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Journal articles on the topic 'Tangent bundle of Lie groups'

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Published: 4 June 2021

Last updated: 7 February 2022

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1

Balan, Vladimir, Maido Rahula, and Nicoleta Voicu. "Iterative calculus on tangent floors." Analele Universitatii "Ovidius" Constanta - Seria Matematica 24, no. 1 (January 1, 2016): 121–52. http://dx.doi.org/10.1515/auom-2016-0007.

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AbstractTangent fibrations generate a “multi-floored tower”, while raising from one of its floors to the next one, one practically reiterates the previously performed actions. In this way, the "tower" admits a ladder-shaped structure. Raising to the first floors suffices for iteratively performing the subsequent steps. The paper mainly studies the tangent functor. We describe the structure of multiple vector bundle which naturally appears on the floors, tangent maps, sector-forms, the lift of vector fields to upper floors. Further, we show how tangent groups of Lie groups lead to gauge theory, and explain in this context the meaning of covariant differentiation. Finally, we will point out within the floors special subbundles - the osculating bundles, which play an essential role in classical theories.

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2

SEIFIPOUR, Davood, and Esmaeil PEYGHAN. "Some properties of Riemannian geometry of the tangent bundle of Lie groups." TURKISH JOURNAL OF MATHEMATICS 43, no. 6 (November 22, 2019): 2842–64. http://dx.doi.org/10.3906/mat-1812-94.

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3

Asgari, Farhad, and Hamid Reza Salimi Moghaddam. "Left invariant Randers metrics of Berwald type on tangent Lie groups." International Journal of Geometric Methods in Modern Physics 15, no. 01 (December 19, 2017): 1850015. http://dx.doi.org/10.1142/s0219887818500159.

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Let [Formula: see text] be a Lie group equipped with a left invariant Randers metric of Berward type [Formula: see text], with underlying left invariant Riemannian metric [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are lifted Randers and Riemannian metrics arising from [Formula: see text] and [Formula: see text] on the tangent Lie group [Formula: see text] by vertical and complete lifts. In this paper, we study the relations between the flag curvature of the Randers manifold [Formula: see text] and the sectional curvature of the Riemannian manifold [Formula: see text] when [Formula: see text] is of Berwald type. Then we give all simply connected three-dimensional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors.

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4

Ivancevic, V., and C. E. M. Pearce. "Topological duality in humanoid robot dynamics." ANZIAM Journal 43, no. 2 (October 2001): 183–94. http://dx.doi.org/10.1017/s144618110001302x.

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AbstractA humanoid robot system may be viewed as a collection of segments coupled at rotational joints which geometrically represent constrained rotational Lie groups. This allows a study of the dynamics of the motion of a humanoid robot. Several formulations are possible. In this paper, dual invariant topological structures are constructed and analyzed on the finite-dimensional manifolds associated with the humanoid motion. Both cohomology and homology structures are examined on the tangent (Lagrangian) as well as on the cotangent (Hamiltonian) bundles on the manifold of the humanoid motion configuration. represented by the toral Lie group. It is established all four topological structures give in essence the same description of humanoid dynamics. Practically this means that whichever of these approaches we use, ultimately we obtain the same mathematical results.

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Machida, Yoshinori, and Hajime Sato. "Twistor theory of manifolds with Grassmannian structures." Nagoya Mathematical Journal 160 (2000): 17–102. http://dx.doi.org/10.1017/s0027763000007698.

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AbstractAs a generalization of the conformal structure of type (2, 2), we study Grassmannian structures of type (n, m) forn, m≥ 2. We develop their twistor theory by considering the complete integrability of the associated null distributions. The integrability corresponds to global solutions of the geometric structures.A Grassmannian structure of type (n, m) on a manifoldMis, by definition, an isomorphism from the tangent bundleTMofMto the tensor productV ⊗ Wof two vector bundlesVandWwith ranknandmoverMrespectively. Because of the tensor product structure, we have two null plane bundles with fibresPm-1(ℝ) andPn-1(ℝ) overM. The tautological distribution is defined on each two bundles by a connection. We relate the integrability condition to the half flatness of the Grassmannian structures. Tanaka’s normal Cartan connections are fully used and the Spencer cohomology groups of graded Lie algebras play a fundamental role.Besides the integrability conditions corrsponding to the twistor theory, the lifting theorems and the reduction theorems are derived. We also study twistor diagrams under Weyl connections.

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6

IVANCEVIC, VLADIMIR G., and TIJANA T. IVANCEVIC. "HUMAN VERSUS HUMANOID ROBOT BIODYNAMICS." International Journal of Humanoid Robotics 05, no. 04 (December 2008): 699–713. http://dx.doi.org/10.1142/s0219843608001595.

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In this paper we compare and contrast modern dynamical methodologies common to both humanoid robotics and human biomechanics. While the humanoid robot's motion is defined on the system of constrained rotational Lie groups SO(3) acting in all major robot joints, human motion is defined on the corresponding system of constrained Euclidean groups SE(3) of the full (rotational + translational) rigid motions acting in all synovial human joints. In both cases the smooth configuration manifolds, Q rob and Q hum , respectively, can be constructed. The autonomous Lagrangian dynamics are developed on the corresponding tangent bundles, TQ rob and TQ hum , respectively, which are themselves smooth Riemannian manifolds. Similarly, the autonomous Hamiltonian dynamics are developed on the corresponding cotangent bundles, T*Q rob and T*Q hum , respectively, which are themselves smooth symplectic manifolds. In this way a full rotational + translational biodynamics simulator has been created with 270 DOFs in total, called the Human Biodynamics Engine, which is currently in its validation stage. Finally, in both the human and the humanoid case, the time-dependent biodynamics generalizing the autonomous Lagrangian (of Hamiltonian) dynamics is naturally formulated in terms of jet manifolds.

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7

Bekar, Murat, and Yusuf Yayli. "Lie Algebra of Unit Tangent Bundle." Advances in Applied Clifford Algebras 27, no. 2 (April 20, 2016): 965–75. http://dx.doi.org/10.1007/s00006-016-0670-1.

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8

Blair, David E. "A Survey of Riemannian Contact Geometry." Complex Manifolds 6, no. 1 (January 1, 2019): 31–64. http://dx.doi.org/10.1515/coma-2019-0002.

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AbstractThis survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

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9

CRASMAREANU, MIRCEA. "DIRAC STRUCTURES FROM LIE INTEGRABILITY." International Journal of Geometric Methods in Modern Physics 09, no. 04 (May 6, 2012): 1220005. http://dx.doi.org/10.1142/s0219887812200058.

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We prove that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi–Cartan–Vranceanu metrics and their Hopf bundles.

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Vulcu, Vlad-Augustin. "Dirac Structures on Banach Lie Algebroids." Analele Universitatii "Ovidius" Constanta - Seria Matematica 22, no. 3 (September 1, 2014): 219–28. http://dx.doi.org/10.2478/auom-2014-0060.

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Abstract In the original definition due to A. Weinstein and T. Courant a Dirac structure is a subbundle of the big tangent bundle T M ⊕ T* M that is equal to its ortho-complement with respect to the so-called neutral metric on the big tangent bundle. In this paper, instead of the big tangent bundle we consider the vector bundle E ⊕ E*, where E is a Banach Lie algebroid and E* its dual. Recall that E* is not in general a Lie algebroid. We define a bilinear and symmetric form on the vector bundle E ⊕ E* and say that a subbundle of it is a Dirac structure if it is equal with its orthocomplement. Our main result is that any Dirac structure that is closed with respect to a type of Courant bracket, endowed with a natural anchor is a Lie algebroid. In the proof the differential calculus on a Lie algebroid is essentially involved. We work in the category of Banach vector bundles.

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11

He, Yong, Xiao Ying Lu, and Wei Na Lu. "Some Properties of Symplectic Manifolds S on the Projective Tangent Bundle PTM." Advanced Materials Research 187 (February 2011): 483–86. http://dx.doi.org/10.4028/www.scientific.net/amr.187.483.

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In this paper, we show the relationship between 2-form of the two projective tangent bundle and the relationship between 2-form on projective tangent bundle and 1-form on by using the theory of fiber bundle and the properties of symplectic manifold of the projective tangent bundle . Moreover, we derived a simpler formula of Lie derivative of a special vector field, which is on the projective tangent bundle.

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12

Arcuş, Constantin M., and Esmaeil Peyghan. "(Pseudo) generalized Kaluza–Klein G-spaces and Einstein equations." International Journal of Mathematics 25, no. 12 (November 2014): 1450116. http://dx.doi.org/10.1142/s0129167x1450116x.

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Introducing the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza–Klein vector bundle, we present the (g, h)-lift of a curve on the base M and we characterize the horizontal and vertical parallelism of the (g, h)-lift of accelerations with respect to a distinguished linear (ρ, η)-connection. Moreover, we study the torsion, curvature and Ricci tensor field associated to a distinguished linear (ρ, η)-connection and we obtain the identities of Cartan and Bianchi type in the general framework of the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle. Finally, we introduce the theory of (pseudo) generalized Kaluza–Klein G-spaces and we develop the Einstein equations in this general framework.

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13

BEKAR, Murat. "Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space." International Electronic Journal of Geometry 12, no. 1 (March 27, 2019): 1–8. http://dx.doi.org/10.36890/iejg.545737.

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14

Agricola, Ilka, and Ana Cristina Ferreira. "Tangent Lie Groups are Riemannian Naturally Reductive Spaces." Advances in Applied Clifford Algebras 27, no. 2 (April 12, 2016): 895–911. http://dx.doi.org/10.1007/s00006-016-0660-3.

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15

Asgari, F., and H. R. Salimi Moghaddam. "On the Riemannian geometry of tangent Lie groups." Rendiconti del Circolo Matematico di Palermo Series 2 67, no. 2 (March 18, 2017): 185–95. http://dx.doi.org/10.1007/s12215-017-0304-z.

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16

Feres, Renato. "Geodesic flows on manifolds of negative curvature with smooth horospheric foliations." Ergodic Theory and Dynamical Systems 11, no. 4 (December 1991): 653–86. http://dx.doi.org/10.1017/s0143385700006416.

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AbstractWe improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φt: V → V on the unit tangent bundle V of M is C∞. Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C∞-isomorphic (i.e., conjugate under a C∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ1, is C∞-isomorphic to the flow , on the quotient Γ\, where Γ is a subgroup of a real Lie group ⊂ Diffeo () with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂHm, m = ½ dim M.

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17

Grady, Ryan, and Owen Gwilliam. "LIE ALGEBROIDS AS SPACES." Journal of the Institute of Mathematics of Jussieu 19, no. 2 (February 13, 2018): 487–535. http://dx.doi.org/10.1017/s1474748018000075.

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In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

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18

Asgari, F., and H. R. Salimi Moghaddam. "Riemannian Geometry of Two Families of Tangent Lie Groups." Bulletin of the Iranian Mathematical Society 44, no. 1 (February 2018): 193–203. http://dx.doi.org/10.1007/s41980-018-0014-0.

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19

Sen, R. N. "The representation of Lie groups by bundle maps." Journal of Mathematical Physics 27, no. 8 (August 1986): 2002–8. http://dx.doi.org/10.1063/1.527018.

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20

GOZZI, E., and M. REUTER. "QUANTUM-DEFORMED GEOMETRY ON PHASE-SPACE." Modern Physics Letters A 08, no. 15 (May 20, 1993): 1433–42. http://dx.doi.org/10.1142/s021773239300115x.

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In this paper we extend the standard Moyal formalism to the tangent and cotangent bundle of the phase-space of any Hamiltonian mechanical system. In this manner we build the quantum analog of the classical Hamiltonian vector-field of time evolution and its associated Lie-derivative. We also use this extended Moyal formalism to develop a quantum analog of the Cartan calculus on symplectic manifolds.

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21

Lichnerowicz, Andr�. "Characterization of Lie groups on the cotangent bundle of a Lie group." Letters in Mathematical Physics 12, no. 2 (August 1986): 111–21. http://dx.doi.org/10.1007/bf00416461.

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22

Androulidakis, Iakovos, and Marco Zambon. "Stefan–Sussmann singular foliations, singular subalgebroids and their associated sheaves." International Journal of Geometric Methods in Modern Physics 13, Supp. 1 (October 2016): 1641001. http://dx.doi.org/10.1142/s0219887816410012.

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We explain and motivate Stefan–Sussmann singular foliations, and by replacing the tangent bundle of a manifold with an arbitrary Lie algebroid, we introduce singular subalgebroids. Both notions are defined using compactly supported sections. The main results of this note are an equivalent characterization, in which the compact support condition is removed, and an explicit description of the sheaf associated to any Stefan–Sussmann singular foliation or singular subalgebroid.

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Goldberg, Vladislav V., and Radu Rosca. "Pseudo-Sasakian manifolds endowed with a contact conformal connection." International Journal of Mathematics and Mathematical Sciences 9, no. 4 (1986): 733–47. http://dx.doi.org/10.1155/s0161171286000881.

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Pseudo-Sasakian manifoldsM˜(U,ξ,η˜,g˜)endowed with a contact conformal connection are defined. It is proved that such manifolds are space formsM˜(K),K<0, and some remarkable properties of the Lie algebra of infinitesimal transformations of the principal vector fieldU˜onM˜are discussed. Properties of the leaves of a co-isotropic foliation onM˜and properties of the tangent bundle manifoldTM˜havingM˜as a basis are studied.

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24

Wang, Changping. "Surfaces in Möbius geometry." Nagoya Mathematical Journal 125 (March 1992): 53–72. http://dx.doi.org/10.1017/s0027763000003895.

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Our purpose in this paper is to give a basic theory of Möbius differential geometay. In such geometry we study the properties of hypersurfaces in unit sphere Sn which are invariant under the Möbius transformation group on Sn.Since any Möbius transformation takes oriented spheres in Sn to oriented spheres, we can regard the Möbius transformation group Gn as a subgroup MGn of the Lie transformation group on the unit tangent bundle USn of Sn. Furthermore, we can represent the immersed hypersurfaces in Sn by a class of Lie geometry hypersurfaces (cf. [9]) called Möbius hypersurfaces. Thus we can use the concepts and the techniques in Lie sphere geometry developed by U. Pinkall ([8], [9]), T. Cecil and S. S. Chern [2] to study the Möbius differential geometry.

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Nikolov, Petko A., and Nikola P. Petrov. "Restriction and dimensional reduction of differential operators." International Journal of Geometric Methods in Modern Physics 16, no. 06 (June 2019): 1950094. http://dx.doi.org/10.1142/s0219887819500944.

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We consider the restriction of a differential operator (DO) [Formula: see text] acting on the sections [Formula: see text] of a vector bundle [Formula: see text] with base [Formula: see text], in the language of jet bundles. When the base of [Formula: see text] is restricted to a submanifold [Formula: see text], all information about derivatives in directions that are not tangent to [Formula: see text] is lost. To restrict [Formula: see text] to a DO [Formula: see text] acting on sections [Formula: see text] of the restricted bundle [Formula: see text] (with [Formula: see text] the natural embedding), one must choose an auxiliary DO [Formula: see text] and express the derivatives non-tangent to [Formula: see text] from the kernel of [Formula: see text]. This is equivalent to choosing a splitting of certain short exact sequence of jet bundles. A property of [Formula: see text] called formal integrability is crucial for restriction’s self-consistency. We give an explicit example illustrating what can go wrong if [Formula: see text] is not formally integrable. As an important application of this methodology, we consider the dimensional reduction of DOs invariant with respect to the action of a connected Lie group [Formula: see text]. The splitting relation comes from the Lie derivative of the action, which is formally integrable. The reduction of the action of another group is also considered.

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Dancer, Andrew, and Andrew Swann. "Hyperkähler metrics associated to compact Lie groups." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 1 (July 1996): 61–69. http://dx.doi.org/10.1017/s0305004100074673.

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It is well known that the cotangent bundle of any manifold has a canonical symplectic structure. If we specialize to the case when the manifold is a compact Lie group G, then this structure is preserved by the actions of G on T*G induced by left and right translation on G. We refer to these as the left and right actions of G on T*G.

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27

Shvedov, Oleg Yu. "Symmetries of semiclassical gauge systems." International Journal of Geometric Methods in Modern Physics 12, no. 10 (October 25, 2015): 1550110. http://dx.doi.org/10.1142/s0219887815501108.

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Semiclassical systems being symmetric under Lie group are studied. A state of a semiclassical system may be viewed as a set (X, f) of a classical state X and a quantum state f in the external classical background X. Therefore, the set of all semiclassical states may be considered as a bundle (semiclassical bundle). Its base {X} is the set of all classical states, while a fiber is a Hilbert space ℱX of quantum states in the external background X. Symmetry transformation of a semiclassical system may be viewed as an automorphism of the semiclassical bundle. Automorphism groups can be investigated with the help of sections of the bundle: to any automorphism of the bundle one assigns a transformation of section of the bundle. Infinitesimal properties of transformations of sections are investigated; correspondence between Lie groups and Lie algebras is discussed. For gauge theories, some points of the semiclassical bundle are identified: a gauge group acts on the bundle. For this case, only gauge-invariant sections of the semiclassical bundle are taken into account.

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Popescu, Paul, Vladimir Rovenski, and Sergey Stepanov. "The Weitzenböck Type Curvature Operator for Singular Distributions." Mathematics 8, no. 3 (March 6, 2020): 365. http://dx.doi.org/10.3390/math8030365.

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We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their L 2 adjoint operators on tensors. Then, we introduce the Weitzenböck type curvature operator on tensors, prove the Weitzenböck type decomposition formula, and derive the Bochner–Weitzenböck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones.

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Janssens, Bas, and Christoph Wockel. "Universal central extensions of gauge algebras and groups." crll 2013, no. 682 (April 5, 2012): 129–39. http://dx.doi.org/10.1515/crelle-2012-0021.

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Abstract. We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed by Neeb and Wockel (2009), is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting.

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RAMADOSS, AJAY C. "THE BIG CHERN CLASSES AND THE CHERN CHARACTER." International Journal of Mathematics 19, no. 06 (July 2008): 699–746. http://dx.doi.org/10.1142/s0129167x08004856.

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Let X be a smooth scheme over a field of characteristic 0. The Atiyah class of the tangent bundle TX of X equips TX[-1] with the structure of a Lie algebra object in the derived category D +(X) of bounded below complexes of [Formula: see text] modules with coherent cohomology [6]. We lift this structure to that of a Lie algebra object [Formula: see text] in the category of bounded below complexes of [Formula: see text] modules in Theorem 2. The "almost free" Lie algebra [Formula: see text] is equipped with Hochschild coboundary. There is a symmetrization map [Formula: see text] where [Formula: see text] is the complex of polydifferential operators with Hochschild coboundary. We prove a theorem (Theorem 1) that measures how I fails to commute with multiplication. Further, we show that [Formula: see text] is the universal enveloping algebra of [Formula: see text] in D +(X). This is used to interpret the Chern character of a vector bundle E on X as the "character of a representation" (Theorem 4). Theorems 4 and 1 are then exploited to give a formula for the big Chern classes in terms of the components of the Chern character.

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Riego, L. Del, and C. T. J. Dodson. "Sprays, universality and stability." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 3 (May 1988): 515–34. http://dx.doi.org/10.1017/s0305004100065130.

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AbstractAn important class of systems of second order differential equations can be represented as sprays on a manifold M with tangent bundle TM↠ M; that is, as certain sections of the second tangent bundle TTM ↠ TM. We consider here quadratic sprays; they correspond to symmetric linear connections on TM ↠ M and hence to principal connections on the frame bundle LM ↠ M. Such connections over M constitute a system of connections, on which there is a universal connection and through which individual connections can be studied geometrically. Correspondingly, we obtain a universal spray-like field for the system of connections and each spray on M arises as a pullback of this ‘universal spray’. The Frölicher-Nijenhuis bracket determines for each spray (or connection) a Lie subalgebra of the Lie algebra of vector fields on M and this subalgebra consists precisely of those morphisms of TTM over TM which preserve the horizontal and vertical distributions; there is a universal version of this result. Each spray induces also a Riemannian structure on LM; it isometrically embeds this manifold as a section of the space of principal connections and gives a corresponding representation of TM as a section of the space of sprays. Such embeddings allow the formulation of global criteria for properties of sprays, in a natural context. For example, if LM is incomplete in a spray-metric then it is incomplete also in the spray-metric induced by a nearby spray, because that spray induces a nearby embedding. For Riemannian manifolds, completeness of LM is equivalent to completeness of M so in the above sense we can say that geodesic incompleteness is stable; it is known to be Whitney stable.

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Asakawa, Tsuguhiko, Hisayoshi Muraki, and Satoshi Watamura. "Topological T-duality via Lie algebroids and Q-flux in Poisson-generalized geometry." International Journal of Modern Physics A 30, no. 30 (October 28, 2015): 1550182. http://dx.doi.org/10.1142/s0217751x15501821.

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It is known that the topological T-duality exchanges H- and F-fluxes. In this paper, we reformulate the topological T-duality as an exchange of two Lie algebroids in the generalized tangent bundle. Then, we apply the same formulation to the Poisson-generalized geometry, which is introduced [T. Asakawa, H. Muraki, S. Sasa and S. Watamura, Int. J. Mod. Phys. A 30, 1550097 (2015), arXiv:1408.2649 [hep-th]] to define R-fluxes as field strength associated with [Formula: see text]-transformations. We propose a definition of Q-flux associated with [Formula: see text]-diffeomorphisms, and show that the topological T-duality exchanges R- and Q-fluxes.

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García-Prada, Oscar, and André Oliveira. "Connectedness of Higgs bundle moduli for complex reductive Lie groups." Asian Journal of Mathematics 21, no. 5 (2017): 791–810. http://dx.doi.org/10.4310/ajm.2017.v21.n5.a1.

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JÓŹWIKOWSKI, MICHAŁ. "JACOBI VECTOR FIELDS FOR LAGRANGIAN SYSTEMS ON ALGEBROIDS." International Journal of Geometric Methods in Modern Physics 10, no. 05 (April 3, 2013): 1350011. http://dx.doi.org/10.1142/s0219887813500114.

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We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler–Lagrange (EL) equations are themselves solutions of the EL equations but considered on a non-standard algebroid (different from the tangent bundle Lie algebroid). As a consequence we obtain a simple non-computational proof of the relation between the null subspace of the second variation of the action and the presence of JVFs (and conjugate points) along an extremal. We work in the framework of skew-symmetric algebroids.

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Schmeding, Alexander. "The Lie group of vertical bisections of a regular Lie groupoid." Forum Mathematicum 32, no. 2 (March 1, 2020): 479–89. http://dx.doi.org/10.1515/forum-2019-0128.

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AbstractIn this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor) for these Lie groups. If the groupoid is locally trivial, i.e., a gauge groupoid, the vertical bisections coincide with the gauge group of the underlying bundle. Hence, the construction recovers the well-known Lie group structure of the gauge groups. To establish the Lie theoretic properties of the vertical bisections of a Lie groupoid over a non-compact base, we need to generalise the Lie theoretic treatment of Lie groups of bisections for Lie groupoids over non-compact bases.

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36

MARTÍNEZ TORRES, DAVID. "NONLINEAR SYMPLECTIC GRASSMANNIANS AND HAMILTONIAN ACTIONS IN PREQUANTUM LINE BUNDLES." International Journal of Geometric Methods in Modern Physics 09, no. 01 (February 2012): 1250001. http://dx.doi.org/10.1142/s0219887812500016.

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In this paper we extend to the Fréchet setting the following well-known fact about finite-dimensional symplectic geometry: if a Lie group G acts on a symplectic manifold in a Hamiltonian fashion with momentum map μ, given x ∈ M the isotropy group Gx acts linearly on the tangent space in a Hamiltonian fashion, with momentum map the Taylor expansion of μ up to degree 2. We use this result to give a conceptual explanation for a formula of Donaldson in [Scalar curvature and projective embeddings. I, J. Differential Geom.59(3) (2001) 479–522], which describes the momentum map of the Hamiltonian infinitesimal action of the Lie algebra of the group of Hamiltonian diffeomorphisms of a closed integral symplectic manifold, on sections of its prequantum line bundle.

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37

Oprea, John, and Daniel Tanré. "Flat circle bundles, pullbacks, and the circle made discrete." International Journal of Mathematics and Mathematical Sciences 2005, no. 21 (2005): 3487–95. http://dx.doi.org/10.1155/ijmms.2005.3487.

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38

Nejadahm, Masumeh, and Hamid Reza Salimi Moghaddam. "Left Invariant Lifted (α, β)-metrics of Douglas Type on Tangent Lie Groups." Zurnal matematiceskoj fiziki, analiza, geometrii 17, no. 2 (March 25, 2021): 201–15. http://dx.doi.org/10.15407/mag17.02.201.

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ERCOLESSI, E., A. IBORT, G. MARMO, and G. MORANDI. "ALTERNATIVE LINEAR STRUCTURES FOR CLASSICAL AND QUANTUM SYSTEMS." International Journal of Modern Physics A 22, no. 18 (July 20, 2007): 3039–64. http://dx.doi.org/10.1142/s0217751x07036890.

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The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative descriptions by changing the linear structure instead. In particular we show how it is possible to construct alternative linear structures on the tangent bundle TQ of some classical configuration space Q that can be considered as "adapted" to the given dynamical system. This fact opens the possibility to use the Weyl scheme to quantize the system in different nonequivalent ways, "evading," so to speak, the von Neumann uniqueness theorem.

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Takahashi, Tomokuni. "Projective Plane Bundles Over an Elliptic Curve." Canadian Mathematical Bulletin 61, no. 1 (March 1, 2018): 201–10. http://dx.doi.org/10.4153/cmb-2017-025-8.

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AbstractWe calculate the dimension of cohomology groups for the holomorphic tangent bundles of each isomorphism class of the projective plane bundle over an elliptic curve. As an application, we construct the families of projective plane bundles, and prove that the families are effectively parametrized and complete.

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41

Sommese, Andrew John, and A. van de Ven. "Homotopy groups of pullbacks of varieties." Nagoya Mathematical Journal 102 (June 1986): 79–90. http://dx.doi.org/10.1017/s002776300000043x.

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In [2, § 9] there is a general result of Fulton and Lazarsfeld relating the homotopy groups of a subvariety of in a certain range of dimensions with those of its pullback under a holomorphic map in the corresponding range of dimensions. It is asked in [2, § 10] whether here is a corresponding result with replaced by a general rational homogeneous manifold, Y, and with the range of dimensions alluded to above shifted by the ampleness of the holomorphic tangent bundle of Y in the sense of [4]. In this paper we use the techniques of [4, 5, 6, 7] to answer this question in the affirmative.

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Hohmann, Manuel, Christian Pfeifer, and Nicoleta Voicu. "Cosmological Finsler Spacetimes." Universe 6, no. 5 (May 5, 2020): 65. http://dx.doi.org/10.3390/universe6050065.

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Applying the cosmological principle to Finsler spacetimes, we identify the Lie Algebra of symmetry generators of spatially homogeneous and isotropic Finsler geometries, thus generalising Friedmann-Lemaître-Robertson-Walker geometry. In particular, we find the most general spatially homogeneous and isotropic Berwald spacetimes, which are Finsler spacetimes that can be regarded as closest to pseudo-Riemannian geometry. They are defined by a Finsler Lagrangian built from a zero-homogeneous function on the tangent bundle, which encodes the velocity dependence of the Finsler Lagrangian in a very specific way. The obtained cosmological Berwald geometries are candidates for the description of the geometry of the universe, when they are obtained as solutions from a Finsler gravity equation.

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GOZZI, E., and M. REUTER. "A PROPOSAL FOR A DIFFERENTIAL CALCULUS IN QUANTUM MECHANICS." International Journal of Modern Physics A 09, no. 13 (May 20, 1994): 2191–227. http://dx.doi.org/10.1142/s0217751x94000911.

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In this paper, using, the Weyl-Wigner-Moyal formalism for quantum mechanics, we develop a quantum-deformed exterior calculus on the phase space of an arbitrary Hamiltonian system. Introducing additional bosonic and fermionic coordinates, we construct a supermanifold which is closely related to the tangent and cotangent bundle over phase space. Scalar functions on the supermanifold become equivalent to differential forms on the standard phase space. The algebra of these functions is equipped with a Moyal superstar product which deforms the pointwise product of the classical tensor calculus. We use the Moyal bracket algebra to derive a set of quantum-deformed rules for the exterior derivative, Lie derivative, contraction, and similar operations of the Cartan calculus.

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Čap, Andreas, and Tomáš Salač. "Parabolic conformally symplectic structures I; definition and distinguished connections." Forum Mathematicum 30, no. 3 (May 1, 2018): 733–51. http://dx.doi.org/10.1515/forum-2017-0018.

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AbstractWe introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type {C_{n}} and admits a contact grading. We show that a structure of each of these types on a smooth manifold M determines a canonical compatible linear connection on the tangent bundle {\mathrm{TM}}. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant’s theorem on Lie algebra cohomology. For each type, we give an explicit description of both the geometric structure and the normalization condition. In particular, the torsion of the canonical connection naturally splits into two components, one of which is exactly the obstruction to the underlying structure being conformally symplectic. This article is the first in a series aiming at a construction of differential complexes naturally associated to these geometric structures.

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Jakimowicz, Grzegorz, Anatol Odzijewicz, and Aneta Sliżewska. "Symmetries of the space of connections on a principal G-bundle and related symplectic structures." Reviews in Mathematical Physics 31, no. 10 (November 2019): 1950039. http://dx.doi.org/10.1142/s0129055x19500399.

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There are two groups which act in a natural way on the bundle [Formula: see text] tangent to the total space [Formula: see text] of a principal [Formula: see text]-bundle [Formula: see text]: the group [Formula: see text] of automorphisms of [Formula: see text] covering the identity map of [Formula: see text] and the group [Formula: see text] tangent to the structural group [Formula: see text]. Let [Formula: see text] be the subgroup of those automorphisms which commute with the action of [Formula: see text]. In the paper, we investigate [Formula: see text]-invariant symplectic structures on the cotangent bundle [Formula: see text] which are in a one-to-one correspondence with elements of [Formula: see text]. Since, as it is shown here, the connections on [Formula: see text] are in a one-to-one correspondence with elements of the normal subgroup [Formula: see text] of [Formula: see text], so the symplectic structures related to them are also investigated. The Marsden–Weinstein reduction procedure for these symplectic structures is discussed.

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Boeijink, Jord, Klaas Landsman, and Walter van Suijlekom. "Quantization commutes with singular reduction: Cotangent bundles of compact Lie groups." Reviews in Mathematical Physics 31, no. 06 (June 25, 2019): 1950016. http://dx.doi.org/10.1142/s0129055x19500168.

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We analyze the ‘quantization commutes with reduction’ problem (first studied in physics by Dirac, and known in the mathematical literature also as the Guillemin–Sternberg Conjecture) for the conjugate action of a compact connected Lie group [Formula: see text] on its own cotangent bundle [Formula: see text]. This example is interesting because the momentum map is not proper and the ensuing symplectic (or Marsden–Weinstein quotient) [Formula: see text] is typically singular.In the spirit of (modern) geometric quantization, our quantization of [Formula: see text] (with its standard Kähler structure) is defined as the kernel of the Dolbeault–Dirac operator (or, equivalently, the spin[Formula: see text]–Dirac operator) twisted by the pre-quantum line bundle. We show that this quantization of [Formula: see text] reproduces the Hilbert space found earlier by Hall (2002) using geometric quantization based on a holomorphic polarization. We then define the quantization of the singular quotient [Formula: see text] as the kernel of the twisted Dolbeault–Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space [Formula: see text].

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Atanasiu, Gheorghe, and Monica Purcaru. "About a class of metrical N-linear connections on the 2-tangent bundle." Filomat 21, no. 1 (2007): 113–28. http://dx.doi.org/10.2298/fil0701113a.

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In the paper herein we treat some problems concerning the metric structure on the 2-tangent bundle: T2M. We determine the set of all metric semi-symmetric N-linear connections, in the case when the nonlinear connection N is fixed. We prove that the sets: TN of the transformations of N-linear connection having the same nonlinear connections N and msT N of the transformations of metric semi-symmetric N-linear connections, having the same nonlinear connection N, together with the composition of mappings are groups. We obtain some important invariants of the group msT N and we give their properties. We also study the transformation laws of the torsion and curvature d-tensor fields, with respect to the transformations of the groups TN and msT N.

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Kishimoto, D., A. Kono, and S. Theriault. "Homotopy commutativity in p-localized gauge groups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 143, no. 4 (July 17, 2013): 851–70. http://dx.doi.org/10.1017/s0308210511000011.

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Let G be a simple, compact Lie group and let $\mathcal{G}_k(G)$ be the gauge group of the principal G-bundle over S4 with second Chern class k. McGibbon classified the groups G that are homotopy commutative when localized at a prime p. We show that in many cases the homotopy commutativity of G, or its failure, determines that of $\mathcal{G}_k(G)$.

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49

BOS, ROGIER. "GEOMETRIC QUANTIZATION OF HAMILTONIAN ACTIONS OF LIE ALGEBROIDS AND LIE GROUPOIDS." International Journal of Geometric Methods in Modern Physics 04, no. 03 (May 2007): 389–436. http://dx.doi.org/10.1142/s0219887807002077.

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We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.

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YOSHINO, TARO. "CRITERION OF PROPER ACTIONS ON HOMOGENEOUS SPACES OF CARTAN MOTION GROUPS." International Journal of Mathematics 18, no. 03 (March 2007): 245–54. http://dx.doi.org/10.1142/s0129167x07004035.

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The Cartan motion group associated to a Riemannian symmetric space X is a semidirect product group acting isometrically on its tangent space. For two subsets in a locally compact group G, Kobayashi introduced the concept of "properness" as a generalization of properly discontinuous actions of discrete subgroups on homogeneous spaces of G. In this paper, we give a criterion of properness for homogeneous spaces of Cartan motion groups. Our criterion has a similar feature to the case where G is a reductive Lie group.

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