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Journal articles on the topic 'Fiber bundles (Mathematics) Geometry, Differential'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

del Hoyo, Matias, and Cristian Ortiz. "Morita Equivalences of Vector Bundles." International Mathematics Research Notices 2020, no. 14 (2018): 4395–432. http://dx.doi.org/10.1093/imrn/rny149.

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Abstract We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and use it to prove the Morita invariance of VB-cohomology, with implications to deformation cohomology of Lie groupoids and of classic geometries. We discuss applications of our theory to Poisson geometry, providing a new insight over Marsden–Weinstein reduction and the integration of Dirac structures. We conclude by proving that the derived category of

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2

MacKenzie, K. C. H. "DIFFERENTIAL GEOMETRY OF FRAME BUNDLES." Bulletin of the London Mathematical Society 22, no. 3 (1990): 311–12. http://dx.doi.org/10.1112/blms/22.3.311.

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3

Nishimura, Hirokazu. "Synthetic differential geometry of jet bundles." Bulletin of the Belgian Mathematical Society - Simon Stevin 8, no. 4 (2001): 639–50. http://dx.doi.org/10.36045/bbms/1102714793.

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4

Donaldson, S. "DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES." Bulletin of the London Mathematical Society 21, no. 1 (1989): 104–6. http://dx.doi.org/10.1112/blms/21.1.104.

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5

Okonek, Christian. "Book Review: Differential geometry of complex vector bundles." Bulletin of the American Mathematical Society 19, no. 2 (1988): 528–31. http://dx.doi.org/10.1090/s0273-0979-1988-15731-x.

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6

Nishimura, H. "Corrigenda to: "Synthetic differential geometry of jet bundles''." Bulletin of the Belgian Mathematical Society - Simon Stevin 9, no. 3 (2002): 473. http://dx.doi.org/10.36045/bbms/1102715071.

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7

Donaldson, Simon. "Atiyah’s work on holomorphic vector bundles and gauge theories." Bulletin of the American Mathematical Society 58, no. 4 (2021): 567–610. http://dx.doi.org/10.1090/bull/1748.

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The first part of the article surveys Atiyah’s work in algebraic geometry during the 1950s, mainly on holomorphic vector bundles over curves. In the second part we discuss his work from the late 1970s on mathematical aspects of gauge theories, involving differential geometry, algebraic geometry, and topology.

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CRAMPIN, M., and D. J. SAUNDERS. "Fefferman-type metrics and the projective geometry of sprays in two dimensions." Mathematical Proceedings of the Cambridge Philosophical Society 142, no. 3 (2007): 509–23. http://dx.doi.org/10.1017/s0305004107000047.

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AbstractA spray is a second-order differential equation field on the slit tangent bundle of a differentiable manifold, which is homogeneous of degree 1 in the fibre coordinates in an appropriate sense; two sprays which are projectively equivalent have the same base-integral curves up to reparametrization. We show how, when the base manifold is two-dimensional, to construct from any projective equivalence class of sprays a conformal class of metrics on a four-dimensional manifold, of signature (2, 2); the Weyl conformal curvature of these metrics is simply related to the projective curvature of

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Kupferman, Raz, Elihu Olami, and Reuven Segev. "Stress theory for classical fields." Mathematics and Mechanics of Solids 25, no. 7 (2017): 1472–503. http://dx.doi.org/10.1177/1081286517723697.

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Classical field theories, together with the Lagrangian and Eulerian approaches to continuum mechanics, are embraced under a geometric setting of a fiber bundle. The base manifold can be either the body manifold of continuum mechanics, the space manifold, or space–time. Differentiable sections of the fiber bundle represent configurations of the system and the configuration space containing them is given the structure of an infinite-dimensional manifold. Elements of the cotangent bundle of the configuration space are interpreted as generalized forces and a representation theorem implies that the

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BRANSON, THOMAS, and OUSSAMA HIJAZI. "IMPROVED FORMS OF SOME VANISHING THEOREMS IN RIEMANNIAN SPIN GEOMETRY." International Journal of Mathematics 11, no. 03 (2000): 291–304. http://dx.doi.org/10.1142/s0129167x00000165.

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We improve the hypotheses on some vanishing theorems for first order differential operators on bundles over a Riemannian spin manifold. The improved hypotheses are uniform, in the sense that they are the same for each of an infinite sequence of bundles in each even dimension. They are also elementary, in the sense that they involve only the bottom eigenvalue of the Yamabe operator on scalars, and the pointwise action of the Weyl conformal curvature tensor on two-forms. In particular, they do not make reference to the higher spin bundles on which the conclusion holds.

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11

Branson, Thomas, and Oussama Hijazi. "Vanishing Theorems and Eigenvalue Estimates in Riemannian Spin Geometry." International Journal of Mathematics 08, no. 07 (1997): 921–34. http://dx.doi.org/10.1142/s0129167x97000433.

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We use the representation theory of the structure group Spin (n), together with the theory of conformally covariant differential operators, to generalize results estimating eigenvalues of the Dirac operator to other tensor-spinor bundles, and to get vanishing theorems for the kernels of first-order differential operators.

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12

Škoda, Zoran. "Some Equivariant Constructions in Noncommutative Algebraic Geometry." gmj 16, no. 1 (2009): 183–202. http://dx.doi.org/10.1515/gmj.2009.183.

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Abstract We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.

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13

Ho, Man-Ho. "Local index theory and the Riemann–Roch–Grothendieck theorem for complex flat vector bundles." Journal of Topology and Analysis 12, no. 04 (2018): 941–87. http://dx.doi.org/10.1142/s1793525319500699.

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The purpose of this paper is to give a proof of the real part of the Riemann–Roch–Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac operator, a variational formula of the Bismut–Cheeger eta form without the kernel bundle assumption in the even dimensional fiber case, and some properties of the Cheeger–Chern–Simons class of complex flat vector bundle.

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14

Bridges, Thomas J. "Canonical multi-symplectic structure on the total exterior algebra bundle." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2069 (2006): 1531–51. http://dx.doi.org/10.1098/rspa.2005.1629.

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The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n -dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M . On the fibre, which has dimension 2 n , the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n -symplectic structures define a partial differential operator, J ∂ , which

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15

BRIDGES, THOMAS J., PETER E. HYDON, and JEFFREY K. LAWSON. "Multisymplectic structures and the variational bicomplex." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 1 (2009): 159–78. http://dx.doi.org/10.1017/s0305004109990259.

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AbstractMultisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formu

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16

Pelayo, Álvaro, and Xiudi Tang. "Moser stability for volume forms on noncompact fiber bundles." Differential Geometry and its Applications 63 (April 2019): 120–36. http://dx.doi.org/10.1016/j.difgeo.2018.12.003.

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17

Banagl, Markus. "Isometric group actions and the cohomology of flat fiber bundles." Groups, Geometry, and Dynamics 7, no. 2 (2013): 293–321. http://dx.doi.org/10.4171/ggd/183.

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18

Cherkashin, A. K. "Geocartographic thinking in modern science." Geodesy and Cartography 961, no. 7 (2020): 27–36. http://dx.doi.org/10.22389/0016-7126-2020-961-7-27-36.

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The purpose of the study is to show how the features of geocartographic way of thinking are manifested in the meta-theory of knowledge based on mathematical formalisms. General cartographic concepts and regularities are considered in the view of metatheoretic analysis using cognitive procedures of fiber bundle from differential geometry. On levels of metainformation generalization, the geocartographic metatheoretic approach to the study of reality is higher than the system-theoretical one. It regulates the type of equations, models, and methods of each intertheory expressed in its own system t

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19

Jeffrey, Lisa C., and Jonathan Weitsman. "Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems." Canadian Journal of Mathematics 52, no. 3 (2000): 582–612. http://dx.doi.org/10.4153/cjm-2000-026-4.

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AbstractThis paper treats the moduli space g,1(Λ) of representations of the fundamental group of a Riemann surface of genus g with one boundary component which send the loop around the boundary to an element conjugate to exp Λ, where Λ is in the fundamental alcove of a Lie algebra. We construct natural line bundles over g,1(Λ) and exhibit natural homology cycles representing the Poincaré dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a part

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20

Antonini, Paolo, Sara Azzali та Georges Skandalis. "Flat bundles, von Neumann algebras andK-theory with ℝ/ℤ-coefficients". Journal of K-Theory 13, № 2 (2014): 275–303. http://dx.doi.org/10.1017/is014001024jkt253.

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AbstractLetMbe a closed manifold andα:π1(M) →Una representation. We give a purelyK-theoretic description of the associated element in theK-theory group ofMwith ℝ/ℤ-coefficients ([α] ∈K1(M; ℝ/ℤ)). To that end, it is convenient to describe the ℝ/ℤ-K-theory as a relativeK-theory of the unital inclusion of ℂ into a finite von Neumann algebraB. We use the following fact: there is, associated withα, a finite von Neumann algebraBtogether with a flat bundleℰ→Mwith fibersB, such thatEα⊗ℰis canonically isomorphic with ℂn⊗ℰ, whereEαdenotes the flat bundle with fiber ℂnassociated withα. We also discuss th

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21

Valdés, Antonio. "Differential invariants of ℝ*-structures". Mathematical Proceedings of the Cambridge Philosophical Society 119, № 2 (1996): 341–56. http://dx.doi.org/10.1017/s030500410007420x.

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A differential invariant of a G-structure is a function which depends on the r-jet of the G-structure and such that it is invariant under the natural action of the pseudogroup of diffeomorphisms of the base manifold. The importance of these objects is clear, since they seem to be the natural obstructions for the equivalence of G-structures. Hopefully, if all the differential invariants coincide over two r–jets of G-structure then they are equivalent under the action of the pseudogroup. If all the differential invariants coincide for every r it is hoped that the G-structures are formally equiva

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22

BRZEZIŃSKI, TOMASZ. "NON-COMMUTATIVE CONNECTIONS OF THE SECOND KIND." Journal of Algebra and Its Applications 07, no. 05 (2008): 557–73. http://dx.doi.org/10.1142/s0219498808002977.

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A connection-like objects, termed hom-connections are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature for a hom-connection is defined, and it is shown that flat hom-connections give rise to a chain complex.

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23

Belova, O. O. "Centered planes in the projective connection space." Differential Geometry of Manifolds of Figures, no. 51 (2020): 29–38. http://dx.doi.org/10.5922/0321-4796-2020-51-4.

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The space of centered planes is considered in the Cartan projective connection space . The space is important because it has connection with the Grassmann manifold, which plays an important role in geometry and topology, since it is the basic space of a universal vector bundle. The space is an n-dimensional differentiable manifold with each point of which an n-dimensional projective space containing this point is associated. Thus, the manifold is the base, and the space is the n-dimensional fiber “glued” to the points of the base. A projective space is a quotient space of a linear space wi

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24

YANG, BO. "A CHARACTERIZATION OF NONCOMPACT KOISO-TYPE SOLITONS." International Journal of Mathematics 23, no. 05 (2012): 1250054. http://dx.doi.org/10.1142/s0129167x12500541.

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We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton's equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics, Vol. 18-I (Academic Press, Boston, MA, 1990), pp. 327–337]. Our examples can be viewed a generalization of previous examples by Cao [Existense of gradient Kähler–

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25

Iezzi, Giovanna, Francesca Di Lillo, Michele Furlani, Marco Degidi, Adriano Piattelli, and Alessandra Giuliani. "The Symmetric 3D Organization of Connective Tissue around Implant Abutment: A Key-Issue to Prevent Bone Resorption." Symmetry 13, no. 7 (2021): 1126. http://dx.doi.org/10.3390/sym13071126.

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Symmetric and well-organized connective tissues around the longitudinal implant axis were hypothesized to decrease early bone resorption by reducing inflammatory cell infiltration. Previous studies that referred to the connective tissue around implant and abutments were based on two-dimensional investigations; however, only advanced three-dimensional characterizations could evidence the organization of connective tissue microarchitecture in the attempt of finding new strategies to reduce inflammatory cell infiltration. We retrieved three implants with a cone morse implant–abutment connection f

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26

Sheu, Albert Jeu-Liang. "A Cancellation Theorem for Modules Over the Group C*-Algebras of Certain Nilpotent Lie Groups." Canadian Journal of Mathematics 39, no. 2 (1987): 365–427. http://dx.doi.org/10.4153/cjm-1987-018-7.

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In recent years, there has been a rapid growth of the K-theory of C*-algebras. From a certain point of view, C*-algebras can be treated as “non-commutative topological spaces”, while finitely generated projective modules over them can be thought of as “non-commutative vector bundles”. The K-theory of C*-algebras [30] then generalizes the classical K-theory of topological spaces [1]. In particular, the K0-group of a unital C*-algebra A is the group “generated” by (or more precisely, the Grothendieck group of) the commutative semigroup of stable isomorphism classes of finitely generated projecti

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27

Giblin, Peter. "Reviews - Differential geometry: bundles, connections, metrics and curvature, by Clifford Henry Taubes. Pp. 298. £27.50. 2011. ISBN 978-0-19-960587-3 (Oxford Graduate Texts in Mathematics No. 23, Oxford University Press)." Mathematical Gazette 97, no. 539 (2013): 373. http://dx.doi.org/10.1017/s0025557200006343.

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28

BOI, LUCIANO. "IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 06, no. 05 (2009): 701–57. http://dx.doi.org/10.1142/s0219887809003783.

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The aim of the first part of this paper is to make some reflections on the role of geometrical and topological concepts in the developments of theoretical physics, especially in gauge theory and string theory, and we show the great significance of these concepts for a better understanding of the dynamics of physics. We will claim that physical phenomena essentially emerge from the geometrical and topological structure of space–time. The attempts to solve one of the central problems in 20th theoretical physics, i.e. how to combine gravity and the other forces into an unitary theoretical explana

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29

Majid, Shahn, and Liam Williams. "Poisson Principal Bundles." Symmetry, Integrability and Geometry: Methods and Applications, January 13, 2021. http://dx.doi.org/10.3842/sigma.2021.006.

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We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also c

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30

Green, Mark, and Phillip Griffiths. "Positivity of vector bundles and Hodge theory." International Journal of Mathematics, July 22, 2021, 2140008. http://dx.doi.org/10.1142/s0129167x21400085.

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Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are [Formula: see text] order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivi

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31

Essig, J. Timo. "Intersection space cohomology of three-strata pseudomanifolds." Journal of Topology and Analysis, April 8, 2019, 1–50. http://dx.doi.org/10.1142/s1793525320500120.

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The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically fla

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32

Tarasov, Vasily E. "Geometric interpretation of fractional-order derivative." Fractional Calculus and Applied Analysis 19, no. 5 (2016). http://dx.doi.org/10.1515/fca-2016-0062.

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AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected

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33

Dervan, Ruadhaí, and Lars Martin Sektnan. "Uniqueness of optimal symplectic connections." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.15.

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Abstract Consider a holomorphic submersion between compact Kähler manifolds, such that each fibre admits a constantscalar curvature Kähler metric. When the fibres admit continuous automorphisms, a choice of fibrewise constant scalarcurvature Kähler metric is not unique. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric satisfying a geometric partial differential equation. The condition generalises the Hermite-Einstein condition for a holomorphic vector bundle through the induced fibrewise Fubini-Study metric on the associated projectivisation. We

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34

Smets, Bart M. N., Jim Portegies, Etienne St-Onge, and Remco Duits. "Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations." Journal of Mathematical Imaging and Vision, September 18, 2020. http://dx.doi.org/10.1007/s10851-020-00991-4.

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Abstract Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multi-orientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for two-dimensional images. In this work, we extend their approach to enhance and denoise images of arbitrary dimension, creating a unified geometric and algorithmic PDE framework, relying on (sub-)Riemannian geometry. In particular, we follow

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