Showing papers on "Frame bundle published in 2014"
TL;DR: In this article, it was shown that a co-Higgs bundle arising from a Schwarzenberger bundle with nonzero Higgs field is rigid, in the sense that a nearby deformation is again Schwartzberger.
Abstract: On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's rank-2 vector bundle on the projective plane, constructed from a line bundle on the double cover CP^1 \times CP^1 \to CP^2, is naturally a co-Higgs bundle, with the twisted endomorphism, or "Higgs field", also descending from the double cover. Allowing the branch conic to vary, we find that Schwarzenberger bundles give rise to an 8-dimensional moduli space of co-Higgs bundles. After studying the deformation theory for co-Higgs bundles on complex manifolds, we conclude that a co-Higgs bundle arising from a Schwarzenberger bundle with nonzero Higgs field is rigid, in the sense that a nearby deformation is again Schwarzenberger.
42 citations
TL;DR: In this paper, the structural structure underlying the Lie 3-algebra of superconformal models in six space-time dimensions has been revealed, and the generalized Bianchi identities can be retrieved concisely from Q 2 = 0, which encode all the essence of the structural identities.
41 citations
TL;DR: In this article, a maximal acceleration invariant quantum field is formulated in terms of the differential geometric structure of the spacetime tangent bundle, and the field is shown to have a physically based Planck-scale effective regularization and a spectral cutoff at the Planck mass.
Abstract: Maximal-acceleration invariant quantum fields are formulated in terms of the differential geometric structure of the spacetime tangent bundle. The simple special case is considered of a flat Minkowski space-time for which the bundle is also flat. The field is shown to have a physically based Planck-scale effective regularization and a spectral cutoff at the Planck mass.
27 citations
TL;DR: In this paper, the Burger-Roblin measure is ergodic for the frame bundle of a non-elementary convex cocompact hyperbolic 3-manifold if and only if δ > 1.
Abstract: Let M be a non-elementary convex cocompact hyperbolic 3-manifold and δ be the critical exponent of its fundamental group. We prove that a one-dimensional unipotent flow for the frame bundle of M is ergodic for the Burger-Roblin measure if and only if δ > 1.
19 citations
TL;DR: The problem of determining the splitting of the normal bundle of rational space curves has been considered in the 80s in a series of papers by Ghione and Sacchiero and by Eisenbud and Van de Ven as mentioned in this paper.
14 citations
TL;DR: The vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler-Einstein manifold of classical type was shown in this article.
Abstract: We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler--Einstein manifold of classical type.
12 citations
TL;DR: In this article, the lifting problem of projectable geometric objects on M to the semi-cotangent bundle is considered and relations between lifted objects and a degenerate symplectic structure are also presented.
Abstract: Using the ber bundle M over a manifold B, we dene a semi-cotangent (pull-back) bundle t B, which has a degenerate symplectic structure. We consider lifting problem of projectable geometric objects on M to the semi-cotangent bundle. Relations between lifted objects and a degenerate symplectic structure are also presented.
12 citations
TL;DR: In this paper, a generalization of Lagrange-Poincare reduction scheme to these types of variational problems allows us to relate it with the Einstein-Hilbert variational problem.
Abstract: The present article deals with a formulation of the so called (vacuum) Palatini gravity as a general variational principle. In order to accomplish this goal, some geometrical tools related to the geometry of the bundle of connections of the frame bundle LM are used. A generalization of Lagrange-Poincare reduction scheme to these types of variational problems allows us to relate it with the Einstein-Hilbert variational problem. Relations with some other variational problems for gravity found in the literature are discussed.
12 citations
TL;DR: In this article, it was shown that there is an analogous picture in the dynamics of objects for which the configuration space is the vector bundle of n-vectors, if we make use of certain graded bundle structures of degree n, i.e. objects generalizing vector bundles.
Abstract: The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration space is TM, is based on the existence of canonical symplectic isomorphisms of double vector bundles T*TM, T*TM, and TT*M. We show that there exist an analogous picture in the dynamics of objects for which the configuration space is the vector bundle of n-vectors, if we make use of certain graded bundle structures of degree n, i.e. objects generalizing vector bundles (for which n=1). For instance, the role of TT*M is played in our approach by the vector bundle of n-vectors on the bundle of n-covectors, which is canonically a graded bundle of degree n over the bundle of n-vectors. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.
10 citations
TL;DR: In this article, the Hilbert curve of a projective bundle over a smooth curve is explicitly described and the reconstruction problem is addressed and the very special geographic shape of Γ allows to recover the adjunction theoretic type of (X, L) for smooth surfaces, for scrolls over smooth curves, as well as for Veronese bundles under additional assumptions.
Abstract: Let X be a projective bundle over a smooth curve and let L be an ample line bundle on X inducing $\mathcal O_{\mathbb P}(r)$ on every fiber. The Hilbert curve Γ of such a polarized manifold (X, L) is explicitly described and the reconstruction problem is addressed. In particular, it is shown that the very special geographic shape of Γ allows to recover the adjunction theoretic type of (X, L) for smooth surfaces, for scrolls over a smooth curve, as well as for Veronese bundles under additional assumptions.
10 citations
TL;DR: The existence of canonical Kahler metrics on the projec- tivisation of strictly Mumford semistable vector bundles over a curve is studied and some partial generalizations are given for higher dimensional base.
Abstract: We study the existence of canonical Kahler metrics on the projec- tivisation of strictly Mumford semistable vector bundles over a curve. Consider the projectivisation P(E) of a holomorphic vector bundle E over a smooth base manifold B polarized by the ample line bundle LB. Various results have established a relationship between the stability of the underlying bundle E and the existence of Kahler metrics with special curva- ture on P(E), at least when c1(LB) can be endowed with an extremal metric. The case of a base manifold of complex dimension 1 has been intensively studied. Building on the work of D. Burns- P. de Bartolomeis, E. Calabi, A. Fujiki, C. Lebrun and many others, V. Apostolov, D. Calderbank, P. Gaudu- chon and C. Tonnesen-Friedman have provided a complete understanding of the situation for stable or polystable bundles over a smooth Riemann surface. They showed that there is a Kahler metric with constant scalar curvature (cscK metric in short) in any Kahler class on P(E) if and only if the bundle E is Mumford polystable (1, 2, 3). Another approach was also carried out in a series of paper of Y.-J. Hong, see (17) who investigated the case of higher dimensional base. Other results for ruled manifolds appeared recently in relationship with extremal Kahler metrics in (7, 21). Up to our knowledge, the case of strictly semistable bundle is still open in complete generality. We expect that for a base manifold of dimension ≥ 2 all the phe- nomena of stability for P(E) could happen when E is Mumford semistable (see for instance (19)). In this note, we essentially study the particular case of a ruled surface given by the projectivisation of a Mumford semistable vector bundle (which is not stable) over a Riemann surface of genus g ≥ 2. Some partial generalizations are given for higher dimensional base (note that the results of Section 3 will be extended in a forthcoming paper). Conventions: If π: E → B is a vector bundle then π: P(E) → B shall de- note the space of complex hyperplanes in the fibres of E. Thus π∗OP(E)(r) = S r E for r ≥ 0.
TL;DR: In this paper, the Lie algebroid generalized tangent bundle of a Kaluza-Klein vector bundle is introduced and the theory of general distinguished linear connections for this space is developed.
Abstract: 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.
TL;DR: In this article, the authors give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle over a Riemannian manifold when $E$ is endowed with a metric connection.
Abstract: We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle $E\rightarrow M$, over a Riemannian manifold $M$, when $E$ is endowed with a metric connection. The tangent bundle of $E$ admits a canonical decomposition and thus it is possible to define an interesting class of two-weights metrics with the weight functions depending on the fibre norm of $E$; hence the generalized concept of spherically symmetric metrics. We study its main properties and curvature equations. Finally we focus on a few applications and compute the holonomy of Bryant-Salamon type $\mathrm{G}_2$ manifolds.
TL;DR: For every line bundle on C, the twisted Petri map is injective as mentioned in this paper, and the injective line bundles are generated by a generic vector bundle on a generic curve.
Abstract: Let C be a generic curve, E a generic vector bundle on C. Then, for every line bundle on C the twisted Petri map
$$P_{E}:H^0(C,L\otimes E)\otimes H^0(C, K\otimes L^*\otimes E^{*})\rightarrow H^0(C, K)$$
is injective.
TL;DR: In this article, it was shown that the tangent bundle of a smooth Banach manifold admits a vector bundle structure over the manifold if and only if the manifold is endowed with a linear connection.
Abstract: The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. For a Banach manifold $M$ and a natural number $k$ first we determine a smooth manifold structure on $T^kM$ which also offers a fiber bundle structure for $(\pi_k,T^kM,M)$. Then we introduce a particular lift of linear connections on $M$ to geometrize $T^kM$ as a vector bundle over $M$. More precisely based on this lifted nonlinear connection we prove that $T^kM$ admits a vector bundle structure over $M$ if and only if $M$ is endowed with a linear connection. As a consequence applying this vector bundle structure we lift Riemannian metrics and Lagrangians from $M$ to $T^kM$. Also, using the projective limit techniques, we declare a generalized Fr\'echet vector bundle structure for $T^\infty M$ over $M$.
TL;DR: In this article, the authors extend the Horrocks correspondence between vector bundles and cohomology modules on the projective plane to the product of two projective lines and prove that there is a one to one correspondence between these sets of invariants and isomorphism classes of vector bundles without line bundle summands.
Abstract: We extend the Horrocks correspondence between vector bundles and cohomology modules on the projective plane to the product of two projective lines. We introduce a set of invariants for a vector bundle on the product of two projective lines, which includes the first cohomology module of the bundle, and prove that there is a one to one correspondence between these sets of invariants and isomorphism classes of vector bundles without line bundle summands.
TL;DR: In this article, the authors define the notion of the $k$'th order differential (T^kg:T^kM\longrightarrow T^kN$) for a given differentiable map $g$ between manifolds $M$ and $N$.
Abstract: The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. In the previous work of the author he proved that $T^kM$, $1\leq k\leq \infty$, admits a vector bundle structure on $M$ if and only if $M $ is endowed with a linear connection or equivalently a connection map on $T^kM$ is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the notion of the $k$'th order differential $T^kg:T^kM\longrightarrow T^kN$ for a given differentiable map $g$ between manifolds $M$ and $N$. As we shall see, $T^kg$ becomes a vector bundle morphism if the base manifolds are endowed with $g$-related connections. In particular, replacing a connection with a $g$-related one, where $g:M\longrightarrow M$ is a diffeomorphism, follows invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combination of connection maps and manifold of $C^r$ maps we offer three examples to support our theory and reveal its interaction with the known problems such as Sasaki lift of metrics.
TL;DR: In this article, the canonical form in the non-holonomic second-order frame bundle of an n-dimensional manifold M is introduced as a restriction of the canonical forms of the bundle.
Abstract: Using horizontal n-bases of the tangent bundle of the linear frame bundle of an n-dimensional manifold M, the canonical form in the non-holonomic second-order frame bundle of M is introduced as a restriction of the canonical form of the bundle . This construction generalizes the ones in the corresponding semi-holonomic and holonomic second-order frame bundles. We prove that the natural projection of the set of all non-holonomic second-order frames of M into defines a principal bundle structure.
TL;DR: In this paper, a Poincare projective bundle on a smooth complex projective curve X×ML is shown to be stable with respect to any polarization on X×MM.
TL;DR: In this article, the slope stability of tautological vector bundles on Hilbert schemes of points on smooth surfaces has been studied and it has been shown that every sufficiently positive semistable vector bundle on a smooth curve arises as the restriction of a tautology vector bundle of the tangent bundle on the projective plane.
Abstract: The purpose of this paper is to explore the geometry and establish the slope stability of tautological vector bundles on Hilbert schemes of points on smooth surfaces. By establishing stability in general we complete a series of results of Schlickewei and Wandel who proved the slope stability of these vector bundles for Hilbert schemes of 2 points or 3 points on K3 or abelian surfaces with Picard group restrictions. In exploring the geometry we show that every sufficiently positive semistable vector bundle on a smooth curve arises as the restriction of a tautological vector bundle on the Hilbert scheme of points on the projective plane. Moreover we show the tautological bundle of the tangent bundle is naturally isomorphic to the sheaf of vector fields tangent to the divisor which consists of nonreduced subschemes.
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TL;DR: In this paper, it was shown that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomology as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch theorem.
Abstract: We show that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomol- ogy as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch theorem. This information has been used to construct a cohomology class of the adjoint bundle of a flat bundle whose restriction to each fiber is the class of the Maurer-Cartan 3-form. Then we use this result to define an invariant of a gauge transformation of a flat bundle which describes the effect of the gauge transformation on a Chern-Simons functional.
TL;DR: In this article, an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold was proved, where the manifold is the unit circle bundle of a positive line bundle over a complex manifold.
Abstract: We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.
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TL;DR: In this paper, the authors considered classical gauge theory with spontaneous symmetry breaking on a principal bundle whose structure group is reducible to a closed subgroup, and treated sections of the quotient bundle as classical Higgs fields.
Abstract: We consider classical gauge theory with spontaneous symmetry breaking on a principal bundle $P\to X$ whose structure group $G$ is reducible to a closed subgroup $H$, and sections of the quotient bundle $P/H\to X$ are treated as classical Higgs fields. In this theory, matter fields with an exact symmetry group $H$ are described by sections of a composite bundle $Y\to P/H\to X$. We show that their gauge $G$-invariant Lagrangian necessarily factorizes through a vertical covariant differential on $Y$ defined by a principal connection on an $H$-principal bundle $P\to P/H$.
TL;DR: In this paper, the structure of a principal bundle on the r-jet prolongation JrFX of the frame bundle FX over a manifold X is introduced, and the associated action of the structure group of JrFX corresponds with the standard actions of differential groups on tensor spaces.
Abstract: In this paper, we introduce the structure of a principal bundle on the r-jet prolongation JrFX of the frame bundle FX over a manifold X. Our construction reduces the well-known principal prolongation WrFX of FX with structure group Gnr. For a structure group of JrFX we find a suitable subgroup of Gnr. We also discuss the structure of the associated bundles. We show that the associated action of the structure group of JrFX corresponds with the standard actions of differential groups on tensor spaces.
TL;DR: In this article, the authors studied the set of points at which the stalks of an ample Banach bundle over an extremally disconnected compact space are finite-dimensional or separable.
Abstract: Topological characteristics are studied of the set of points at which the stalks of an ample Banach bundle over an extremally disconnected compact space are finite-dimensional or separable. Some connection is established between finite dimensionality or separability of the stalks of a bundle and the analogous properties of the stalks of the ample hull of the bundle. We obtain a new criterion for existence of a dual bundle.
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TL;DR: In this paper, the first two articles aiming to introduce symplectic spinors into the field of symplectic topology and the subject of Frobenius structures are presented, and a classification of the irreducibles and the indecomposables of the latter in terms of certain U(n)-reductions of the G-extension of the metaplectic frame bundle and a certain connection on it is given.
Abstract: This is the first of two articles aiming to introduce symplectic spinors into the field of symplectic topology and the subject of Frobenius structures. After exhibiting a (tentative) axiomating setting for Frobenius structures resp. ’Higgs pairs’ in the context of symplectic spinors, we present immediate observations concerning a local Schroedinger equation, the first structure connection and the existence of ’spectrum’, its topological interpretation and its connection to ’formality’ which are valid for the case of standard Frobenius structures. We give a classification of the irreducibles and the indecomposables of the latter in terms of certain U(n)-reductions of the G-extension of the metaplectic frame bundle and a certain connection on it, where G is the semi-direct product of the metaplectic group and the Heisenberg group, while the indecomposable case involves in addition the combinatorial structure of the eigenstates of the n-dimensional harmonic oscillator. In the second ∗
TL;DR: In this paper, a star product on the sections of the endomorphism bundle of the dual bundle of a holomorphic Hermitian vector bundle was constructed, which also has the appropriately generalized property of separation of variables.
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TL;DR: In this article, Dey and Mathai showed that the corresponding Quillen determinant line bundle is isomorphic to the given G$-equivariant holomorphic Hermitian line bundle.
Abstract: Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,H)$ on $X\times M$ (the action of $G$ on $X$ is trivial), such that the corresponding Quillen determinant line bundle $({\mathcal Q}, h_Q)$, which is a $G$--equivariant holomorphic Hermitian line bundle on $M$, is isomorphic to the given $G$--equivariant holomorphic Hermitian line bundle $({\mathcal L}\,,h)$. This proves a conjecture of Dey and Mathai in \cite{DM}.
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TL;DR: In this paper, a version of Grothendieck's descent theorem on an enriched principal fiber bundle with an action of a larger group scheme was proved for the equivariant Picard and the class groups arising from such bundles.
Abstract: We prove a version of Grothendieck's descent theorem on an `enriched' principal fiber bundle, a principal fiber bundle with an action of a larger group scheme Using this, we prove the isomorphisms of the equivariant Picard and the class groups arising from such a principal fiber bundle
01 Sep 2014
TL;DR: In this paper, the authors consider the vector bundle E E E, where E is a Banach Lie algebroid and E its dual, and show that any Dirac structure that is closed with respect to a type of Courant bracket, endowed with a natural anchor is a Lie Algebra.
Abstract: In the original denition due to A. Weinstein and T. Courant a Dirac structure is a subbundle of the big tangent bundle TM 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 dene 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 dierential calculus on a Lie algebroid is essentially involved. We work in the category of Banach vector bundles.