Showing papers on "Frame bundle published in 2012"
TL;DR: In this article, it was shown that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative.
Abstract: We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of " movable curves " , which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1, 1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.
461 citations
TL;DR: In this paper, the concept of a graded bundle is introduced, which is a natural generalization of a vector bundle and whose standard examples are higher tangent bundles T n Q playing a fundamental role in higher order Lagrangian formalisms.
64 citations
TL;DR: In this article, the authors study a model of geometry of vision due to Petitot, Citti, and Sarti, where the primary visual cortex V1 lifts an image from a corrupted image to the bundle of directions of the plane, and then the corrupted image is reconstructed by minimizing the energy necessary for activation of the orientation columns corresponding to regions in which the image is corrupted.
Abstract: In this paper we study a model of geometry of vision due to Petitot, Citti, and Sarti. One of the main features of this model is that the primary visual cortex V1 lifts an image from $\mathbb{R}^2$ to the bundle of directions of the plane. Neurons are grouped into orientation columns, each of them corresponding to a point of this bundle. In this model a corrupted image is reconstructed by minimizing the energy necessary for the activation of the orientation columns corresponding to regions in which the image is corrupted. The minimization process intrinsically defines a hypoelliptic heat equation on the bundle of directions of the plane. In the original model, directions are considered both with and without orientation, giving rise, respectively, to a problem on the group of rototranslations of the plane $SE(2)$ or on the projective tangent bundle of the plane $PT\mathbb{R}^2$. We provide a mathematical proof of several important facts for this model. We first prove that the model is mathematically consis...
61 citations
TL;DR: In this article, the Tian-Zelditch expansion on the circle bundle associated to a positive line bundle on a complex projective manifold is considered. But the authors focus on the case of compact tori.
Abstract: Let X be the circle bundle associated to a positive line bundle on a complex projective (or, more generally, compact symplectic) manifold. The Tian-Zelditch expansion on X may be seen as a local manifestation of the decomposition of the (generalized) Hardy space H(X) into isotypes for the S
1-action. More generally, given a compatible action of a compact Lie group, and under general assumptions guaranteeing finite dimensionality of isotypes, we may look for asymptotic expansions locally reflecting the equivariant decomposition of H(X) over the irreducible representations of the group. We focus here on the case of compact tori.
38 citations
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TL;DR: In this article, it was shown that the Forni bundle of the Hodge bundle is always flat and orthogonal to the tangent space of the moduli space of a curve.
Abstract: Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M We show that the Forni bundle of N (ie the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N As a corollary, it follows that the Hodge bundle of N is semisimple
35 citations
TL;DR: In this paper, it was shown that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions, meaning that the family may be taken to be a finite collection.
Abstract: A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle.
29 citations
TL;DR: In this paper, the geometric properties of the manifold of states described as (uniform) matrix product states are studied, and the main interest is in the states living in the tangent space to the base manifold, which have been shown to be interesting in relation to time dependence and elementary excitations.
Abstract: We study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e. the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a K\"ahler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a well-chosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the Levi-Civita connection) and the Riemann curvature tensor.
20 citations
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TL;DR: In this article, the authors extend the notion of a Parseval frame for a fixed Hilbert space to that of a movingParseval Frame for a vector bundle over a manifold, and prove that a sequence of vector fields is a moving Parsevel Frame for the tangent bundle of a Riemannian manifold if and only if the vector fields are the orthogonal projection of the moving orthonormal basis for a larger vector bundle.
Abstract: Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for a vector bundle over a manifold. Many vector bundles do not have a moving basis, but in contrast to this every vector bundle over a paracompact manifold has a moving Parseval frame. We prove that a sequence of sections of a vector bundle is a moving Parseval frame if and only if the sections are the orthogonal projection of a moving orthonormal basis for a larger vector bundle. In the case that our vector bundle is the tangent bundle of a Riemannian manifold, we prove that a sequence of vector fields is a Parseval frame for the tangent bundle of a Riemannian manifold if and only if the vector fields are the orthogonal projection of a moving orthonormal basis for the tangent bundle of a larger Riemannian manifold.
18 citations
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TL;DR: In this article, a quasi-principle frame bundle is proposed for the equivalence problem of distributions with fixed submanifolds of flags on each fiber, which is a generalization of the Tanaka prolongation procedure for filtered structures.
Abstract: Motivated by the geometric theory of differential equations and the variational ap- proach to the equivalence problem for geometric structures on manifolds, we consider the problem of equivalence for distributions with fixed submanifolds of flags on each fiber. We call them flag structures. The construction of the canonical frames for these structures can be given in the two prolongation steps: the first step, based on our previous works (18, 19), gives the canonical bundle of moving frames for the fixed submanifolds of flags on each fiber and the second step consists of the prolongation of the bundle obtained in the first step. The bundle obtained in the first step is not as a rule a principal bundle so that the classical Tanaka prolongation procedure for filtered structures can not be applied to it. However, under natural assumptions on submanifolds of flags and on the ambient distribution, this bundle satisfies a nice weaker property. The main goal of the present paper is to formalize this property, introducing the so-called quasi-principle frame bundles, and to generalize the Tanaka prolongation procedure to these bundles. Applications to the equivalence problems for systems of differential equations of mixed order, bracket generating distributions, sub-Riemannian and more general structures on distributions are given.
17 citations
TL;DR: In this article, the authors studied the paraholomorphy property of two Riemannian metrics of Cheeger Gromoll type depending on one parameter and two parameters by using compatible paracomplex structures J a and J a, b on the tangent bundle T M.
13 citations
TL;DR: In this paper, the Lagrangian is defined in a fiber bundle with base space and fibers F(x) which can be treated as Lorentzian manifolds equipped with the Wheeler-DeWitt metric.
Abstract: We apply the ADM approach to obtain a Hamiltonian description of the Einstein-Hilbert action. In doing so we add four new ingredients: (i) We eliminate the diffeomorphism constraints. (ii) We replace the densities $\sqrt g$ by a function $\f(x,g_{ij})$ with the help of a fixed metric $\chi$ such that the Lagrangian and hence the Hamiltonian are functions. (iii) We consider the Lagrangian to be defined in a fiber bundle with base space $\so$ and fibers F(x) which can be treated as Lorentzian manifolds equipped with the Wheeler-DeWitt metric. It turns out that the fibers are globally hyperbolic. (iv) The Hamiltonian operator $H$ is a normally hyperbolic operator in the bundle acting only in the fibers and the Wheeler-DeWitt equation $Hu=0$ is a hyperbolic equation in the bundle. Since the corresponding Cauchy problem can be solved for arbitrary smooth data with compact support, we then apply the standard techniques of QFT which can be naturally modified to work in the bundle.
TL;DR: In this paper, the existence of symbolic extensions for partially hyperbolic diffeomorphisms with a 2-dimensional center bundle has been shown in the context of local center manifolds.
Abstract: We relate the symbolic extension entropy of a partially hyperbolic dynamical system to the entropy appearing at small scales in local center manifolds. In particular, we prove the existence of symbolic extensions for $\mathcal{C}^2$ partially hyperbolic diffeomorphisms with a $2$-dimensional center bundle.
200 words.
TL;DR: In this paper, the authors generalize the criterion of Weil and Atiyah for a principal bundle to have an algebraic connection and investigate principal bundles over a root stack for dimension one.
Abstract: We investigate principal bundles over a root stack. In case of dimension one, we generalize the criterion of Weil and Atiyah for a principal bundle to have an algebraic connection.
TL;DR: In this paper, the authors give conditions under which the tangent bundle endowed with such a structure and with a general natural lifted metric is a Riemannian almost product (locally product) or an (almost) para-Hermitian manifold.
Abstract: We find the almost product (locally product) structures of general natural lift type on the tangent bundle of a Riemannian manifold. We get the conditions under which the tangent bundle endowed with such a structure and with a general natural lifted metric is a Riemannian almost product (locally product) or an (almost) para-Hermitian manifold. We give a characterization of the general natural (almost) para-Hermitian structures, which are (almost) para-K a hlerian on the tangent bundle.
TL;DR: In this paper, the index bundle gerbe of a family of self-adjoint Dirac-type operators was constructed for compact Riemann surfaces, whose 3-curvature corresponds to the third de Rham cohomology groups of these moduli spaces.
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TL;DR: In this paper, the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three was studied.
Abstract: Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space FTM on the bundle of frames in the tangent bundle of M. We show that FTM is the group completion of LM, and prove that it is obtained by localizing LM with respect to adding a "small twist".
TL;DR: In this paper, the authors extend the notion of tangent bundle to a $z$ graded bundle, which has a Lie algebroid structure and they can develop notions semi-riemannian metrics, Levi-civita connection, and curvature on it.
Abstract: In this paper we will extend the notion of tangent bundle to a $\z$ graded tangent bundle. This graded bundle has a Lie algebroid structure and we can develop notions semi-riemannian metrics, Levi-civita connection, and curvature, on it. In case of space-times manifolds, even part of the tangent bundle is related to space and time structures(gravity) and odd part is related to mass distribution in space-time. In this structure, mass becomes part of the geometry, and Einstein field equation can be reconstructed in a new simpler form. The new field equation is purely geometric.
TL;DR: In this paper, a geometric theory for bundle shifts and its duality to the geometric theory of Cowen-Douglas operators is proposed. But this theory is not applicable to the case of Hilbert space operators.
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TL;DR: In this paper, a spinor bundle is constructed from an oriented smooth vector bundle of even rank with a fiberwise inner product over a compact connected orientable smooth manifold with Riemannian metric.
Abstract: A bundle gerbe is constructed from an oriented smooth vector bundle of even rank with a fiberwise inner product, over a compact connected orientable smooth manifold with Riemannian metric. From a trivialization of the bundle gerbe is constructed an irreducible Clifford module bundle, a spinor bundle over the smooth free loop space of the manifold.
First, a Clifford algebra bundle over the loop space is constructed from the vector bundle. A polarization class bundle is constructed, choosing continuously over each point of the loop space a polarization class of Lagrangian subspaces of the complexification of the real vector space from which the Clifford algebra is made. Being unable to choose a Lagrangian subspace continuously from the polarization class over each point, the thesis constructs a bundle gerbe over the loop space of the base manifold to encode over each loop all such subspaces, along with the isomorphisms between the Fock spaces made from them, resulting from their being in the same polarization class.
The vanishing of the Dixmier-Douady class of the bundle gerbe implies that the latter has a trivialization, from which is constructed a spinor bundle.
TL;DR: In this article, a generalization of Lagrange-Poincar\'e 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-Poincar\'e 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.
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TL;DR: In this paper, the splitting of the normal bundle of rational curves was studied with apolarity theory and some particular subvarieties in some Hilbert scheme of rational curve, defined by splitting type of normal bundle and the restricted tangent bundle.
Abstract: As in our previous work [1] we address the problem to determine the splitting of the normal bundle of rational curves. With apolarity theory we are able to characterize some particular subvarieties in some Hilbert scheme of rational curves, defined by the splitting type of the normal bundle and the restricted tangent bundle.
TL;DR: In this article, it was shown that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable.
Abstract: We prove that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi–Cartan–Vranceanu metrics and their Hopf bundles.
TL;DR: A semistability algorithm for vector bundles that are given as a kernel of a surjective morphism between splitting bundles on the projective space over an algebraically closed field K.
Abstract: We develop a semistability algorithm for vector bundles that are given as a kernel of a surjective morphism between splitting bundles on the projective space over an algebraically closed field K. This class of bundles is a generalization of syzygy bundles. We show how to implement this algorithm in a computer algebra system. Further, we give applications, mainly concerning the computation of Tannaka dual groups of stable vector bundles of degree 0 on and on certain smooth complete intersection curves. We also use our algorithm to close a case left open in a recent work of L. Costa, P. Macias Marques, and R. M. Miro-Roig regarding the stability of the syzygy bundle of general forms. Finally, we apply our algorithm to provide a computational approach to tight closure. All algorithms are implemented in the computer algebra system CoCoA.
TL;DR: In this article, the curvature of the diagonal lift from an affine manifold to the linear frame bundle is investigated, and the authors obtain analogous results in the affine setting.
Abstract: We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de Leon in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.
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TL;DR: In this paper, the authors provide abstract conditions which imply the existence of a robustly invariant neighbourhood of a global section of a fibre bundle flow and apply such a result to the bundle flow generated by an Anosov flow when the fibre is the space of jets.
Abstract: We provide abstract conditions which imply the existence of a robustly invariant neighbourhood of a global section of a fibre bundle flow. We then apply such a result to the bundle flow generated by an Anosov flow when the fibre is the space of jets (which are described by local manifolds). As a consequence we obtain sets of manifolds (e.g. approximations of stable manifolds) that are left invariant, {\bf for all} negative times, by the flow and its small perturbations. Finally, we show that the latter result can be used to easily fix a mistake recently uncovered in the paper {\em Smooth Anosov flows: correlation spectra and stability}, \cite{BuL}, by the present authors.
TL;DR: In this paper, the authors introduced a class of g-natural metrics Ga,b on the tangent bundle of a Finsler manifold (M, F) which generalizes the associated Sasaki-Matsumoto metric and Miron metric.
Abstract: Recently the third author studied horizontal Laplacians in real Finsler vector bundles and complex Finsler manifolds. In this paper, we introduce a class of g-natural metrics Ga,b on the tangent bundle of a Finsler manifold (M, F) which generalizes the associated Sasaki–Matsumoto metric and Miron metric. We obtain the Weitzenbock formula of the horizontal Laplacian associated to Ga,b, which is a second-order differential operator for general forms on tangent bundle. Using the horizontal Laplacian associated to Ga,b, we give some characterizations of certain objects which are geometric interest (e.g. scalar and vector fields which are horizontal covariant constant) on the tangent bundle. Furthermore, Killing vector fields associated to Ga,b are investigated.
Journal Article•
TL;DR: In this article, it was shown that the fourfold Whitney sum of the tangent bundle of real projective planes of any three dimensional nontrivial real G-representation is equivariantly a product bundle.
Abstract: let G be a nontrivial cyclic group of odd order. In the present paper, we will prove that the fourfold Whitney sum of the tangent bundle of real projective plane of any three dimensional nontrivial real G-representation is equivariantly a product bundle.
TL;DR: In this paper, the authors have discussed the method to make a discrete twist in a large and much discussed class of vector bundles such that the generation number gets new contributions (which can be tuned suitably) and at the same time the space of bundle moduli of the new, twisted bundle is a proper subspace (where the non-generic twist class exists) of the original bundle modulus space; one thus gets a model, closely related to the original model one started with, but with enhanced flexibility in the generator number and where on the other hand the number of bundle
Abstract: A major problem in discussing heterotic string models is the stabilisation of the many vector bundle moduli via the superpotential generated by world-sheet instantons. In arXiv:1110.6315 we have discussed the method to make a discrete twist in a large and much discussed class of vector bundles such that the generation number gets new contributions (which can be tuned suitably) and at the same time the space of bundle moduli of the new, twisted bundle is a proper subspace (where the ’new’, non-generic twist class exists) of the original bundle moduli space; one thus gets a model, closely related to the original model one started with, but with enhanced flexibility in the generation number and where on the other hand the number of bundle moduli is somewhat reduced. Whereas in the previous paper the emphasis was on examples for the new flexibility in the generation number we here classify and describe explicitly the twists and give the precise reduction formula (for the number of moduli) for SU(5) bundles leading to an SU(5) GUT group in four dimensions. Finally we give various examples where the bundle moduli space is reduced completely: the superpotential for such rigid bundles becomes a function of the complex structure moduli alone (besides the exponential Kahler moduli contribution).
01 Jan 2012
TL;DR: In this article, the authors extend the methods of Banagl to link bundles of two types: (1) Fibers CW-complexes with (amongst other conditions) evenly graded homology and base space a sphere.
Abstract: Banagl defines a spatial version of intersection homology. A key step is fiberwise homology truncation of the link bundle of a pseudomanifold. The difficulty of extending said results to more general link bundles is informed by two factors: firstly, the type of fiber (which is also the link of the pseudomanifold), and secondly, the base space of the bundle (which is the singular set of the pseudomanifold). We extend the methods of Banagl to link bundles of two types: (1) Fibers CW-complexes with (amongst other conditions) evenly graded homology and base space a sphere. (2) Using a fiber admitting truncation only in selected degrees and base space such that the bundle is glued from two trivial bundles. Different methods are required in each setting. In the first setting, truncation of the fiberwise gluing homeomorphisms yields only homotopy equivalences. Hence homotopy theory is necessary to build a truncated bundle with the right properties. In the second case, this difficulty is not encountered, and no homotopy theory is necessary. Here, we use sheaf theory. In both cases we require the link bundle to be glued from trivial bundles by means of cellular homeomorphisms. Generalized Poincare duality is shown for pseudomanifolds with each type of link bundle.