arXiv research

Defines connections on parabolic vector bundles for Lie algebroids.

Study of generalized vector bundles and their geometric tools.

New dHYM connections found on complex vector bundles.

The study examines connections and their curvatures on different types of bundles.

A linear connection is associated to a nonlinear connection on a vector bundle by a linearization procedure. Our definition is intrinsic in terms of vector fields on the bundle. For a connection on an affine bundle our procedure can be applied after homogenization and restriction. Several applications in Classical Mech…

Criterion for Lie algebroid connections on compact Riemann surfaces.

The paper extends Laplacian spectra approximations to vector bundles.

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding…

A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is `parametrised' in some specific way by a vector bundle map from a presc…

Extends optimal regularity and compactness to vector bundles over non-Riemannian manifolds.

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalisation of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids we define the notion of a weighted $A$-connection on a graded bundle. In a natural sense weighted $A$-connections are adapte…

Study on hermitian Yang-Mills connections on blown-up manifolds.

In a fibre bundle, natural derivatives of a section are defined as tangent vector fields on the image of a section of the fibre bundle. A local extension to vector fields in the tangent bundle leads to a direct proof of the formula expressing the curvature of a connection in terms of covariant derivatives. The result i…

Holomorphic Lie algebroid connections on Riemann surfaces are characterized.

Uniqueness proof for Calderón's problem on real-analytic vector bundles.

Symmetries of bundle gerbes modeled using multiplicative vector fields.

Computes the decomposition of rank-three bundles over the projective line with three marked points.

Criterion found for Lie algebroid connections on parabolic bundles.

Develops theory of d-holomorphic connections on Klein surfaces.

New algebraic structure for vector bundles with special properties.

We show that a unipotent vector bundle on a non-Kaehler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.

We consider one possible definition of a diffeological connection on a diffeological vector pseudo-bundle. It is different from the one proposed in [7] and is in fact simpler, since it is obtained by a straightforward adaption of the standard definition of a connection as an operator on the space of all smooth sections…

The paper defines a Chern-Simons invariant for stably trivial vector bundles and uses it to obstruct conformal immersions.

We introduce $Z$-critical connections for holomorphic vector bundles and prove their existence under stability conditions.

Torsors over moduli spaces of vector bundles with fixed determinant.

Develops combinatorial theory of vector bundles on simplicial complexes.

Equivalence of second order differential operators in vector bundles studied.

Solves Dirac equation coupled to vector bundles.

A diffeological connection on a diffeological vector pseudo-bundle is defined just the usual one on a smooth vector bundle; this is possible to do, because there is a standard diffeological counterpart of the cotangent bundle. On the other hand, there is not yet a standard theory of tangent bundles, although there are …

We characterize all LVMB manifolds X such that the holomorphic tangent bundle TX is spanned at the generic point by a family of global holomorphic vector fields, each of them having non-empty zero locus. We deduce that holomorphic connections on semi-stable holomorphic vector bundles over LVMB manifolds with this previ…

Finite vector bundles over complex manifolds are trivializable via finite covers.

Study perturbations of submodules in Drury-Arveson space, finding smooth vector bundles with Hermitian connections.

In this article we consider the continuity of the eigenvalues of the connection Laplacian of $G$-connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically $G$-equivariant measured Gromov-Hausdorff topology on the space of metric measure spaces with isometric $G$-…

In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new approach to connections on a bundle as a consistent way to lift the dynamics of…

This paper addresses Cheeger and Gromoll's question of which vector bundles admit a complete metric of nonnegative curvature, and relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle which admits a metric of nonnegative curvature must admit a c…

New findings on biquotient bundles lacking inverses in various dimensions.

We build metrized quantum vector bundles, over a generically transcendental quantum torus, from Riemannian metrics, using Rosenberg's Levi-Civita connections for these metrics. We also prove that two metrized quantum vector bundles, corresponding to positive scalar multiples of a Riemannian metric, have distance zero b…

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 f…

A vector bundle with connection over a supermanifold leads naturally to a notion of parallel transport along superpaths. In this note we show that {\it every} such parallel transport along superpaths comes form a vector bundle with connection, at least when the base supermanifold is a manifold.

We make evident a curvature tensor for every vector sub-bundle of an arbitrary manifold tangent bundle which reduces to the curvature tensor of an Ehresmann connection in the case of the horizontal sub-bundle of the tangent bundle to the total space of the nonlinear fiber bundle on which the connection is defined. Then…

Study on topological rigidity of ALE vector bundles with specific conditions.

Narasihman and Ramanan proved that an arbitrary connection in a vector bundle over a base space B can be obtained as the pull-back (via a correctly chosen classifying map from B into the appropriate Grassmannian) of the universal connection in the universal bundle over the Grassmannian. The purpose of this paper is to …

We give a proof of the existence of radial (smooth) parallel sections of vector bundles endowed with a linear connection.

The paper contains a review on the general connection theory on differentiable fibre bundles. Particular attention is paid to (linear) connections on vector bundles. The (local) representations of connections in frames adapted to holonomic and arbitrary frames is considered.

Establishes a connection between Kähler metrics and vector bundle sections.

Holomorphic connections on Calabi-Yau manifolds are flat.

Paper constructs connections on curves with specific Galois groups.

This paper studies generic properties of connections on vector bundles, solving cohomological equations and proving opaque connections.