Cotangent Bundle Research Articles

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

In this survey paper, we give an overview of our recent works on the study of the $W$-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the $W$-entropy formula for the heat equation associated with the Witten Laplacian on $n$-dimensional complete Riemannian manifolds with the $CD(K, m)$-condition, and the $W$-entropy formula for the heat equation associated with the time dependent Witten Laplacian on $n$-dimensional compact manifolds equipped with a $(K, m)$-super Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$. Furthermore, we prove an analogue of the $W$-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result recaptures an important result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two $W$-entropy formulas, we introduce the Langevin deformation of geometric flows on the cotangent bundle over the Wasserstein space and prove an extension of the $W$-entropy formula for the Langevin deformation. Finally, we make a discussion on the $W$-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory.

The Graf Product: A Clifford Structure Framework on the Exterior Bundle

The geometric product, defined by Graf on the space of differential forms, endows the sections of the exterior bundle by a structure that is necessary to construct a Clifford algebra. The Graf product is introduced and revisited with a suitable underlying framework that naturally encompasses a coframe in the cotangent bundle, besides the volume element centrality, the Hodge operator and the so called truncated subalgebra as well.

The regular semisimple locus of the affine quotient of the cotangent bundle of the Grothendieck–Springer resolution

Let G = G L n ( C ) , the general linear group over the complex numbers, and let B be the set of invertible upper triangular matrices in G . Let b = Lie ( B ) . For μ : T ∗ ( b × C n ) → b ∗ , where b ∗ ≅ g ∕ u and u being strictly upper triangular matrices in g = Lie ( G ) , we prove that the Hamiltonian reduction μ − 1 ( 0 ) r s s ∕ ∕ B of the extended regular semisimple locus b r s s of the Borel subalgebra is smooth, affine, reduced, and scheme-theoretically isomorphic to a dense open locus of C 2 n . We also show that the B -invariant functions on the regular semisimple locus of the Hamiltonian reduction of b × C n arise as the trace of a certain product of matrices.

Diagonal lift in the semi-cotangent bundle and its applications

The present paper is devoted to some results concerning the diagonal lift of tensor fields of type (1,1) from manifold M to its semi-cotangent bundle t*M. In this context, cross-sections in the semi-cotangent (pull-back) bundle t*M of cotangent bundle T*M by using projection (submersion) of the tangent bundle TM can be also defined.

On Semi-Tensor Bundle

We investigate some lifts of tensor fields of type (1,0) on a cross-section in the semi-tensor (pullback) bundle tM of tensor bundle TM of type (p,q) by using projection (submersion) of the cotangent bundle T*M and we find some relation for them.

Derivatives of Sasakian metric ^{S}g on Cotangent Bundle

In this paper, we define a Sasakian metric ^{S}g on cotangent bundle T^{∗}Mⁿ, which is completely determined by its action on complete lifts of vector fields. Later, we obtain the covariant and Lie derivatives applied to Sasakian metrics with respect to the complete and vertical lifts of vector and kovector fields, respectively

Geodesics on the Momentum Phase Space with metric $^{{C}}{g}$

In this paper, a system of the differential equations giving geodesics on the momentum phase space with pseudo Riemann metric $^{C}g$ of a Hamilton space is found by using the Euler Lagrange equations. Then, space like geodesics on pseudo hyperbolic 2-space $H_{1}^{2}$ are obtained. Finally, a system of the differential equations giving geodesics on the cotangent bundle with pseudo Riemann metric $^{C}g$ of $H_{1}^{2}$ is get.

Mean curvature flows in manifolds of special holonomy

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.

On the existence of closed magnetic geodesics via symplectic reduction

Let (M, g) be a closed Riemannian manifold and $$\sigma $$ be a closed 2-form on M representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic flow of the pair $$(g,\sigma )$$ can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle $$T^*E$$ of a suitable $$S^1$$ -bundle E over M or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of E. We thenstudy the relation between the stability property of energy hypersurfacesin $$(T^*M,dp\wedge dq+\pi ^*\sigma )$$ and of the corresponding codimension2 coisotropic submanifolds in $$(T^*E,dp\wedge dq)$$ arising via symplecticreduction. Finally, we reprove the main result of Asselle and Benedetti (J Topol Anal 8(3):545–570, 2016) in this setting.

Maximal proper force, black hole horizons and matter as curvature in momentum space

Starting with the study of the geometry on the cotangent bundle (phase space), it is shown that the maximal proper force condition, in the case of a uniformly accelerated observer of mass [Formula: see text] along the [Formula: see text] axis, leads to a minimum value of [Formula: see text] lying [Formula: see text] the Rindler wedge and given by the black hole horizon radius [Formula: see text]. Whereas in the uniform circular motion case, we find that the maximal proper force condition implies that the radius of the circle cannot exceed the value of the horizon radius [Formula: see text]. A correspondence is found between the black hole horizon radius and a singularity in the curvature of momentum space. The fact that the geometry (metric) in phase spaces is observer-dependent (on the momentum of the massive particle/observer) indicates further that the matter stress energy tensor and vacuum energy in the underlying space-time may admit an interpretation in terms of the curvature in momentum spaces. Some final comments are made pertaining to the Asymptotic Safety program in gravity and why phase space geometry seems to be a proper arena for a space–time–matter unification.

Nearby Lagrangian fibers and Whitney sphere links

Let $n>3$, and let $L$ be a Lagrangian embedding of $\mathbb{R}^{n}$ into the cotangent bundle $T^{\ast }\mathbb{R}^{n}$ of $\mathbb{R}^{n}$ that agrees with the cotangent fiber $T_{x}^{\ast }\mathbb{R}^{n}$ over a point $x\neq 0$ outside a compact set. Assume that $L$ is disjoint from the cotangent fiber at the origin. The projection of $L$ to the base extends to a map of the $n$-sphere $S^{n}$ into $\mathbb{R}^{n}\setminus \{0\}$. We show that this map is homotopically trivial, answering a question of Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in $T^{\ast }\mathbb{R}^{n}$, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016), 1–59] and Ekholm and Smith [Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), 195–240] with symplectic field theory.

Focal Points at Infinity for Short-Range Scattering Trajectories

Classical scattering trajectories are known to form a Lagrangian manifold in euclidean phase space, which allows the classification of local focal points for sufficiently small dimensions. For the case of a short-range potential, we show that the natural description of focal points at infinity is a Lagrangian manifold in the cotangent bundle of the sphere and establish the relationship between focal points at infinity and the projection singularities of that manifold.

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

Let M be an n-dimensional differentiable manifold equipped with a torsion-free linear connection $$\nabla $$ and $$T^{*}M$$ its cotangent bundle. The present paper aims to study a metric connection $$\widetilde{ \nabla }$$ with nonvanishing torsion on $$T^{*}M$$ with modified Riemannian extension $${}\overline{g}_{\nabla ,c}$$ . First, we give a characterization of fibre-preserving projective vector fields on $$(T^{*}M,{}\overline{g} _{\nabla ,c})$$ with respect to the metric connection $$\widetilde{\nabla }$$ . Secondly, we study conditions for $$(T^{*}M,{}\overline{g}_{\nabla ,c})$$ to be semi-symmetric, Ricci semi-symmetric, $$\widetilde{Z}$$ semi-symmetric or locally conharmonically flat with respect to the metric connection $$ \widetilde{\nabla }$$ . Finally, we present some results concerning the Schouten–Van Kampen connection associated to the Levi-Civita connection $$ \overline{\nabla }$$ of the modified Riemannian extension $$\overline{g} _{\nabla ,c}$$ .

On effective existence of symmetric differentials of complex hyperbolic space forms

For a noncompact complex hyperbolic space form of finite volume $$X=\mathbb {B}^n/\varGamma $$ , we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification $$\overline{X}$$ of X similar to the case of producing cusp forms on hyperbolic Riemann surfaces. We introduce a natural geometric measurement which measures the size of the infinity $$\overline{X}-X$$ called canonical radius of a cusp of $$\varGamma $$ . The main result in the article is that there is a constant $$r^*=r^*(n)$$ depending only on the dimension, so that if the canonical radii of all cusps of $$\varGamma $$ are larger than $$r^*$$ , then there exist symmetric differentials of $$\overline{X}$$ vanishing at infinity. As a corollary, we show that the cotangent bundle $$T_{\overline{X}}$$ is ample modulo the infinity if moreover the injectivity radius in the interior of $$\overline{X}$$ is larger than some constant $$d^*=d^*(n)$$ which depends only on the dimension.

Spectral asymptotics for sub-Riemannian Laplacians, I: Quantum ergodicity and quantum limits in the 3-dimensional contact case

This is the first paper of a series in which we plan to study spectral asymptotics for sub-Riemannian Laplacians and to extend results that are classical in the Riemannian case concerning Weyl measures, quantum limits, quantum ergodicity, quasi-modes, trace formulae.Even if hypoelliptic operators have been well studied from the point of view of PDEs, global geometrical and dynamical aspects have not been the subject of much attention. As we will see, already in the simplest case, the statements of the results in the sub-Riemannian setting are quite different from those in the Riemannian one. Let us consider a sub-Riemannian (sR) metric on a closed three-dimensional manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We establish a Quantum Ergodicity (QE) theorem for the eigenfunctions of any associated sR Laplacian under the assumption that the Reeb flow is ergodic. The limit measure is given by the normalized Popp measure.This is the first time that such a result is established for a hypoelliptic operator, whereas the usual Shnirelman theorem yields QE for the Laplace-Beltrami operator on a closed Riemannian manifold with ergodic geodesic flow.To prove our theorem, we first establish a microlocal Weyl law, which allows us to identify the limit measure and to prove the microlocal concentration of the eigenfunctions on the characteristic manifold of the sR Laplacian. Then, we derive a Birkhoff normal form along this characteristic manifold, thus showing that, in some sense, all 3D contact structures are microlocally equivalent. The quantum version of this normal form provides a useful microlocal factorization of the sR Laplacian. Using the normal form, the factorization and the ergodicity assumption, we finally establish a variance estimate, from which QE follows.We also obtain a second result, which is valid without any ergodicity assumption: every Quantum Limit (QL) can be decomposed in a sum of two mutually singular measures: the first measure is supported on the unit cotangent bundle and is invariant under the sR geodesic flow, and the second measure is supported on the characteristic manifold of the sR Laplacian and is invariant under the lift of the Reeb flow. Moreover, we prove that the first measure is zero for most QLs.

Stabilisation of a relative equilibrium of an underactuated AUV on SE(3)

ABSTRACTThis paper presents stabilisation of a relative equilibrium of an underactuated autonomous underwater vehicle (AUV) with three independent control inputs on Lie group SE(3), since SE(3) can describe the evolution of a rigid body without singularities or ambiguities. First, by using the interconnection and damping assignment (IDA) method through kinetic energy shaping, a controller is proposed to stabilise the relative equilibrium in the sense of Lyapunov stability in cotangent bundle T*SE(3) modulo an isotropy subgroup when a marine vehicle immerses in an ideal fluid. Then, for an AUV submersed in a viscous fluid, an asymptotical static control law consisting of three terms is designed to stabilise the relative equilibrium via the IDA method, which these three control terms are derived from kinetic energy shaping, potential energy shaping and damping reassignment, respectively. The control term derived from kinetic energy shaping preserves stability of this motion modulo translations along the shortest semi-axis in any direction and rotations about the same axis. Moreover, the static controller is developed to a dynamic one via adaptive neural network approach, which is robust for uncertainties in the system's damping and mass. Simulation results complete our work.

Asymptotic safety in quantum gravity and diffeomorphic non-isometric metric solutions to the Schwarzschild metric

We revisit the construction of diffeomorphic but not isometric metric solutions to the Schwarzschild metric. These solutions require the introduction of non-trivial areal–radial functions and are characterized by the key property that the radial horizon’s location is displaced continuously towards the singularity (r = 0). In the limiting case scenario the location of the singularity and horizon merges and any in-falling observer hits a null singularity at the very moment they cross the horizon. This fact may have important consequences for the resolution of the fire wall problem and the complementarity controversy in black holes. This construction allows us to borrow the results over the past two decades pertaining to the study of the renormalization group improvement of Einstein’s equations, which was based on the possibility that quantum Einstein gravity may be non-perturbatively renormalizable and asymptotically safe because of the presence of interacting (non-Gaussian) ultraviolet fixed points. The particular areal–radial function that eliminates the interior of a black hole, and furnishes a truly static metric solution everywhere, is used to establish the desired energy-scale relation k = k(r), which is obtained from the k (energy) dependent modifications to the running Newtonian coupling G(k), cosmological constant Λ(k), and space–time metric gij,(k)(x). (Anti) de Sitter – Schwarzschild metrics are also explored as examples. We conclude with a discussion of the role that asymptotic safety may have in the geometry of phase spaces (cotangent bundles of space–time) (i.e., in establishing a quantum space–time geometry or classical phase geometry correspondence gij,(k)(x) ↔ gij(x, E)).

Morita equivalence and the generalized Kähler potential

We solve the problem of determining the fundamental degrees of freedom underlying a generalized Kähler structure of symplectic type. For a usual Kähler structure, it is well-known that the geometry is determined by a complex structure, a Kähler class, and the choice of a positive $(1, 1)$-form in this class, which depends locally on only a single real-valued function: the Kähler potential. Such a description for generalized Kähler geometry has been sought since it was discovered in 1984. We show that a generalized Kähler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a holomorphic symplectic Morita equivalence between them, and the choice of a positive Lagrangian brane bisection, which depends locally on only a single real-valued function, which we call the generalized Kähler potential. Our solution draws upon, and specializes to, the many results in the physics literature which solve the problem under the assumption (which we do not make) that the Poisson structures involved have constant rank. To solve the problem we make use of, and generalize, two main tools: the first is the notion of symplectic Morita equivalence, developed by Weinstein and Xu to study Poisson manifolds; the second is Donaldson’s interpretation of a Kähler metric as a real Lagrangian submanifold in a deformation of the holomorphic cotangent bundle.

On the positivity of the logarithmic cotangent bundle

The aim of this work is to construct examples of pairs whose logarithmic cotangent bundles have strong positivity properties. These examples are constructed from any smooth n-dimensional complex projective varieties by considering the sum of at least n general sufficiently ample hypersurfaces.

Elliptic stable envelopes and finite-dimensional representations of elliptic quantum group

We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ on the Gelfand-Tsetlin basis of the tensor product of the $n$-vector representations. The result is described in a combinatorial way by using the partitions of $[1,n]$. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we show a correspondence of the Gelfand-Tsetlin bases (resp. the standard bases) to the fixed point classes (resp. the stable classes) in the equivariant elliptic cohomology $E_T(X)$ of the cotangent bundle $X$ of the partial flag variety. As a result we obtain a geometric representation of the elliptic quantum group on $E_T(X)$.