Cotangent Bundle Research Articles

Almost Complex and Hypercomplex Norden Structures Induced by Natural Riemann Extensions

The Riemann extension, introduced by E. K. Patterson and A. G. Walker, is a semi-Riemannian metric with a neutral signature on the cotangent bundle T∗M of a smooth manifold M, induced by a symmetric linear connection ∇ on M. In this paper we deal with a natural Riemann extension g¯, which is a generalization (due to M. Sekizawa and O. Kowalski) of the Riemann extension. We construct an almost complex structure J¯ on the cotangent bundle T∗M of an almost complex manifold (M,J,∇) with a symmetric linear connection ∇ such that (T∗M,J¯,g¯) is an almost complex manifold, where the natural Riemann extension g¯ is a Norden metric. We obtain necessary and sufficient conditions for (T∗M,J¯,g¯) to belong to the main classes of the Ganchev–Borisov classification of the almost complex manifolds with Norden metric. We also examine the cases when the base manifold is an almost complex manifold with Norden metric or it is a complex manifold (M,J,∇′) endowed with an almost complex connection ∇′ (∇′J=0). We investigate the harmonicity with respect to g¯ of the almost complex structure J¯, according to the type of the base manifold. Moreover, we define an almost hypercomplex structure (J¯1,J¯2,J¯3) on the cotangent bundle T∗M4n of an almost hypercomplex manifold (M4n,J1,J2,J3,∇) with a symmetric linear connection ∇. The natural Riemann extension g¯ is a Hermitian metric with respect to J¯1 and a Norden metric with respect to J¯2 and J¯3.

Bourgeois contact structures: Tightness, fillability and applications

Given a contact structure on a manifold V together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on V times {mathbb {T}}^2. We prove that all such structures are universally tight in dimension 5, independent of whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact 5-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for {mathbb {S}}^1-invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the n-torus has a unique symplectically aspherical strong filling up to diffeomorphism.

Euler characteristic of stable envelopes

In this paper we prove a formula relating the equivariant Euler characteristic of K-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the index vertex is the power series expansion of a rational function. This result is a consequence of the 3d mirror self-symmetry of the variety considered here. In general, one expects an analogous result to hold for any two varieties related by 3d mirror symmetry.

Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems

Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics. The isospectral property thereby often corresponds to mathematically or physically important conservation laws. The most prominent feature of these systems, i.e. the conservation of the eigenvalues of the matrix state variable, should therefore be retained when discretizing these systems. Recently, it was shown how isospectral Runge-Kutta methods can in the Lie-Poisson case be obtained through Hamiltonian reduction of symplectic Runge-Kutta methods on the cotangent bundle of a Lie group. We provide the Lagrangian analogue and, in the case of symplectic diagonally implicit Runge-Kutta methods, derive the methods through a discrete Euler-Poincaré reduction. Our derivation relies on a formulation of diagonally implicit isospectral Runge-Kutta methods in terms of the Cayley transform, generalizing earlier work that showed this for the implicit midpoint rule. Our work is also a generalization of earlier variational Lie group integrators that, interestingly, appear when these are interpreted as update equations for intermediate time points. From a practical point of view, our results allow for a simple implementation of higher order isospectral methods and we demonstrate this with numerical experiments where both the isospectral property and energy are conserved to high accuracy.

Some Results On Silver Riemannian Structures

Our aim in this paper is to study of silver Riemannian structures on manifold and bundle. An integrability condition and curvature properties for silver Riemannian structure are investigated via the Tachibana operator. Twin silver Riemannian metric is defined and some properties of twin silver Riemannian metric are investigated. Examples of silver structure are given on tangent and cotangent bundles.

Virtual cycles on projective completions and quantum Lefschetz formula

For a compact quasi-smooth derived scheme M with (−1)-shifted cotangent bundle N, there are at least two ways to localise the virtual cycle of N to M via torus and cosection localisations, introduced by Jiang-Thomas [17]. We produce virtual cycles on both the projective completion N‾:=P(N⊕OM) and projectivisation P(N) and show the ones on N‾ push down to Jiang-Thomas cycles and the one on P(N) computes the difference.Using similar ideas we give an expression for the difference of the quintic and t-twisted quintic GW invariants of Guo-Janda-Ruan [15].

Retraction Maps: A Seed of Geometric Integrators

The classical notion of retraction map used to approximate geodesics is extended and rigorously defined to become a powerful tool to construct geometric integrators and it is called discretization map. Using the geometry of the tangent and cotangent bundles, we are able to tangently and cotangent lift such a map so that these lifts inherit the same properties as the original one and they continue to be discretization maps. In particular, the cotangent lift of a discretization map is a natural symplectomorphism, what plays a key role for constructing geometric integrators and symplectic methods. As a result, a wide range of (higher-order) numerical methods are recovered and canonically constructed by using different discretization maps, as well as some operations with Lagrangian submanifolds.

Numerically non-special varieties

Campana introduced the class of special varieties as the varieties admitting no Bogomolov sheaves, i.e. rank-one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Campana raised the question of whether one can replace the Kodaira dimension by the numerical dimension in this characterization. We answer partially this question showing that a projective manifold admitting a rank-one coherent subsheaf of the cotangent bundle with numerical dimension one is not special. We also establish the analytic characterization with the non-existence of Zariski dense entire curve and the arithmetic version with non-potential density in the (split) function field setting. Finally, we conclude with a few comments for higher codimensional foliations which may provide some evidence towards a generalization of the aforementioned results.

Bi-Hamiltonian structure of Sutherland models coupled to two -valued spins from Poisson reduction

We introduce a bi-Hamiltonian hierarchy on the cotangent bundle of the real Lie group , and study its Poisson reduction with respect to the action of the product group U(n) × U(n) arising from left- and right-multiplications. One of the pertinent Poisson structures is the canonical one, while the other is suitably transferred from the real Heisenberg double of . When taking the quotient of we focus on the dense open subset of whose elements have pairwise distinct singular values. We develop a convenient description of the Poisson algebras of the U(n) × U(n) invariant functions, and show that one of the Hamiltonians of the reduced bi-Hamiltonian hierarchy yields a hyperbolic Sutherland model coupled to two -valued spins. Thus we obtain a new bi-Hamiltonian interpretation of this model, which represents a special case of Sutherland models coupled to two spins obtained earlier from reductions of cotangent bundles of reductive Lie groups equipped with their canonical Poisson structure. Upon setting one of the spins to zero, we recover the bi-Hamiltonian structure of the standard hyperbolic spin Sutherland model that was derived recently by a different method.

Global hypersurfaces of section in the spatial restricted three-body problem

We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of a circle family of global hypersurfaces of section for the regularized dynamics. Below the first critical value, these hypersurfaces are diffeomorphic to the unit disk cotangent bundle of the two-sphere, and they carry symplectic forms on their interior, which are each deformation equivalent to the standard symplectic form. The boundary of the global hypersurface of section is an invariant set for the regularized dynamics that is equal to a level set of the Hamiltonian describing the regularized planar problem. The first return map is Hamiltonian, and restricts to the boundary as the time-1 map of a positive reparameterization of the Reeb flow in the planar problem. This construction holds for any choice of mass ratio, and is therefore non-perturbative. We illustrate the technique in the completely integrable case of the rotating Kepler problem, where the return map can be studied explicitly.

A Note on Partial Quasi-Morphisms and Products in Lagrangian Floer Homology in Cotangent Bundles

We define partial quasi-morphisms on the group of Hamiltonian diffeomorphisms of the cotangent bundle using the spectral invariants in Lagrangian Floer homology with conormal boundary conditions, where the product compatible with the PSS isomorphism and the homological intersection product is lacking.

A note on quadratic Poisson brackets on mathrm {gl}(n,{mathbb {R}}) related to Toda lattices

It is well known that the compatible linear and quadratic Poisson brackets of the full symmetric and of the standard open Toda lattices are restrictions of linear and quadratic r-matrix Poisson brackets on the associative algebra mathrm {gl}(n,{mathbb {R}}). We here show that the quadratic bracket on mathrm {gl}(n,{mathbb {R}}), corresponding to the r-matrix defined by the splitting of mathrm {gl}(n,{mathbb {R}}) into the direct sum of the upper triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a quadratic Poisson structure on the cotangent bundle T^* mathrm {GL}(n,{mathbb {R}}). This complements the interpretation of the linear r-matrix bracket as a reduction of the canonical Poisson bracket of the cotangent bundle.

Cotangent bundle and microsupports in the mixed characteristic case

For a regular scheme and a prime number $p$, we define the FW-cotangent bundle as a vector bundle on the closed subscheme defined by $p=0$, under a certain finiteness condition. For a constructible complex on the etale site of the scheme, we introduce the condition to be micro-supported on a closed conical subset in the FW-cotangent bundle. At the end of the article, we compute the singular supports in some cases.

Frobenius–Witt differentials and regularity

T. Dupuy, E. Katz, J. Rabinoff, D. Zureick-Brown introduced the module of total $p$-differentials for a ring over $Z/p^2Z$. We study the same construction for a ring over $Z_{(p)}$ and prove a regularity criterion. For a local ring, the tensor product with the residue field is constructed in a different way by O. Gabber, L. Ramero. In another article arXiv:2006.00448, we use the sheaf of FW-differentials to define the cotangent bundle and the micro-support of an etale sheaf.

On a class of globally analytic hypoelliptic sums of squares

We consider sums of squares operators globally defined on the torus. We show that if some assumptions are satisfied the operators are globally analytic hypoelliptic. The purpose of the assumptions is to rule out the existence of a Hamilton leaf on the characteristic variety lying along the fiber of the cotangent bundle, i.e. the case of the (global) Métivier operator.

Nonassociative analogs of Lie groupoids

We introduce nonassociative geometric objects generalizing naturally Lie groupoids and called (smooth) quasiloopoids and loopoids. We prove that the tangent bundles of smooth loopoids are canonically smooth loopoids again (it is nontrivial in the case of loopoids). We show also that this is not true if the cotangent bundles are concerned. After providing a few natural constructions, we show how the Lie-like functor associates with loopoids skew algebroids and almost Lie algebroids and how discrete mechanics on Lie groupoids can be reformulated in the nonassociative case.

Differential operators on G/U and the Gelfand-Graev action

Let G be a complex semisimple group and U its maximal unipotent subgroup. We study the algebra D(G/U) of algebraic differential operators on G/U and also its quasi-classical counterpart: the algebra of regular functions on T⁎(G/U), the cotangent bundle. A long time ago, S. Gelfand and M. Graev have constructed an action of the Weyl group on D(G/U) by algebra automorphisms. The Gelfand-Graev construction was not algebraic, it involved analytic methods in an essential way. We give a new algebraic construction of the Gelfand-Graev action, as well as its quasi-classical counterpart. Our approach is based on Hamiltonian reduction and involves the ring of Whittaker differential operators on G/U, a twisted analogue of D(G/U).Our main result has an interpretation, via geometric Satake, in terms of spherical perverse sheaves on the affine Grassmannian for the Langlands dual group.

An Arnold-type principle for non-smooth objects

In this article, we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point shows that the conjecture does not admit a direct generalization to continuous settings. However, it appears that the following Arnold-type principle continues to hold in \(C^0\) settings: suppose that X is a non-smooth object for which one can define spectral invariants. If the number of spectral invariants associated to X is smaller than the number predicted by the (homological) Arnold conjecture, then the set of fixed/intersection points of X is homologically non-trivial, hence it is infinite. We recently proved that the above principle holds for Hamiltonian homeomorphisms of closed and aspherical symplectic manifolds. In this article, we verify this principle in two new settings: \(C^0\) Lagrangians in cotangent bundles and Hausdorff limits of Legendrians in 1-jet bundles which are isotopic to 0-section. An unexpected consequence of the result on Legendrians is that the classical Arnold conjecture does hold for Hausdorff limits of Legendrians in 1-jet bundles.

Exploring black hole mechanics in cotangent bundle geometries

The classical and continuum limit of a quantum gravitational setting could lead, at mesoscopic regimes, to a very different notion of geometry with respect to the pseudo-Riemannian one of special and general relativity. A possible way to characterize this modified spacetime notion is by a momentum-dependent metric, in such a way that particles with different energies could probe different spacetimes. Indeed, doubly special relativity theories, deforming the special relativistic kinematics while maintaining a relativity principle, have been understood within a geometrical context, by considering a curved momentum space. The extension of these momentum spaces to curved spacetimes and its possible phenomenological implications have been recently investigated. Following this line of research, we address here the first two laws of black holes thermodynamics in the context of a cotangent bundle metric, depending on both momentum and spacetime coordinates, compatible with the relativistic deformed kinematics of doubly special relativity.

A dual pair for the contact group

AbstractGeneralizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, this dual pair provides a conceptual identification of non-linear Grassmannians of weighted submanifolds with certain coadjoint orbits of the contact group. Moreover, the EPContact dual pair gives rise to singular solutions for the geodesic equation on the group of contact diffeomorphisms. For the projectivized cotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pair due to Holm and Marsden.