Showing papers on "Frame bundle published in 2016"

Negative holomorphic curvature and positive canonical bundle

Abstract: In this note we show that if a projective manifold admits a Kahler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

On the behavior of sequences of solutions to U(1) Seiberg-Witten systems in dimension 4

Abstract: This paper studies the behavior of sequences of solutions to Seiberg-Witten like equations for a pair consisting of a Hermitian connection on a line bundle over a 4-dimensional manifold and a section of the self-dual spinor bundle of a complex Clifford module on the manifold Examples include the cases where the Clifford module is a direct sum of C2 bundles associated to SpinC structures; and the case of the SU(2) Vafa-Witten equations with an Abelian ansatz

Small Mass Limit of a Langevin Equation on a Manifold

Abstract: We study damped geodesic motion of a particle of mass $m$ on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as $m \to 0$, its solutions converge to solutions of a limiting equation which includes a {\it noise-induced drift} term. A very special case of the main result presents Brownian motion on the manifold as a limit of inertial systems.

On bundles that admit fiberwise hyperbolic dynamics

Abstract: This paper is devoted to rigidity of smooth bundles which are equipped with fiberwise geometric or dynamical structure. We show that the fiberwise associated sphere bundle to a bundle whose leaves are equipped with (continuously varying) metrics of negative curvature is a topologically trivial bundle when either the base space is simply connected or, more generally, when the bundle is fiber homotopically trivial. We present two very different proofs of this result: a geometric proof and a dynamical proof. We also establish a number of rigidity results for bundles which are equipped with fiberwise Anosov dynamical systems. Finally, we present several examples which show that our results are sharp in certain ways or illustrate necessity of various assumptions.

Pure spinors, intrinsic torsion and curvature in odd dimensions

Abstract: We study the geometric properties of a ( 2 m + 1 ) -dimensional complex manifold M admitting a holomorphic reduction of the frame bundle to the structure group P ⊂ Spin ( 2 m + 1 , C ) , the stabiliser of the line spanned by a pure spinor at a point. Geometrically, M is endowed with a holomorphic metric g, a holomorphic volume form, a spin structure compatible with g, and a holomorphic pure spinor field ξ up to scale. The defining property of ξ is that it determines an almost null structure, i.e. an m-plane distribution N ξ along which g is totally degenerate. We develop a spinor calculus, by means of which we encode the geometric properties of N ξ and of its rank- ( m + 1 ) orthogonal complement N ξ ⊥ corresponding to the algebraic properties of the intrinsic torsion of the P-structure. This is the failure of the Levi-Civita connection ∇ of g to be compatible with the P-structure. In a similar way, we examine the algebraic properties of the curvature of ∇. Applications to spinorial differential equations are given. Notably, we relate the integrability properties of N ξ and N ξ ⊥ to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we conjecture a Goldberg–Sachs-type theorem on the existence of a certain class of almost null structures when ( M , g ) has prescribed curvature. We discuss applications of this work to the study of real pseudo-Riemannian manifolds.

A note on actions of some monoids

Abstract: Smooth actions of the multiplicative monoid $(\mathbb{R},\cdot)$ of real numbers on manifolds lead to an alternative, and for some reasons simpler, definition of a vector bundle, a double vector bundle and related structures like a graded bundle [Grabowski and Rotkiewicz, J. Geom. Phys. 2011]. For these reasons it is natural to study smooth actions of certain monoids closely related with the monoid $(\mathbb{R},\cdot)$ . Namely, we discuss geometric structures naturally related with: smooth and holomorphic actions of the monoid of multiplicative complex numbers, smooth actions of the monoid of second jets of punctured maps $(\mathbb{R},0)\rightarrow (\mathbb{R},0)$, smooth action of the monoid of real 2 by 2 matrices and smooth actions of multiplicative reals on a supermanifold. In particular cases we recover the notions of a holomorphic vector bundle, a complex vector bundle and a non-negatively graded manifold.

Symmetric differentials on complex hyperbolic manifolds with cusps

Abstract: Let $(X, D)$ be a logarithmic pair, and let $h$ be a singular metric on the tangent bundle, smooth on the open part of $X$. We give sufficient conditions on the curvature of $h$ for the logarithmic and the standard cotangent bundles to be big. As an application, we give a metric proof of the bigness of logarithmic cotangent bundle on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of the standard tangent bundle in the more specific case of the ball. We obtain effective ramification orders for a cover $X' \longrightarrow X$, etale outside the boundary, to have all its subvarieties with big cotangent bundle. We also prove that the standard tangent bundle of such a cover is nef if the ramification is high enough. Moreover, the ramification orders we obtain do not depend on the dimension of the quotient of the ball we consider.

Spin structures on loop spaces that characterize string manifolds

Abstract: Classically, a spin structure on the loop space of a manifold is a lift of the structure group of the looped frame bundle from the loop group to its universal central extension. Heuristically, the loop space of a manifold is spin if and only if the manifold itself is a string manifold, against which it is well-known that only the if-part is true in general. In this article we develop a new version of spin structures on loop spaces that exists if and only if the manifold is string, as desired. This new version consists of a classical spin structure plus a certain fusion product related to loops of frames in the manifold. We use the lifting gerbe theory of Carey-Murray, recent results of Stolz-Teichner on loop spaces, and some own results about string geometry and Brylinski-McLaughlin transgression.

The 2-Hilbert Space of a Prequantum Bundle Gerbe

Abstract: We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Waldorf's version of the 2-category of line bundle gerbes. We show that these morphism categories carry a monoidal structure under which they are semisimple and abelian. We introduce a dual functor on the sections, which yields a closed structure on the morphisms between bundle gerbes and turns the category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert spaces fit various expectations from higher prequantisation. We then extend the transgression functor to the full 2-category of bundle gerbes and demonstrate its compatibility with the additional structures introduced. We discuss various aspects of Kostant-Souriau prequantisation in this setting, including its dimensional reduction to ordinary prequantisation.

Polarisation of Graded Bundles

Abstract: We construct the full linearisation functor which takes a graded bundle of degreek (a particular kind of graded manifold) and produces ak-fold vector bundle. We fully characterise the image of the full linearisation functor and show that we obtain a subcategory ofk-fold vector bundles consisting of symmetric k-fold vector bundles equipped with a family of morphisms indexed by the symmetric group Sk. Interestingly, for the degree 2 case this additional structure gives rise to the notion of a symplectical double vector bundle, which is the skew-symmetric analogue of a metric double vector bundle. We also discuss the related case of fully linearising N-manifolds, and how one can use the full linearisation functor to \superise" a graded bundle.

The singularities of the invariant metric on the Jacobi line bundle

Abstract: A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, equipped with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way that the metric acquires only logarithmic singularities. This result is the key of being able to compute arithmetic intersection numbers from these line bundles. Hence, it is natural to ask whether Mumford’s result remains valid for line bundles on mixed Shimura varieties. In this paper we examine the simplest case, namely the Jacobi line bundle on the universal elliptic curve, whose sections are the Jacobi forms. We will show that Mumford’s result cannot be extended directly to this case and that a new type of singularity appears. By using the theory of b-divisors, we show that an analogue of Mumford’s extension theorem can be obtained. We also show that this extension is meaningful because it satisfies Chern-Weil theory and a Hilbert-Samuel type of formula.

Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds

Abstract: We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as most probable paths for a driving semi-martingale that through stochastic development is mapped to the manifold. We discuss how the paths are projections of geodesics for a sub-Riemannian metric on the frame bundle of the manifold, and how the curvature of the underlying connection appears in the sub-Riemannian Hamilton–Jacobi equations. Evolution equations for both metric and cometric formulations of the sub-Riemannian metric are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we relate the paths to geodesics and polynomials in Riemannian geometry. Examples from the family of paths are visualized on embedded surfaces, and we explore computational representations on finite dimensional landmark manifolds with geometry induced from right-invariant metrics on diffeomorphism groups.

A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds

Abstract: We give a short proof of the unique ergodicity of the strong stable foliation of the geodesic flow on the frame bundle of a hyperbolic manifold admitting a finite measure of maximal entropy. Equivalently, let G = S0o(n, 1), $\Gamma$ \textless{} G be a discrete subgroup of G, and G = N AK the Iwasawa decomposition of G. If the geodesic flow on $\Gamma$\G admits a finite measure of maximal entropy, we prove that the action of N on $\Gamma$\G by right multiplication admits a unique invariant measure supported on points whose A-orbit does not diverge.

Self-adjoint extensions of differential operators on Riemannian manifolds

Abstract: We study \(H=D^*D+V\), where D is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold M, and V is a Hermitian bundle endomorphism. In the case when M is geodesically complete, we establish the essential self-adjointness of positive integer powers of H. In the case when M is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of H, expressed in terms of the behavior of V relative to the Cauchy boundary of M.

Construction of Categorical Bundles from Local Data

Abstract: A categorical principal bundle is a structure comprised of categories that is analogous to a classical principal bundle; examples arise from geometric contexts involving bundles over path spaces. We show how a categorical principal bundle can be constructed from local data specified through transition functors and natural transformations.

Mean Topological Dimension for random bundle transformations

Abstract: We introduce the mean topological dimension for random bundle transformations, and show that continuous bundle random dynamical systems with finite topological entropy, or the small boundary property have zero mean topological dimensions.

Tangent Bundle of the Hypersurfaces in a Euclidean Space

Abstract: We consider an immersed orientable hypersurface f : M ! R n+1 of the Euclidean space (f an immersion), and observe that the tan- gent bundle TM of the hypersurface M is an immersed submanifold of the Euclidean space R 2n+2 . Then we show that in general the induced metric on TM is not a natural metric and obtain expressions for the horizontal and vertical lifts of the vector fields on M. We also study the special case in which the induced metric on TM becomes a natural metric and show that in this case the tangent bundle TM is trivial.

The Lagrange-Poincaré equations for a mechanical system with symmetry on the principal fiber bundle over the base represented by the bundle space of the associated bundle

Abstract: The Lagrange--Poincare equations for a mechanical system which describes the interaction of two scalar particles that move on a special Riemannian manifold, consisting of the product of two manifolds, the total space of a principal fiber bundle and the vector space, are obtained. The derivation of equations is performed by using the variational principle developed by Poincare for the mechanical systems with a symmetry. The obtained equations are written in terms of the dependent variables which, as in gauge theories, are implicitly determined by means of equations representing the local sections of the principal fiber bundle.

Cohomology of Cryo-Electron Microscopy

Abstract: The goal of cryo-electron microscopy (EM) is to reconstruct the 3-dimensional structure of a molecule from a collection of its 2-dimensional projected images. In this article, we show that the basic premise of cryo-EM --- patching together 2-dimensional projections to reconstruct a 3-dimensional object --- is naturally one of Cech cohomology with SO(2)-coefficients. We deduce that every cryo-EM reconstruction problem corresponds to an oriented circle bundle on a simplicial complex, allowing us to classify cryo-EM problems via principal bundles. In practice, the 2-dimensional images are noisy and a main task in cryo-EM is to denoise them. We will see how the aforementioned insights can be used towards this end.

A study on fuzzy tangent bundle of fuzzy Banach manifold

Abstract: In this paper, we study some of fuzzy topological and analytical properties of fuzzy tangent bundle and fuzzy cotangent bundle of fuzzy Banach manifold.

Interior proximal bundle algorithm with variable metric for nonsmooth convex symmetric cone programming

Abstract: This paper is devoted to the study of a bundle proximal-type algorithm for solving the problem of minimizing a nonsmooth closed proper convex function subject to symmetric cone constraints, which include the positive orthant in , the second-order cone, and the cone of positive semidefinite symmetric matrices. On the one hand, the algorithm extends the proximal algorithm with variable metric described by Alvarez et al. to our setting. We show that the sequence generated by the proposed algorithm belongs to the interior of the feasible set by an appropriate choice of a regularization parameter. Also, it is proven that each limit point of the sequence generated by the algorithm solves the problem. On the other hand, we provide a natural extension of bundle methods for nonsmooth symmetric cone programs. We implement and test numerically our bundle algorithm with some instances of Euclidean Jordan algebras.

Stability of Secant Bundles on Second Symmetric Power of Curves

Abstract: Given a rank $r$ stable bundle over a smooth irreducible projective curve $C,$ there is an associated rank $2r$ bundle over $S^2(C),$ the second symmetric power of $C.$ In this article we study the stability of this bundle. As a consequence we get an immersion from the moduli space of stable bundles over $C$ to the associated moduli space of stable bundles over $S^2(C).$

Discrete free Abelian central stabilizers in a higher order frame bundle

Abstract: Let a real Lie group $G$ have a $C^\infty$ action on a real manifold $M$. Assume every nontrivial element of $G$ has nowhere dense fixpoint set in $M$. First, we show, in every frame bundle, except possibly the $0$th, that each stabilizer admits no nontrivial compact subgroups. Second, we show that, if $G$ is connected, then there is a dense open $G$-invariant subset of some higher order frame bundle of $M$ such that, for any point $x$ in that subset, the stabilizer in $G$ of $x$ is a discrete, finitely-generated, free-Abelian, central subgroup of $G$. We derive several corollaries of these two results.

On Complex Supermanifolds with Trivial Canonical Bundle

Abstract: We give an algebraic characterisation for the triviality of the canonical bundle of a complex supermanifold in terms of a certain Batalin-Vilkovisky superalgebra structure. As an application, we study the Calabi-Yau case, in which an explicit formula in terms of the Levi-Civita connection is achieved. Our methods include the use of complex integral forms and the recently developed theory of superholonomy.