Showing papers on "Frame bundle published in 2013"
TL;DR: In this paper, it was shown that any smooth arithmetically Gorenstein surface whose rank-r Ulrich bundles admit a generalized theta-divisor admits del Pezzo surfaces.
60 citations
TL;DR: In this article, a twistor version of the hyperkahler/quaternion Kahler correspondence is established and the corresponding holomorphic line bundle on twistor space is described and many examples computed, including monopole and Higgs bundle moduli spaces.
Abstract: A hyperkahler manifold with a circle action fixing just one complex structure admits a natural hyperholomorphic line bundle. This observation forms the basis for the construction of a corresponding quaternionic Kahler manifold in the work of A.Haydys. In this paper the corresponding holomorphic line bundle on twistor space is described and many examples computed, including monopole and Higgs bundle moduli spaces. Finally a twistor version of the hyperkahler/quaternion Kahler correspondence is established.
58 citations
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TL;DR: In this article, it was shown that an algebraic stack with affine stabilizer groups satisfies the resolution property if and only if it is a quotient of a quasi-affine scheme by the action of the general linear group.
Abstract: We show that an algebraic stack with affine stabilizer groups satisfies the resolution property if and only if it is a quotient of a quasi-affine scheme by the action of the general linear group, or equivalently, if there exists a vector bundle whose associated frame bundle has quasi-affine total space. This generalizes a result of B. Totaro to non-normal and non-noetherian schemes and algebraic stacks. Also, we show that the vector bundle induces such a quotient structure if and only if it is a tensor generator in the category of quasi-coherent sheaves.
33 citations
28 Aug 2013
TL;DR: A dimensionality reduction procedure for data in Riemannian manifolds that moves the analysis from a center point to local distance measurements, providing a natural view of data generated by multimodal distributions and stochastic processes.
Abstract: In Euclidean vector spaces, dimensionality reduction can be centered at the data mean. In contrast, distances do not split into orthogonal components and centered analysis distorts inter-point distances in the presence of curvature. In this paper, we define a dimensionality reduction procedure for data in Riemannian manifolds that moves the analysis from a center point to local distance measurements. Horizontal component analysis measures distances relative to lower-order horizontal components providing a natural view of data generated by multimodal distributions and stochastic processes. We parametrize the non-local, low-dimensional subspaces by iterated horizontal development, a constructive procedure that generalizes both geodesic subspaces and polynomial subspaces to Riemannian manifolds. The paper gives examples of how low-dimensional horizontal components successfully approximate multimodal distributions.
30 citations
TL;DR: In this paper, the authors investigated the properties of stratified-algebraic vector bundles on real algebraic variety X and showed that they have many surprising properties, which distinguish them from algebraic and topological vector bundles.
Abstract: We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X. A topological vector bundle on X is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification of X such that the restriction of the bundle to each stratum is an algebraic vector bundle. In particular, every algebraic vector bundle on X is stratified-algebraic. It turns out that stratified-algebraic vector bundles have many surprising properties, which distinguish them from algebraic and topological vector bundles.
28 citations
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TL;DR: In this article, the authors introduce the $b^k$-tangent bundle, whose sections are vector fields with "order $k$ tangency" to the hypersurface of a manifold.
Abstract: Let $Z$ be a hypersurface of a manifold $M$. The $b$-tangent bundle of $(M, Z)$, whose sections are vector fields tangent to $Z$, is used to study pseudodifferential operators and stable Poisson structures on $M$. In this paper we introduce the $b^k$-tangent bundle, whose sections are vector fields with "order $k$ tangency" to $Z$. We describe the geometry of this bundle and its dual, generalize the celebrated Mazzeo-Melrose theorem of the de Rham theory of $b$-manifolds, and apply these tools to classify certain Poisson structures on compact oriented surfaces.
22 citations
TL;DR: In this paper, it was shown that when the 3-form is specified in terms of a line bundle on a spectral divisor, which is a global extension of the Higgs bundle spectral cover, the gluing data of the local model is uniquely determined in a way that ensures agreement with Heterotic results whenever a heterogeneous dual exists.
Abstract: Local models that capture the 7-brane physics of F-theory compactifications for supersymmetric GUTs are conveniently described in terms of an E
8 gauge theory in the presence of a Higgs bundle. Though the Higgs bundle data is usually determined by the local geometry and G-flux, additional gluing data must be specified whenever the Higgs bundle spectral cover is not smooth. In this paper, we argue that this additional information is determined by data of the M-theory 3-form that is not necessarily captured by the cohomology class of the G-flux. More specifically, we show that when the 3-form is specified in terms of a line bundle on a spectral divisor, which is a global extension of the Higgs bundle spectral cover, the gluing data of the local model is uniquely determined in a way that ensures agreement with Heterotic results whenever a Heterotic dual exists.
20 citations
TL;DR: In this article, the authors characterize the almost product and locally product structures of general natural lift type on the cotangent bundle of a Riemannian manifold and study the closedness of the 2-form associated to the obtained (almost) para-Hermitian structure.
Abstract: We characterize the almost product and locally product structures of general natural lift type on the cotangent bundle of a Riemannian manifold. We find the conditions under which the cotangent bundle endowed with the constructed almost product (locally product) structure and with a pseudo-Riemannian metric obtained as a general natural lift of the metric from the base manifold, is a Riemannian almost product (locally product) or an (almost) para-Hermitian manifold. Finally, by studying the closedness of the 2-form associated to the obtained (almost) para-Hermitian structure, we characterize the general natural (almost) para-Kahlerian structures on the cotangent bundle.
16 citations
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TL;DR: In this paper, the existence of continuous Hermitian metrics with semi-positive curvatures on the so-called Zariski's example of a line bundle defined over the blow-up of $\mathbb{P}^2$ at some twelve points.
Abstract: Our interest is a regularity of a minimal singular metric of a line bundle. One main conclusion of our general result in this paper is the existence of continuous Hermitian metrics with semi-positive curvatures on the so-called Zariski's example of a line bundle defined over the blow-up of $\mathbb{P}^2$ at some twelve points. This is an example of a line bundle which is nef, big, not semi-ample, and whose section ring is not finitely generated. We generalize this result to the higher dimensional case when the stable base locus of a line bundle is a smooth hypersurface with a holomorphic tubular neighborhood.
15 citations
TL;DR: In this article, a new proof of the commutativity of the Dunkl operators among themselves is presented, as a consequence of a geometric property, namely that the connection has curvature zero.
Abstract: A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean spaceE, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a program to generalize harmonic analysis in Euclidean spaces. This gives us a new, geometric way of viewing the Dunkl operators. In particular, we present a new proof of the commutativity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero.
15 citations
TL;DR: In this article, the authors studied the equivalence problem of time scale preserving equivalence of ordinary differential equations and of Veronese webs and derived basic invariants of such objects.
Abstract: We study dynamic pairs $(X,V)$ where $X$ is a vector field on a smooth manifold $M$ and $V\subset TM$ is a vector distribution, both satisfying certain regularity conditions. We construct basic invariants of such objects and solve the equivalence problem. In particular, we assign to $(X,V)$ a canonical connection and a canonical frame on a certain frame bundle. We compute the curvature and torsion. The results are applied to the problem of time scale preserving equivalence of ordinary differential equations and of Veronese webs. The framework of dynamic pairs $(X,V)$ is shown to include sprays, control-affine systems, mechanical control systems, Veronese webs and other structures.
30 Oct 2013
TL;DR: In this paper, the authors considered a deformation of Sasaki metric on the tangent bundle of a Riemannian manifold and derived conditions for the deformation to be recurrent or pseudo symmetric.
Abstract: In the present paper, we consider a deformation (in the horizontal bundle) of Sasaki metric on the tangent bundle TM over an n dimensional Riemannian manifold (M;g): We rstly study some properties of deformed Sasaki metric which is pure with respect to some paracomplex structures on TM: Finally conditions for deformed Sasaki metric to be recurrent or pseudo symmetric are given.
TL;DR: SCISM is developed, an algorithm based on relative frame bundle adjustment, which splits the recovered map of 3D landmarks and keyframes poses so that the camera can continue to grow and explore a local map in real time while, at the same time, a bulk map is optimized in the background.
Abstract: The recovery of structure from motion in real time over extended areas demands methods that mitigate the effects of computational complexity and arithmetical inconsistency. In this paper, we develop SCISM, an algorithm based on relative frame bundle adjustment, which splits the recovered map of 3D landmarks and keyframes poses so that the camera can continue to grow and explore a local map in real time while, at the same time, a bulk map is optimized in the background. By temporarily excluding certain measurements, it ensures that both maps are consistent, and by using the relative frame representation, new results from the bulk process can update the local process without disturbance. The paper first shows how to apply this representation to the parallel tracking and mapping (PTAM) method, a real-time bundle adjuster, and compares results obtained using global and relative frames. It then explains the relative representation's use in SCISM and describes an implementation using PTAM. The paper provides evidence of the algorithm's real-time operation in outdoor scenes, and includes comparison with a more conventional submapping approach.
TL;DR: In this paper, the twisted versions of Kirchhoff's network theorem and matrix-tree theorem on connected finite graphs are proved.Twisting here refers to chains with coefficients in a flat unitary line bundle.
Abstract: We prove ‘twisted’ versions of Kirchhoff’s network theorem and Kirchhoff’s matrix-tree theorem on connected finite graphs. Twisting here refers to chains with coefficients in a flat unitary line bundle.
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TL;DR: In this paper, the authors established a version of the Miyaoka generic semi-positivity theorem in the context of log-canonical orbifold pairs, and showed that the canonical bundle associated to a lc pair is big as soon as there exists a generically injective morphism from an ample line bundle to some symmetric power of the cotangent bundle.
Abstract: In this article we establish a version of Y. Miyaoka generic semi-positivity theorem in the context of log-canonical orbifold pairs. As an application, we show that the canonical bundle associated to a lc pair is big as soon as there exists a generically injective morphism from an ample line bundle to some symmetric power of the cotangent bundle associated to the orbifold pair.
TL;DR: In this paper, it was shown that at the presence of a (possibly nonlinear) connection on ( p, E. M ), T E on M admits a v.b. structure.
TL;DR: In this paper, the authors give a new criterion ensuring that a surface of general type with canonical singularities has a minimal resolution with big cotangent bundle, and provide many examples of surfaces with negative second Segre number and big cotsent bundle.
Abstract: Surfaces of general type with positive second Segre number are known to have big cotangent bundle. We give a new criterion ensuring that a surface of general type with canonical singularities has a minimal resolution with big cotangent bundle. This provides many examples of surfaces with negative second Segre number and big cotangent bundle.
TL;DR: In this paper, it was shown that the extendability of singular positive metrics on a line bundle along a sub-variety implies the ampleness of the line bundle, which implies that the amplifyability of the bundle can be improved.
Abstract: Coman, Guedj and Zeriahi proved that, for an ample line bundle L on a projective manifold X, any singular positive metric on the line bundle L|V along a subvariety \({V \subset X}\) can be extended to a global singular positive metric on L. In this paper, we prove that the extendability of singular positive metrics on a line bundle along a subvariety implies the ampleness of the line bundle.
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TL;DR: In this article, it was shown that if the holomorphic vector field defining the extremal Kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal KG metrics on the projectivization of the dual vector bundle.
Abstract: In this paper, we consider a compact Kahler manifold with extremal Kahler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kahler metrics on the projectivization of the dual vector bundle.
TL;DR: In this article, the authors introduced Cheeger-Gromoll type metric on the cotangent bundle of Riemannian manifold and investigated curvature properties and geodesics on the COTANGENT bundle with respect to the Cheeger Gromoll metric.
Abstract: The purpose of this paper is to introduce Cheeger-Gromoll type metricon the cotangent bundle of Riemannian manifold and investigate curvature properties and geodesics on the cotangent bundle with respectto the Cheeger-Gromoll metric.
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TL;DR: In this article, the authors give a simple proof of a triviality criterion due to I.Biswas and J.Pedro and P.Dos Santos, and show that a vector bundle on a homogenous space is trivial if and only if the restrictions of the vector bundle to Schubert lines are trivial.
Abstract: In this paper, we give a simple proof of a triviality criterion due to I.Biswas and J.Pedro and P.Dos Santos. We also prove a vector bundle on a homogenous space is trivial if and only if the restrictions of the vector bundle to Schubert lines are trivial. Using this result and Chern classes of vector bundles, we give a general criterion of a uniform vector bundle on a homogenous space to be splitting. As an application, we prove a uniform vector bundle on classical Grassmannians and quadrics of low rank is splitting.
TL;DR: In this article, it was shown that every real analytic orbifold has a real analytic Riemannian metric and that every reduced real analytic Orbifold can be expressed as a quotient of the real analytic manifold by a real analytical almost free action of a compact Lie group.
Abstract: We begin by showing that every real analytic orbifold has a real analytic Riemannian metric. It follows that every reduced real analytic orbifold can be expressed as a quotient of a real analytic manifold by a real analytic almost free action of a compact Lie group. We then extend a well-known result of Nomizu and Ozeki concerning Riemannian metrics on manifolds to the orbifold setting: Let X be a smooth (real analytic) orbifold and let be a smooth (real analytic) Riemannian metric on X . Then X has a complete smooth (real analytic) Riemannian metric conformal to .
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TL;DR: In this article, the authors show that a smooth projective variety with a torsion canonical bundle over complex numbers is unobstructed if one of the following conditions is satisfied: (1) the line bundle L is ample, (2) the fundamental group of X is finite.
Abstract: Let X be a smooth projective variety with a torsion canonical bundle over complex numbers and L be a line bundle on X. We prove the pair (X,L) is unobstructed if one of the following conditions is satisfied, (1)the line bundle L is ample, (2)the fundamental group $\pi_1(X)$ of X is finite.
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TL;DR: In this article, the authors studied the space of oriented genus g subsurfaces of a fixed manifold M and its homological properties, and constructed a "scanning map" which compared this space to the spaces of sections of a certain fibre bundle over M associated to its tangent bundle.
Abstract: We study the space of oriented genus g subsurfaces of a fixed manifold M, and in particular its homological properties. We construct a "scanning map" which compares this space to the space of sections of a certain fibre bundle over M associated to its tangent bundle, and show that this map induces an isomorphism on homology in a range of degrees.
Our results are analogous to McDuff's theorem on configuration spaces, extended from 0-manifolds to 2-manifolds.
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TL;DR: In this paper, the equivalence of the existence of a Killing vector field on a Riemannian manifold with a non-degenerated g-natural metric was proved.
Abstract: The tangent bundle of a Riemannian manifold (M,g) with non-degenerated g-natural metric G that admits a Killing vector field is investigated. Using Taylor's formula (TM,G) is decomposed into four classes that are investigated separately. The equivalence of the existence of Killing vector field on M and TM is proved.
Key words: Riemannian manifold, tangent bundle, g-natural metric, Killing vector field, non-degenerate metric.
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TL;DR: The geometry of a submanifold of a Riemannian manifold was studied in this paper, where the authors showed that minimality is equivalent to harmonicity of the Gauss map.
Abstract: Let $M$ be a submanifold of a Riemannian manifold $(N,g)$. $M$ induces a subbundle $O(M,N)$ of adapted frames over $M$ of the bundle of orthonormal frames $O(N)$. Riemannian metric $g$ induces natural metric on $O(N)$. We study the geometry of a submanifold $O(M,N)$ in $O(N)$. We characterize the horizontal distribution of $O(M,N)$ and state its correspondence with the horizontal lift in $O(N)$ induced by the Levi--Civita connection on $N$.
In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold $M$ with deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures.
TL;DR: In this article, the authors studied natural lifting operations from a bundle τ : E → R to the bundle π : J 1 τ ∗ → E which is the dual of the first-jet bundle J 1τ.
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TL;DR: In this article, a generalisation of Sylvester's law of inertia to real non-degenerate quadratic forms on a fixed real vector bundle over a connected locally connected paracompact Hausdorff space is presented.
Abstract: This paper presents a generalisation of Sylvester's law of inertia to real non-degenerate quadratic forms on a fixed real vector bundle over a connected locally connected paracompact Hausdorff space. By interpreting the classical inertia as a complete discrete invariant for the natural action of the general linear group on quadratic forms, the simplest generalisation consists in substituting such group with the group of gauge transformations of the bundle. Contrary to the classical law of inertia, here the full action and its restriction to the identity path component typically have different orbits, leading to two invariants: a complete invariant for the full action is given by the isomorphism class of the orthonormal frame bundle associated to a quadratic form, while a complete invariant for the restricted action is the homotopy class of any maximal positive-definite subbundle associated to a quadratic form. The latter invariant is finer than the former, which in turn is finer than inertia. Moreover, the orbit structure thus obtained might be used to shed light on the topology of the space of non-degenerate quadratic forms on a vector bundle.
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TL;DR: In this article, the problem of description of matter fields with an exact symmetry group with respect to a closed subgroup was addressed. But the problem was not addressed in this paper, since it is essential that matter fields admit an action of a gauge group.
Abstract: Higgs fields are attributes of classical gauge theory on a principal bundle $P\to X$ whose structure Lie group $G$ if is reducible to a closed subgroup $H$. They are represented by sections of the quotient bundle $P/H\to X$. A problem lies in description of matter fields with an exact symmetry group $H$. They are represented by sections of a composite bundle which is associated to an $H$-principal bundle $P\to P/H$. It is essential that they admit an action of a gauge group $G$.
TL;DR: In this article, the authors investigated submanifolds in a tangent bundle endowed with g-natural metric G, defined by a vector field on a base manifold, and gave a sufficient condition for a vector fields on M to define a totally geodesic sub-manifold in (TM,G).
Abstract: In the paper we investigate submanifolds in a tangent bundle endowed with g-natural metric G, defined by a vector field on a base manifold. We give a sufficient condition for a vector field on M to defined totally geodesic submanifold in (TM,G). The parallel vector field is discussed in more detail.