a Theorem of Mohammadi and Oh.

Topologically twisted SUSY gauge theory, gauge-Bethe correspondence and quantum cohomology

We calculate the partition function and correlation functions in A-twisted 2d mathcal{N} = (2, 2) U(N) gauge theories and topologically twisted 3d mathcal{N} = 2 U(N) gauge theories containing an adjoint chiral multiplet with particular choices of R-charges and the magnetic fluxes for flavor symmetries. According to the Gauge-Bethe correspondence, they correspond to the Heisenberg XXX1/2 and XXZ1/2 spin chain models, respectively. We identify the partition function with the inverse of the norm of the Bethe eigenstate. Correlation functions are identified to coefficients of the expectation value of Baxter Q-operator. In addition, we consider correlation functions of 2d mathcal{N} = (2, 2)* theories and their relations to the equivariant integration of the equivariant quantum cohomology classes of the cotangent bundle of Grassmann manifolds and the equivariant quantum cohomology ring. Also, we study the twisted chiral ring relations of supersymmetric Wilson loops in 3d mathcal{N} = 2* theories and the Bethe subalgebra of the XXZ1/2 spin chain models.

Tautological classes and smooth bundles over BSU(2)

For a Lie group G and a smooth manifold W, we study the difference between smooth actions of G on W and bundles over the classifying space of G with fiber W and structure group Diff(W). In particular, we exhibit smooth manifold bundles over BSU(2) that are not induced by an action. The main tool for reaching this goal is a technical result that gives a constraint for the values of tautological classes pulled back to the cohomology of BSU(2) along a map induced by an action.

Integrable Dynamic Systems with Dissipation and Finitely Many Degrees of Freedom

We establish the integrability for some classes of dynamic systems on the tangent bundles of multidimensional manifolds. We consider the case where the force fields possess variable dissipation. An example of a four-dimensional manifold is discussed in detail.

New cases of integrable systems with dissipation on the tangent bundles of multi-dimensional manifolds

New cases of integrable systems with dissipation on the tangent bundles of multi-dimensional manifolds