Cotangent Bundle Research Articles

Contact non-squeezing and orderability via the shape invariant

In this paper, we prove a contact non-squeezing result for a class of embeddings between starshaped domains in the contactization of the symplectization of the unit cotangent bundle of certain manifolds. The class of embeddings includes embeddings which are not isotopic to the identity. This yields a new proof that there is no positive loop of contactomorphisms in the unit cotangent bundles under consideration. The arguments are inspired by the shape invariant introduced by Sikorav and Eliashberg.

Remarks on the geometry of the variety of planes of a cubic fivefold

This note presents some properties of the variety of planes $F_2(X)\subset G(3,7)$ of a cubic $5$-fold $X\subset \mathbb P^6$. A cotangent bundle exact sequence is first derived from the remark made by Iliev and Manivel that $F_2(X)$ sits as a Lagrangian subvariety of the variety of lines of a cubic $4$-fold, which is a hyperplane section of $X$. Using the sequence, the Gauss map of $F_2(X)$ is then proven to be an embedding. The last section is devoted to the relation between the variety of osculating planes of a cubic $4$-fold and the variety of planes of the associated cyclic cubic $5$-fold.

Classification of momentum proper exact Hamiltonian group actions and the equivariant Eliashberg cotangent bundle conjecture

Let G be a compact and connected Lie group. The Hamiltonian G-model functor maps the category of symplectic representations of closed subgroups of G to the category of exact Hamiltonian G-actions. Based on previous joint work with Y. Karshon, the restriction of this functor to the momentum proper subcategory on either side induces a bijection between the sets of isomorphism classes. This classifies all momentum proper exact Hamiltonian G-actions (of arbitrary complexity). As an extreme case, we obtain a version of the Eliashberg cotangent bundle conjecture for transitive smooth actions. As another extreme case, the momentum proper Hamiltonian G-actions on contractible manifolds are exactly the symplectic G-representations, up to isomorphism.

The double Gelfand–Cetlin system, invariance of polarization, and the Peter–Weyl theorem

The bundle map T⁎U(n)⟶U(n) provides a real polarization of the cotangent bundle T⁎U(n), and yields the geometric quantization Q1(T⁎U(n))=L2(U(n)). We use the Gelfand–Cetlin systems of Guillemin and Sternberg to show that T⁎U(n) has a different real polarization with geometric quantization Q2(T⁎U(n))=⨁αVα⊗Vα⁎, where the sum is over all dominant integral weights α of U(n). The Peter–Weyl theorem, which states that these two quantizations are isomorphic, may therefore be interpreted as an instance of “invariance of polarization” in geometric quantization.

The symplectic cohomology of magnetic cotangent bundles

We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles, without any restrictions on the magnetic form, using the dissipative method for compactness introduced by Groman (2023). As an application, we deduce that if N is a closed orientable manifold and is a magnetic form that is not weakly exact, then the \pi_1 1-sensitive Hofer–Zehnder capacity of any compact set in the magnetic cotangent bundle determined by \sigma is finite.

Covariant action for conformal higher spin gravity

Conformal higher spin (HS) gravity is a HS extension of Weyl gravity and is a family of local HS theories, which was put forward by Segal and Tseytlin. We propose a manifestly covariant and coordinate-independent action for these theories. The result is based on an interplay between HS symmetries and deformation quantization: a locally equivalent but manifestly background-independent reformulation, known as the parent system, of the off-shell multiplet of conformal HS fields (Fradkin–Tseytlin fields) can be interpreted in terms of Fedosov deformation quantization of the underlying cotangent bundle. This brings into the game the invariant quantum trace, induced by the Feigin–Felder–Shoikhet cocycle of Weyl algebra, which extends Segal’s action into a gauge invariant and globally well-defined action functional on the space of configurations of the parent system. The same action can be understood within the worldline approach as a correlation function in the topological quantum mechanics on the circle.

Note on geodesics of cotangent bundle with Berger-type deformed Sasaki metric over K\"ahlerian manifold

In this paper, first, we introduce the Berger-type deformed Sasaki metric on the cotangent bundle $T^{\ast}M$ over a K\"{a}hlerian manifold $(M^{2m}, J, g)$ and investigate the Levi-Civita connection of this metric. Secondly, we present the unit cotangent bundle equipped with Berger-type deformed Sasaki metric, and we investigate the Levi-Civita connection. Finally, we study the geodesics on the cotangent bundle and on unit cotangent bundle with respect to the Berger-type deformed Sasaki metric.

An Intrinsic Version of the k-Harmonic Equation

The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle TkM to the cotangent bundle T*Tk−1M. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves.

Tangent, cotangent, normal and conormal bundles are almost never instanton bundles

In this very short note we give an elementary characteristic free proof of the result claimed in the title (see Theorem 1.2 for a more precise formulation), generalizing a recent result proved by Benedetti et al. 2023 (see reference list) for Ulrich bundles over the complex field. Moreover, we also give a similar result about the twists of the cotangent bundle and make some comments about the possibility to obtain an analogous result for twists of the tangent bundle.

Canonical domains for coadjoint orbits

AbstractThis paper describes two real analytic symplectomorphisms defined on appropriate dense open subsets of any coadjoint orbit of a compact semisimple Lie algebra. The first symplectomorphism sends the open dense subset to a bounded subset of a standard cotangent bundle. The second symplectomorphism has as target a bounded subset of a hyperbolic coadjoint orbit of an associated non‐compact semisimple Lie algebra. Therefore, coadjoint orbits of compact Lie algebras are symplectic compactifications of domains of cotangent bundles, and are in symplectic correspondence with hyperbolic orbits of non‐compact semisimple Lie algebras.

Wigner function for the quantum mechanics on a sphere

The Wigner quasiprobability function for a particle on a sphere is introduced and its properties investigated. In opposition to alternative approaches this Wigner function depends on the points of the classical phase space, that is the cotangent bundle T∗S2.

Symmetries of κ-Minkowski space-time: a possibility of exotic momentum space geometry?

The quest for a quantum gravity phenomenology has inspired a quantum notion of space-time, which motivates us to study the fate of the relativistic symmetries of a particular model of quantum space-time, as well as its intimate connection with the plausible emergent curved “physical momentum space”. We here focus on the problem of Poincare symmetry of κ-Minkowski type non-commutative (quantum) space-time, where the Poincare algebra, on its own, remains undeformed, but in order to preserve the structure of the space-time non-commutative (NC) algebra, the actions of the algebra generators on the operator-valued space-time manifold must be enveloping algebra valued that lives in entire phase space i.e. the cotangent bundle on the space-time manifold (at classical level). Further, we constructed a model for a spin-less relativistic massive particle enjoying the deformed Poincare symmetry, using the first order form of geometric Lagrangian, that satisfies a new deformed dispersion relation and explored a feasible regime of a future Quantum Gravity theory in which the momentum space becomes curved. In this scenario there is only a mass scale (Planck mass mp), but no length scale. Finally, we relate the deformed mass shell to the geodesic distance in this curved momentum space, where the mass of the particle gets renormalized as a result of noncommutativity. We show, that under some circumstances, the Planck mass provides an upper bound for the observed renormalized mass.

The cotangent bundle of K3 surfaces of degree two

K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the projectivised cotangent bundle of a very general polarised K3 surface $S$ of degree two. In particular, we describe the geometry of a surface $D_S \subset \mathbb{P}(\Omega_S)$ that plays a similar role to the surface of bitangents for a quartic in $\mathbb{P}^3$.

Integrable Multi-Hamiltonian Systems from Reduction of an Extended Quasi-Poisson Double of {text {U}}(n)

We construct a master dynamical system on a {text {U}}(n) quasi-Poisson manifold, {mathcal {M}}_d, built from the double {text {U}}(n) times {text {U}}(n) and dge 2 open balls in mathbb {C}^n, whose quasi-Poisson structures are obtained from T^* mathbb {R}^n by exponentiation. A pencil of quasi-Poisson bivectors P_{underline{z}} is defined on {mathcal {M}}_d that depends on d(d-1)/2 arbitrary real parameters and gives rise to pairwise compatible Poisson brackets on the {text {U}}(n)-invariant functions. The master system on {mathcal {M}}_d is a quasi-Poisson analogue of the degenerate integrable system of free motion on the extended cotangent bundle T^*!{text {U}}(n) times mathbb {C}^{ntimes d}. Its commuting Hamiltonians are pullbacks of the class functions on one of the {text {U}}(n) factors. We prove that the master system descends to a degenerate integrable system on a dense open subset of the smooth component of the quotient space {mathcal {M}}_d/{text {U}}(n) associated with the principal orbit type. Any reduced Hamiltonian arising from a class function generates the same flow via any of the compatible Poisson structures stemming from the bivectors P_{underline{z}}. The restrictions of the reduced system on minimal symplectic leaves parameterized by generic elements of the center of {text {U}}(n) provide a new real form of the complex, trigonometric spin Ruijsenaars–Schneider model of Krichever and Zabrodin. This generalizes the derivation of the compactified trigonometric RS model found previously in the d=1 case.

Sheaves and symplectic geometry of cotangent bundles

Sheaves and symplectic geometry of cotangent bundles

Notes on some properties of the natural Riemann extension

Let $(M,\nabla)$ be an $n$-dimensional differentiable manifold with a torsion-free linear connection and $T^{*}M$ its cotangent bundle. In this context we study some properties of the natural Riemann extension (M. Sekizawa (1987), O. Kowalski and M. Sekizawa (2011)) on the cotangent bundle $T^{*}M$. First, we give an alternative definition of the natural Riemann extension with respect to horizontal and vertical lifts. Secondly, we investigate metric connection for the natural Riemann extension. Finally, we present geodesics on the cotangent bundle $T^{*}M$ endowed with the natural Riemann extension.

On singular generalizations of the Singer–Hopf conjecture

AbstractThe Singer–Hopf conjecture predicts the sign of the topological Euler characteristic of a closed aspherical manifold. In this note, we propose singular generalizations of the Singer–Hopf conjecture, formulated in terms of the Euler–Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove these new conjectures under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle. The main ingredients in the proof are the semi‐positivity properties of nef vector bundles together with a topological version of the Riemann–Roch theorem, proved by Kashiwara.

Intrinsic incremental evolution of hypoelastic discrete mechanical systems

AbstractThis paper provides an explicit geometric and coordinate‐free formulation of incremental discrete mechanics in the framework of potentially non integrable hypoelasticity. First, the general framework is developed in order to tackle hypoelasticity as an Ehresmann connection on the cotangent bundle . Two types of incremental evolutions may be distinguished, the weak or integrable incremental evolutions and the strong or non integrable incremental evolutions, according to the nature of the hypoelastic constitutive law. The geometric structure of the double tangent bundle is fully used in order to get the geometric counterpart κ of the so‐called tangent stiffness matrix. Subject to specific conditions in , the incremental evolution is then a well‐founded question. An hypoelastic four‐grains granular system illustrates in detail these general results.

Loop coproduct in Morse and Floer homology

By a well-known theorem of Viterbo, the symplectic homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. In this paper, we extend the scope of this isomorphism in several directions. First, we give a direct definition of Rabinowitz loop homology in terms of Morse theory on the loop space and prove that its product agrees with the pair-of-pants product on Rabinowitz Floer homology. The proof uses compactified moduli spaces of punctured annuli. Second, we prove that, when restricted to positive Floer homology, resp. loop space homology relative to the constant loops, the Viterbo isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. Third, we introduce reduced loop homology, which is a common domain of definition for a canonical reduction of the loop product and for extensions of the loop homology coproduct which together define the structure of a commutative cocommutative unital infinitesimal anti-symmetric bialgebra. Along the way, we show that the Abbondandolo–Schwarz quasi-isomorphism going from the Floer complex of quadratic Hamiltonians to the Morse complex of the energy functional can be turned into a filtered chain isomorphism using linear Hamiltonians and the square root of the energy functional.

An almost Complex Structure with Norden Metric on the Phase Space

On the total space of the cotangent bundle of a Riemannian manifold, we construct a semi-Riemannian metric $G$, with respect to which an almost complex structure $J$ introduced by Oproiu and Poro\cb{s}niuc is self-adjoint. The structure $(J,G)$ turnes out to be an almost complex structure with Norden metric (this notion is known in the literature from Norden's papers). The semi-Riemannian context is different from the Riemannian one, as it is pointed out by Duggal and Bejancu in their monograph. We study this structure and provide some necessary and sufficient conditions for it to be a K\"ahler structure with Norden metric.