Cotangent Bundle Research Articles

Projective Vector Fields on the Cotangent Bundle with Modified Riemannian Extension

Let be the cotangent bundle of an dimensional Riemannian manifold . The purpose of the present paper is give a characterization of fibre-preserving projective vector fields with respect to modified Riemannian extension.

Degeneration of globally hyperbolic maximal anti-de Sitter structures along pinching sequences

Let S be a closed oriented surface of genus at least 2. Using the parameterisation of the deformation space of globally hyperbolic maximal anti-de Sitter structures on S×R by the cotangent bundle over the Teichmüller space of S, we study the behaviour of these geometric structures along pinching sequences. We show, in particular, that the regular globally hyperbolic anti-de Sitter structures introduced in [15] naturally appear as limiting points.

Elliptic and K-theoretic stable envelopes and Newton polytopes

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $$K$$ -theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the $$K$$ -theoretic stable envelopes and our elliptic stable envelopes. We show that the $$K$$ -theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the $${\mathfrak {gl}}_2$$ case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the $$K$$ -theoretic stable envelopes.

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.

Birational geometry of compactifications of Drinfeld half-spaces over a finite field

We study compactifications of Drinfeld half-spaces over a finite field. In particular, we construct a purely inseparable endomorphism of Drinfeld's half-space Ω(V) over a finite field k that does not extend to an endomorphism of the projective space P(V). This should be compared with theorem of Rémy, Thuillier and Werner that every k-automorphism of Ω(V) extends to a k-automorphism of P(V). Our construction uses an inseparable analogue of the Cremona transformation. We also study foliations on Drinfeld's half-spaces. This leads to various examples of interesting varieties in positive characteristic. In particular, we show a new example of a non-liftable projective Calabi–Yau threefold in characteristic 2 and we show examples of rational surfaces with klt singularities, whose cotangent bundle contains an ample line bundle.

Representing the Big tilting sheaves as holomorphic Morse Branes

We introduce Morse branes in the Fukaya category of a holomorphic symplectic manifold, with the goal of constructing tilting objects in the category. We give a construction of a class of Morse branes in the cotangent bundles, and apply it to give the holomorphic branes that represent the big tilting sheaves on flag varieties.

On integrability conditions, operators, and the purity conditions of the Sasakian metric with respect to lifts of F l (7 ; 1) -structure on the cotangent bundle

There are many structures in the cotangent bundle. These include the complete and horizontal lifts of the $F_{\lambda }(7,1)$-structure. The $F_{\lambda }(7,1)$-structure was first extended in $M^{n}$ to $T^{\ast }(M^{n})$ by Das, Nivas, and Pathak. Later, the horizontal and complete lift of the $F_{a}(K,1)$-structure in the tangent bundle was given by Prasad and Chauhan. This paper consists of two main sections. In the first part, we find the integrability conditions by calculating Nijenhuis tensors of the complete lifts of the $F_{\lambda }(7,1)$-structure. Later, we get the results of the Tachibana operators applied to vector and covector fields according to the complete lifts of the $% F_{\lambda }(7,1)$-structure in the cotangent bundle $T^{\ast }(M_{n})$. Finally, we study the purity conditions of the Sasakian metric with respect to the complete lifts of the $F_{\lambda }(7,1)$-structure. In the second part, all results obtained in the first section are obtained according to the horizontal lifts of the $F_{\lambda }(7,1)$-structure in cotangent bundle $T^{\ast }(M_{n})$.

Homothetic congruences in general relativity

The kinematical characteristics of distinct infalling homothetic fields are discussed by specifying the transverse subspace of their generated congruences to the energy–momentum deposit of the chosen gravitational system. This is pursued through the inclusion of the base manifold’s cotangent bundle in a generalized Raychaudhuri equation and its kinematical expressions. Exploiting an electromagnetic energy–momentum tensor as the source of non-gravitational effects, I investigate the evolution of the mentioned homothetic congruences, as they fall onto a Reissner–Nordström black hole. The results show remarkable differences to the common expectations from infalling congruences of massive particles.

On the Relation of Lie Algebroids to Constrained Systems and their BV/BFV Formulation

We observe that a system of irreducible, fiber-linear, first class constraints on T*M is equivalent to the definition of a foliation Lie algebroid over M. The BFV formulation of the constrained system is given by the Hamiltonian lift of the Vaintrob description (E[1],Q) of the Lie algebroid to its cotangent bundle T*E[1]. Affine deformations of the constraints are parametrized by the first Lie algebroid cohomology H^1_Q and lead to irreducible constraints also for much more general Lie algebroids such as Dirac structures; the modified BFV function follows by the addition of a representative of the deformation charge. Adding a Hamiltonian to the system corresponds to a metric g on M. Evolution invariance of the constraint surface introduces a connection nabla on E and one reobtains the compatibility of g with (E,rho,nabla) found previously in the literature. The covariantization of the Hamiltonian to a function on T*E[1] serves as a BFV-Hamiltonian, iff, in addition, this connection is compatible with the Lie algebroid structure, turning (E, rho, [ , ], nabla) into a Cartan-Lie algebroid. The BV formulation of the system is obtained from BFV by a (time-dependent) AKSZ procedure.

Projective geometry and the quaternionic Feix–Kaledin construction

Starting from a complex manifold S with a real-analytic c-projective structure whose curvature has type (1,1), and a complex line bundle L with a connection whose curvature has type (1,1), we construct the twistor space Z of a quaternionic manifold M with a quaternionic circle action which contains S as a totally complex submanifold fixed by the action. This extends a construction of hypercomplex manifolds, including hyperkaehler metrics on cotangent bundles, obtained independently by B. Feix and D. Kaledin. When S is a Riemann surface, M is a self-dual conformal 4-manifold, and the quotient of M by the circle action is an Einstein-Weyl manifold with an asymptotically hyperbolic end, and our construction coincides with a construction presented by the first author in a previous paper. The extension also applies to quaternionic Kaehler manifolds with circle actions, as studied by A. Haydys and N. Hitchin.

On Cotangent Bundles Hamiltonian Tubes Theorem and Its Some Applications in Reduction Theory

This paper aims to study the Cotangent Bundles Hamiltonian Tubes theorem and its applications in reduction theory. The mathematical analysis method used. And found some results; The theory of reduction of cotangent bundles developed playing an important role in solution of the general problem for reduction a single or bit type cotangent bundles for base manifolds, possibility study of Hamiltonian tubes when the simplistic manifolds is a cotangent bundles, in the concrete case of cotangent bundles there is a strong motivation coming from geometric mechanics and geometric quantization that makes it desirable to obtain explicit fiber local models.

Discrete Optimal Control of Interconnected Mechanical Systems

This article develops variational integrators for a class of underactuated mechanical systems using the theory of discrete mechanics. A discrete optimal control problem is then formulated for this class of system on the phase spaces of the actuated and unactuated subsystems separately. Exploiting the left-trivialization of the cotangent bundle, and assuming the time-step of discrete evolution is small enough to exploit the diffeomorphism feature of the exponential map in a neighbourhood of the identity of the Lie group, that enables a mapping of the group variables to the Lie algebra, a variational approach is adopted to obtain the first order necessary conditions that characterise optimal trajectories. The proposed approach is then demonstrated on two benchmark underactuated systems through numerical experiments.

Mather theory and symplectic rigidity

Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $\phi_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,\omega)$. These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures and we construct an example showing that their support can be extremely unstable when $H$ fails to be convex, even for nearly integrable $H$. Parts of these results extend work by Viterbo and Vichery. Using ideas due to Entov-Polterovich we also detect interesting invariant measures for $\phi_H$ by studying a generalization of the symplectic shape of sublevel sets of $H$. This approach differs from the first one in that it works also for $(M,\omega)$ in which every compact subset can be displaced. We present applications to Hamiltonian systems on $\mathbb{R}^{2n}$ and twisted cotangent bundles.

On the jumping phenomenon of $\operatorname{dim}_{\mathbb{C}} H^q (\mathcal{X}_t,\mathcal{E}_t)$

Let $X$ be a compact complex manifold and $E$ be a holomorphic vector bundle on $X$. Given a deformation $(\mathcal{X},\mathcal{E})$ of the pair $(X,E)$ over a small polydisk $B$ centered at the origin, we study the jumping phenomenon of the cohomology groups $\dim_{\mathbb{C}}H^q(\mathcal{X}_t,\mathcal{E}_t)$ near $t = 0$. Generalizing previous results of X. Ye for the tangent bundle $E = T_{\mathcal{X}_t}$ and exterior powers of the cotangent bundle $E = \Omega^p_{\mathcal{X}_t}$, we show that there are precisely two cohomological obstructions to the stability of $\dim_{\mathbb{C}}H^q(\mathcal{X}_t,\mathcal{E}_t)$, which can be expressed explicitly in terms of the Maurer-Cartan element associated to the deformation $(\mathcal{X},\mathcal{E})$. As an application, we study the jumping phenomenon of the dimension of the cohomology group $H^1(\mathcal{X}_t,\text{End}(T_{\mathcal{X}_t}))$ which is related to a question raised by physicists.

Relations Between Laplace Spectra and Geometric Quantization of Reimannian Symmetric Spaces

We consider a modified Kostant-Souriau geometric quantization scheme due to Czyz and Hess for Hamiltonian systems on the cotangent bundles of compact rank-one Riemannian symmetric spaces (CROSS). It is used, together with a symplectic reduction process, to relate its energy spectrum to the spectrum of the Laplace-Beltrami operator. Moreover, the corresponding eigenspaces have real dimension equal to the complex dimension of the space of the holomorphic sections of the quantum bundle which is obtained after the quantization. The relation between the two constructions was first noticed by Mladenov and Tsanov for the case of the spheres. In addition to the CROSS case, we announce preliminary results related to the case of compact Riemannian symmetric spaces of higher rank.

Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the first case we also discuss a distinguished contact cone structure, arising as the variety of minimal rational tangents. We discuss the repercussion of the aforementioned classification theorems for the geometry of quaternion-K\"ahler manifolds with positive scalar curvature and for the geometry of second-order PDEs imposed on hypersurfaces.

Note on a ball rolling over a sphere: Integrable Chaplygin system with an invariant measure without Chaplygin hamiltonization

In this note we consider the nonholonomic problem of rolling without slipping and twisting of an ??-dimensional balanced ball over a fixed sphere. This is a ????(??)?Chaplygin system with an invariant measure that reduces to the cotangent bundle ??*?????1. For the rigid body inertia operator r I? = I? + ?I, I = diag(I1,...,In) with a symmetry I1 = I2 = ... =Ir ? Ir+1 = Ir+2 = ... = In, we prove that the reduced system is integrable, general trajectories are quasi-periodic, while for ?? ? 1, ?? ? 1 the Chaplygin reducing multiplier method does not apply.

An underactuated drift-free left-invariant control system on the Lie group ISO(3,1)

The purpose of our paper is to study a class of left-invariant, drift-free optimal control problem on the Lie group ISO(3,1). The left-invariant, drift-free optimal control problems involves finding a trajectory-control pair on ISO(3,1), which minimize a cost function and satisfies the given dynamical constrains and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*G using the optimal Hamiltonian on G*, where the maximum principle yields the optimal control. We use energy methods (Arnold’s method, in this case) to give sufficient conditions fornonlinear stability of the equilibrium states. Around this equilibrium states we might be able to find the periodical orbits using Moser's theorem, as future work. For the some unstable equilibrium states, a quadratic control is considered in order to stabilize the dynamics.

Systems of Hamilton-Jacobi equations

In this article we develop a generalization of the Hamilton-Jacobi theory, by considering in the cotangent bundle an involutive system of dynamical equations.

Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology

We define an elliptic version of the stable envelope of Maulik and Okounkov for the equivariant elliptic cohomology of cotangent bundles of Grassmannians. It is a version of the construction proposed by Aganagic and Okounkov and is based on weight functions and shuffle products. We construct an action of the dynamical elliptic quantum group associated with $\mathfrak{gl}_2$ on the equivariant elliptic cohomology of the union of cotangent bundles of Grassmannians. The generators of the elliptic quantum groups act as difference operators on sections of admissible bundles, a notion introduced in this paper.