Relevant bibliographies by topics / Symplectic bundles

Academic literature on the topic 'Symplectic bundles'

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Published: 14 March 2023

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Journal articles on the topic "Symplectic bundles"

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Choe, Insong, and George H. Hitching. "Non-defectivity of Grassmannian bundles over a curve." International Journal of Mathematics 27, no. 07 (June 2016): 1640002. http://dx.doi.org/10.1142/s0129167x16400024.

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Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.
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Choe, Insong, and G. H. Hitching. "A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve." International Journal of Mathematics 25, no. 05 (May 2014): 1450047. http://dx.doi.org/10.1142/s0129167x14500475.

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A symplectic or orthogonal bundle V of rank 2n over a curve has an invariant t(V) which measures the maximal degree of its isotropic subbundles of rank n. This invariant t defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on t(V), which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case.
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Benedetti, Gabriele, and Alexander F. Ritter. "Invariance of symplectic cohomology and twisted cotangent bundles over surfaces." International Journal of Mathematics 31, no. 09 (July 30, 2020): 2050070. http://dx.doi.org/10.1142/s0129167x20500706.

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We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be nonexact and noncompactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology.
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CHOE, INSONG, and GEORGE H. HITCHING. "Lagrangian subbundles of symplectic bundles over a curve." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 2 (February 22, 2012): 193–214. http://dx.doi.org/10.1017/s0305004112000096.

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AbstractA symplectic bundle over an algebraic curve has a natural invariantsLagdetermined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound onsLagwhich is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced bysLagon moduli spaces of symplectic bundles, and get a full picture for the case of rank four.
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Bakuradze, M. "On the Buchstaber Subring in MSp∗." gmj 5, no. 5 (October 1998): 401–14. http://dx.doi.org/10.1515/gmj.1998.401.

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Abstract A formula is given to calculate the last n number of symplectic characteristic classes of the tensor product of the vector Spin(3)- and Sp(n)-bundles through its first 2n number of characteristic classes and through characteristic classes of Sp(n)-bundle. An application of this formula is given in symplectic cobordisms and in rings of symplectic cobordisms of generalized quaternion groups.
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HITCHING, GEORGE H. "RANK FOUR SYMPLECTIC BUNDLES WITHOUT THETA DIVISORS OVER A CURVE OF GENUS TWO." International Journal of Mathematics 19, no. 04 (April 2008): 387–420. http://dx.doi.org/10.1142/s0129167x08004716.

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The moduli space [Formula: see text] of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier–Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from [Formula: see text] to the space of even 4Θ divisors on the Jacobian variety of the curve.
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de Araujo, Artur. "Generalized quivers, orthogonal and symplectic representations, and Hitchin–Kobayashi correspondences." International Journal of Mathematics 30, no. 03 (March 2019): 1850085. http://dx.doi.org/10.1142/s0129167x18500854.

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We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group [Formula: see text]. We first study the case when [Formula: see text] or [Formula: see text], interpreting them as orthogonal (respectively symplectic) bundle representations of the symmetric quivers introduced by Derksen–Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. In particular, we completely characterize the polystable forms of such representations. Finally, we discuss Hitchin–Kobayashi correspondences for these objects.
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BISWAS, INDRANIL, and SARBESWAR PAL. "ON MODULI SPACE OF HIGGS Gp(2n, ℂ)-BUNDLES OVER A RIEMANN SURFACE." International Journal of Geometric Methods in Modern Physics 07, no. 02 (March 2010): 311–22. http://dx.doi.org/10.1142/s0219887810004002.

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Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by KX. Let [Formula: see text] denote the moduli space of semistable Higgs Gp (2n, ℂ)-bundles over X of fixed topological type. The complex variety [Formula: see text] has a natural holomorphic symplectic structure. On the other hand, for any ℓ ≥ 1, the Liouville symplectic from on the total space of KX defines a holomorphic symplectic structure on the Hilbert scheme Hilb ℓ(KX) parametrizing the zero-dimensional subschemes of KX. We relate the symplectic form on Hilb ℓ(KX) with the symplectic form on [Formula: see text].
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BISWAS, INDRANIL, TOMAS L. GÓMEZ, and VICENTE MUÑOZ. "AUTOMORPHISMS OF MODULI SPACES OF SYMPLECTIC BUNDLES." International Journal of Mathematics 23, no. 05 (May 2012): 1250052. http://dx.doi.org/10.1142/s0129167x12500528.

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Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let MSp(L) be the moduli space of semistable symplectic bundles (E, φ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of MSp(L) is generated by the automorphisms of the form E ↦ E ⊗ M, where [Formula: see text], together with the automorphisms induced by automorphisms of X.
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Otiman, Alexandra. "Locally conformally symplectic bundles." Journal of Symplectic Geometry 16, no. 5 (2018): 1377–408. http://dx.doi.org/10.4310/jsg.2018.v16.n5.a5.

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Dissertations / Theses on the topic "Symplectic bundles"

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at, Andreas Cap@esi ac. "Equivariant Symplectic Geometry of Cotangent Bundles." Moscow Math. J. 1, No.2 (2001) 287-299, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi996.ps.

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Hitching, George H. "Moduli of symplectic bundles over curves." Thesis, Durham University, 2005. http://etheses.dur.ac.uk/2351/.

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Let Х be a complex projective smooth irreducible curve of genus g. We begin by giving background material on symplectic vector bundles and principal bundles over X and introduce the moduli spaces we will be studying, In Chapter 2 we describe the stable singular locus and semistable boundary of the moduli space Mx(Sp2 C) of semistable principal Sp2 C-bundles over X. In Chapter 3 we give results on symplectic extensions and Lagrangian subbundles. In Chapter 4, we assemble some results on vector bundles of rank 2 and degree 1 over a curve of genus 2, which are needed in what follows. Chapter 5 describes a generically finite cover of Aix(Sp2C) for a curve of genus 2. In the last chapter, we give some results on theta-divisors of rank 4 symplectic vector bundles over curves: we prove that the general such bundle over a curve of genus 2 possesses a theta-divisor, and characterise those stable bundles with singular theta-divisors. Many results on symplectic bundles admit analogues in the orthogonal case, which we have outlined where possible.
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Henriksen, Tobias Våge. "Symplectic Homology and Shape of Cotangent Bundles." Thesis, Uppsala universitet, Algebra och geometri, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-451813.

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Plummer, Michael. "Stratified fibre bundles and symplectic reduction on coadjoint orbits of SU(n)." Thesis, University of Surrey, 2008. http://epubs.surrey.ac.uk/842671/.

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The problem of classifying the reduced phase spaces of the natural torus action on a generic coadjoint orbit of SU(n) is considered. The concept of a stratified fibre bundle is defined. It is proved that the orbit map of an equivariant fibre bundle is a stratified fibre bundle. This result is then used to give an iterative description of the reduced phase spaces of the torus action on a generic coadjoint orbit of SU(n). The theory is illustrated with a detailed examination of the n = 3 case, that of the two torus action on a coadjoint orbit of SU(3).
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Kirchhoff-Lukat, Charlotte Sophie. "Aspects of generalized geometry : branes with boundary, blow-ups, brackets and bundles." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/283007.

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This thesis explores aspects of generalized geometry, a geometric framework introduced by Hitchin and Gualtieri in the early 2000s. In the first part, we introduce a new class of submanifolds in stable generalized complex manifolds, so-called Lagrangian branes with boundary. We establish a correspondence between stable generalized complex geometry and log symplectic geometry, which allows us to prove results on local neighbourhoods and small deformations of this new type of submanifold. We further investigate Lefschetz thimbles in stable generalized complex Lefschetz fibrations and show that Lagrangian branes with boundary arise in this context. Stable generalized complex geometry provides the simplest examples of generalized complex manifolds which are neither complex nor symplectic, but it is sufficiently similar to symplectic geometry for a multitude of symplectic results to generalize. Our results on Lefschetz thimbles in stable generalized complex geometry indicate that Lagrangian branes with boundary are part of a potential generalisation of the Wrapped Fukaya category to stable generalized complex manifolds. The work presented in this thesis should be seen as a first step towards the extension of Floer theory techniques to stable generalized complex geometry, which we hope to develop in future work. The second part of this thesis studies Dorfman brackets, a generalisation of the Courant- Dorfman bracket, within the framework of double vector bundles. We prove that every Dorfman bracket can be viewed as a restriction of the Courant-Dorfman bracket on the standard VB-Courant algebroid, which is in this sense universal. Dorfman brackets have previously not been considered in this context, but the results presented here are reminiscent of similar results on Lie and Dull algebroids: All three structures seem to fit into a more general duality between subspaces of sections of the standard VB-Courant algebroid and brackets on vector bundles of the form T M ⊕ E ∗ , E → M a vector bundle. We establish a correspondence between certain properties of the brackets on one, and the subspaces on the other side.
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Choy, Jaeyoo. "Moduli spaces of framed symplectic and orthogonal bundles on P2 and the K-theoretic Nekrasov partition functions." 京都大学 (Kyoto University), 2015. http://hdl.handle.net/2433/198873.

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Karlsson, Cecilia. "Orienting Moduli Spaces of Flow Trees for Symplectic Field Theory." Doctoral thesis, Uppsala universitet, Algebra och geometri, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-269551.

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This thesis consists of three scientific papers dealing with invariants of Legendrian and Lagrangian submanifolds. Besides the scientific papers, the thesis contains an introduction to contact and symplectic geometry, and a brief outline of Symplectic field theory with focus on Legendrian contact homology. In Paper I we give an orientation scheme for moduli spaces of rigid flow trees in Legendrian contact homology. The flow trees can be seen as the adiabatic limit of sequences of punctured pseudo-holomorphic disks with boundary on the Lagrangian projection of the Legendrian. So to equip the trees with orientations corresponds to orienting the determinant line bundle of the dbar-operator over the space of Lagrangian boundary conditions on the punctured disk. We define an  orientation of this line bundle and prove that it is well-defined in the limit. We also prove that the chosen orientation scheme gives rise to a combinatorial algorithm for computing the orientation of the trees, and we give an explicit description of this algorithm. In Paper II we study exact Lagrangian cobordisms with cylindrical Legendrian ends, induced by Legendrian isotopies. We prove that the combinatorially defined DGA-morphisms used to prove invariance of Legendrian contact homology for Legendrian knots over the integers can be derived analytically.  This is proved using the orientation scheme from Paper I together with a count of abstractly perturbed flow trees  of the Lagrangian cobordisms. In Paper III we prove a flexibility result for closed, immersed Lagrangian submanifolds in the standard symplectic plane.
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Monzner, Alexandra [Verfasser], Karl Friedrich [Akademischer Betreuer] Siburg, and Lorenz Johannes [Akademischer Betreuer] Schwachhöfer. "Partial quasi-morphisms and symplectic quasi-integrals on cotangent bundles / Alexandra Monzner. Betreuer: Karl Friedrich Siburg. Gutachter: Lorenz Johannes Schwachhöfer." Dortmund : Universitätsbibliothek Dortmund, 2012. http://d-nb.info/1099912598/34.

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MOSSA, ROBERTO. "Balanced metrics on complex vector bundles and the diastatic exponential of a symmetric space." Doctoral thesis, Università degli Studi di Cagliari, 2011. http://hdl.handle.net/11584/266274.

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This thesis deals with two different subjects: balanced metrics on complex vector bundles and the diastatic exponential of a symmetric space. Correspondingly we have two main results. In the first one we prove that if a holomorphic vector bundle E over a compact Kähler manifold (M,ω) admits a ω-balanced metric then this metric is unique. In the second one, after defining the diastatic exponential of a real analytic Kähler manifold, we prove that for every point p of an Hermitian symmetric space of noncompact type there exists a globally defined diastatic exponential centered in p which is a diffeomorphism and it is uniquely determined by its restriction to polydisks.
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Kennedy, Chris A. "Construction of Maps by Postnikov Towers." The Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1533034197206461.

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Books on the topic "Symplectic bundles"

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Abe, Takeshi. Strange duality for parabolic symplectic bundles on a pointed projective line. Kyoto, Japan: Kyōto Daigaku Sūri Kaiseki Kenkyūjo, 2008.

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Clay Mathematics Institute Workshop on Moduli Spaces of Vector Bundles, with a View toward Coherent Sheaves (2006 Cambridge, Mass.). Grassmannians, moduli spaces, and vector bundles: Clay Mathematics Institute Workshop on Moduli Spaces of Vector Bundles, with a View towards Coherent Sheaves, October 6-11, 2006, Cambridge, Massachusetts. Edited by Ellwood D. (David) 1966- and Previato Emma. Providence, RI: American Mathematical Society, 2011.

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Diffeology. Providence, Rhode Island: American Mathematical Society, 2013.

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editor, Donagi Ron, Katz Sheldon 1956 editor, Klemm Albrecht 1960 editor, and Morrison, David R., 1955- editor, eds. String-Math 2012: July 16-21, 2012, Universität Bonn, Bonn, Germany. Providence, Rhode Island: American Mathematical Society, 2015.

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McDuff, Dusa, and Dietmar Salamon. Symplectic fibrations. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198794899.003.0007.

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This chapter begins with a general discussion of symplectic fibrations and symplectic forms on the total space. The next section describes in detail symplectic 2-sphere bundles over Riemann surfaces. Subsequent sections develop the notions of symplectic connection and holonomy, explain the Sternberg–Weinstein universal construction for fibre bundles, discuss Seidel’s construction of generalized Dehn twists, and introduce the Guillemin–Lerman–Sternberg coupling form. The final section studies Hamiltonian fibrations.
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McDuff, Dusa, and Dietmar Salamon. Linear symplectic geometry. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198794899.003.0003.

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The second chapter introduces the basic concepts of symplectic topology in the linear algebra setting, such as symplectic vector spaces, the linear symplectic group, Lagrangian subspaces, and the Maslov index. In the section on linear complex structures particular emphasis is placed on the homotopy equivalence between the space of symplectic forms and the space of linear complex structures. The chapter includes sections on symplectic vector bundles and the first Chern class.
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McDuff, Dusa, and Dietmar Salamon. Symplectic group actions. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198794899.003.0006.

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The chapter begins with a discussion of circle actions and their relation to 2-sphere bundles. It continues with a section on general Hamiltonian group actions and moment maps, then proceeds to discuss various explicit examples in both finite and infinite dimensions, and introduces the Marsden–Weinstein quotient, together with new examples that explain its relation to the construction of generating functions for Lagrangians. Further sections give a proof of the Atiyah–Guillemin–Sternberg convexity theorem about the image of the moment map in the case of torus actions, and use equivariant cohomology to prove the Duistermaat–Heckman localization formula for circle actions. It closes with an overview of geometric invariant theory which grows out of the interplay between the actions of a real Lie group and its complexification.
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Mann, Peter. Constrained Hamiltonian Dynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0021.

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This chapter focuses on autonomous geometrical mechanics, using the language of symplectic geometry. It discusses manifolds (including Kähler manifolds, Riemannian manifolds and Poisson manifolds), tangent bundles, cotangent bundles, vector fields, the Poincaré–Cartan 1-form and Darboux’s theorem. It covers symplectic transforms, the Marsden–Weinstein symplectic quotient, presymplectic and symplectic 2-forms, almost symplectic structures, symplectic leaves and foliation. It also discusses contact structures, musical isomorphisms and Arnold’s theorem, as well as integral invariants, Nambu structures, the Nambu bracket and the Lagrange bracket. It describes Poisson bi-vector fields, Poisson structures, the Lie–Poisson bracket and the Lie–Poisson reduction, as well as Lie algebra, the Lie bracket and Lie algebra homomorphisms. Other topics include Casimir functions, momentum maps, the Euler–Poincaré equation, fibre derivatives and the geodesic equation. The chapter concludes by looking at deformation quantisation of the Poisson algebra, using the Moyal bracket and C*-algebras to develop a quantum physics.
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McDuff, Dusa, and Dietmar Salamon. The arnold conjecture. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198794899.003.0012.

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This chapter contains a proof of the Arnold conjecture for the standard torus, which is based on the discrete symplectic action. The symplectic part of this proof is very easy. However, for completeness of the exposition, one section is devoted to a fairly detailed discussion of the relevant Conley index theory and of Ljusternik–Schnirelmann theory. Closely related to the problem of finding symplectic fixed points is the Lagrangian intersection problem. The chapter outlines a proof of Arnold’s conjecture for cotangent bundles that again uses the discrete symplectic action, this time to construct generating functions for Lagrangian submanifolds. The chapter ends with a brief outline of the construction and applications of Floer homology.
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McDuff, Dusa, and Dietmar Salamon. Generating functions. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198794899.003.0010.

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This chapter discusses generating functions in more detail. It shows how generating functions give rise to discrete-time analogues of the symplectic action functional and hence lead to discrete variational problems. The results of this chapter form the basis for the proofs in Chapter 11 of the Arnold conjecture for the torus and in Chapter 12 of the existence of the Hofer–Zehnder capacity. The final section examines generating functions for exact Lagrangian submanifolds of cotangent bundles.
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Book chapters on the topic "Symplectic bundles"

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Koszul, Jean-Louis, and Yi Ming Zou. "Cotangent Bundles." In Introduction to Symplectic Geometry, 57–73. Singapore: Springer Singapore, 2019. http://dx.doi.org/10.1007/978-981-13-3987-5_3.

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Libermann, Paulette, and Charles-Michel Marle. "Symplectic vector spaces and symplectic vector bundles." In Symplectic Geometry and Analytical Mechanics, 1–52. Dordrecht: Springer Netherlands, 1987. http://dx.doi.org/10.1007/978-94-009-3807-6_1.

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Blair, David E. "Tangent Bundles and Tangent Sphere Bundles." In Riemannian Geometry of Contact and Symplectic Manifolds, 137–55. Boston, MA: Birkhäuser Boston, 2002. http://dx.doi.org/10.1007/978-1-4757-3604-5_9.

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Blair, David E. "Tangent Bundles and Tangent Sphere Bundles." In Riemannian Geometry of Contact and Symplectic Manifolds, 169–93. Boston: Birkhäuser Boston, 2010. http://dx.doi.org/10.1007/978-0-8176-4959-3_9.

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Benenti, Sergio. "Symplectic Relations on Cotangent Bundles." In Hamiltonian Structures and Generating Families, 57–76. New York, NY: Springer New York, 2011. http://dx.doi.org/10.1007/978-1-4614-1499-5_4.

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Blair, David E. "Principal S 1-bundles." In Riemannian Geometry of Contact and Symplectic Manifolds, 11–16. Boston, MA: Birkhäuser Boston, 2002. http://dx.doi.org/10.1007/978-1-4757-3604-5_2.

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Blair, David E. "Principal S 1-bundles." In Riemannian Geometry of Contact and Symplectic Manifolds, 15–21. Boston: Birkhäuser Boston, 2010. http://dx.doi.org/10.1007/978-0-8176-4959-3_2.

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Crumeyrolle, Albert. "Symplectic Spinor Bundles—The Maslov Index." In Orthogonal and Symplectic Clifford Algebras, 256–67. Dordrecht: Springer Netherlands, 1990. http://dx.doi.org/10.1007/978-94-015-7877-6_18.

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Losev, Ivan. "Procesi Bundles and Symplectic Reflection Algebras." In Algebraic and Analytic Microlocal Analysis, 3–61. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-01588-6_1.

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Tralle, Aleksy, and John Oprea. "Symplectic structures in total spaces of bundles." In Symplectic Manifolds with no Kähler Structure, 137–72. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/bfb0092613.

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Conference papers on the topic "Symplectic bundles"

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Gothen, Peter B., Carlos Herdeiro, and Roger Picken. "Higgs bundles and the real symplectic group." In XIX INTERNATIONAL FALL WORKSHOP ON GEOMETRY AND PHYSICS. AIP, 2011. http://dx.doi.org/10.1063/1.3599126.

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Andrianopoli, Laura, Riccardo D'Auria, and Sergio Ferrara. "Flat Symplectic Bundles of N-Extended Supergravities, Central Charges and Black-Hole Entropy." In Proceedings of the APCTP Winter School. WORLD SCIENTIFIC, 1998. http://dx.doi.org/10.1142/9789814447287_0007.

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Uzun, N. "Symplectic evolution of an observed light bundle." In Proceedings of the MG16 Meeting on General Relativity. WORLD SCIENTIFIC, 2023. http://dx.doi.org/10.1142/9789811269776_0319.

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Cioroianu, Eugen-Mihaita. "Locally conformal symplectic structures: From standard to line bundle approach." In TIM 19 PHYSICS CONFERENCE. AIP Publishing, 2020. http://dx.doi.org/10.1063/5.0001020.

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Yoshimura, Hiroaki. "A Geometric Approach to Constraint Stabilization for Holonomic Lagrangian Systems." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35429.

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In this paper, we develop a geometric approach to constraint stabilization for holonomic mechanical systems in the context of Lagrangian formulation. We first show that holonomic mechanical systems, for the case in which a given Lagrangian is hyperregular, can be formulated by using the Lagrangian two-form, namely, a symplectic structure on the tangent bundle of a configuration manifold that is induced from the cotangent bundle via the Legendre transformation. Then, we present an idea of geometric constraint stabilization and we show that a holonomic Lagrangian system with geometric constraint stabilization can be formulated by the Lagrange-d’Alembert principle, together with its local coordinate expression for the sake of numerical computations. Finally, we illustrate the numerical verification that the proposed method enables to stabilize constraint violations effectively in comparison with the Baumgarte and Gear–Gupta–Leimkuhler methods together with an example of a linkage mechanism.
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Yoshimura, Hiroaki. "On the Lagrangian Formalism of Nonholonomic Mechanical Systems." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84273.

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The paper illustrates the Lagrangian formalism of mechanical systems with nonholonomic constraints using the ideas of geometric mechanics. We first review a Lagrangian system for a conservative mechanical system in the context of variational principle of Hamilton, and we investigate the case that a given Lagrangian is hyperregular, which can be illustrated in the context of the symplectic structure on the tangent bundle of a configuration space by using the Legendre transformation. The Lagrangian system is denoted by the second order vector field and the Lagrangian one- and two-forms associated with a given hyperregular Lagrangian. Then, we demonstrate that a mechanical system with nonholonomic constraints can be formulated on the tangent bundle of a configuration manifold by using Lagrange multipliers. To do this, we investigate the Lagrange-d’Alembert principle from geometric points of view and we also show the intrinsic expression of the Lagrange-d’Alembert equations of motion for nonholonomic mechanical systems with nonconservative force fields.
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