Infinitesimal Isometries Along Curves and Generalized Jacobi Equations

On a Riemannian manifold, a solution of the Killing equation is an infinitesimal isometry. Since the Killing equation is overdetermined, infinitesimal isometries do not exist in general. A completely determined prolongation of the Killing equation is a PDE on the bundle of 1-jets of vector fields. Restricted to a curve, this becomes an ODE that generalizes the Jacobi equation. A solution of this ODE is called an infinitesimal isometry along the curve, which we show to be an infinitesimal rigid variation of the curve. We define Killing transport to be the associated linear isometry between fibers of the bundle along the curve, and show that it is parallel translation for a connection on the bundle related to the Riemannian connection. Restricting to dimension two, we study the holonomy of this connection, prove the Gauss–Bonnet theorem by means of Killing transport, and determine the criteria for local existence of infinitesimal isometries.     

How the Curvature Generates the Holonomy of a Connection in an Arbitrary Fibre Bundle

For an arbitrary fibre bundle with a connection, the holonomy group of which is a Lie transformation group, it is shown how the parallel displacement along a null-homotopic loop can be obtained from the curvature by integration. The result also sheds some new light on the situation for vector bundles and principal fibre bundles. The Theorem of Ambrose–Singer is derived as a corollary in our general setting. The curvature of the connection is interpreted as a differential 2-form with values in the holonomy algebra bundle, the elements of which are special vector fields on the fibres of the given bundle. Received: May 16, 2006; Revised: July 30, 2006; Accepted: August 2, 2006     

The conformal Killing equation on forms—prolongations and applications

We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on k-forms to a twisting of the conformal Killing equation on (k−?)-forms for various integers ?. These tools are used to develop a helicity raising and lowering construction in the general setting and on conformally Einstein manifolds.     

Parallel connections over symmetric spaces

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 Riemannian (or unitary) vector bundles with parallel curvature then reduces to finding representations of the structure group of the canonical principal bundle.     

Symplectic spinors,holonomy and Maslov index

In this note it is shown that the Maslov index for pairs of Lagrangian paths as introduced by Leray and later canonized by Cappell, Lee and Miller appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinor bundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold \(L\), along closed or non-closed paths therein. In especially, the CLM-Index mod \(4\) determines the holonomy group of this line bundle w.r.t. the Levi-Civita-connection on \(L\), hence its vanishing mod 4 is equivalent to the existence of a trivializing parallel section. Moreover, it is shown that the CLM-Index determines parallel transport in that line-bundle along arbitrary paths when compared to the parallel transport w.r.t. to the canonical flat connection of Euclidean space, if the Lagrangian tangent planes at the endpoints either coincide or are orthogonal. This is derived from a result on parallel transport of certain elements of the dual spinor bundle along closed or endpoint-transversal paths.     

Lightlike Hypersurfaces in Indefinite Trans-Sasakian Manifolds

This paper deals with lightlike hypersurfaces of indefinite trans-Sasakian manifolds of type (α, β), tangent to the structure vector field. Characterization Theorems on parallel vector fields, integrable distributions, minimal distributions, Ricci-semi symmetric, geodesibility of lightlike hypersurfaces are obtained. The geometric configuration of lightlike hypersurfaces is established. We prove, under some conditions, that there are no parallel and totally contact umbilical lightlike hypersurfaces of trans-Sasakian space forms, tangent to the structure vector field. We show that there exists a totally umbilical distribution in an Einstein parallel lightlike hypersurface which does not contain the structure vector field. We characterize the normal bundle along any totally contact umbilical leaf of an integrable screen distribution. We finally prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle and its image under ${\overline{\phi}}$ .     

A probabilistic approach to the Yang–Mills heat equation

We construct a parallel transport U in a vector bundle E, along the paths of a Brownian motion in the underlying manifold, with respect to a time dependent covariant derivative ∇ on E, and consider the covariant derivative ∇₀U of the parallel transport with respect to perturbations of the Brownian motion. We show that the vertical part U⁻¹∇₀U of this covariant derivative has quadratic variation twice the Yang–Mills energy density (i.e., the square norm of the curvature 2-form) integrated along the Brownian motion, and that the drift of such processes vanishes if and only if ∇ solves the Yang–Mills heat equation. A monotonicity property for the quadratic variation of U⁻¹∇₀U is given, both in terms of change of time and in terms of scaling of U⁻¹∇₀U. This allows us to find a priori energy bounds for solutions to the Yang–Mills heat equation, as well as criteria for non-explosion given in terms of this quadratic variation.     

Connections and restrictions to curves

We construct a vector bundle E on a smooth complex projective surface X with the property that the restriction of E to any smooth closed curve in X admits an algebraic connection while E does not admit any algebraic connection.     

The determinant bundle on the moduli space of stable triples over a curve

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,?), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, and?: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.     

A vanishing result for the Spin c Dirac operator defined along the leaves of a foliation

In the setting of a closed Riemannian manifold endowed with a smooth, non-necessarily integrable distribution, we extend a Lichnerowicz type formula which is known to work in the particular case of a transverse bundle associated to a Riemannian foliation. Interesting settings in which non-integrable distributions appear naturally are emphasized. As an application, we consider the distribution as being even dimensional and integrable; we consider also a hermitian line bundle, with a hermitian connection, such that the induced curvature tensor is non-degenerate, and an arbitrary hermitian bundle endowed also with a hermitian connection. Taking the k power of the line bundle and canonically constructing a Spin ^c Dirac operator defined along the leaves of the foliation generated by the distribution, we prove a vanishing result for the half kernel of this operator.     

Vector bundles on p-adic curves and parallel transport

We define functorial isomorphisms of parallel transport along étale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.     

Stability of rank 2 vector bundles along isomonodromic deformations

We are interested in the stability of holomorphic rank 2 vector bundles of degree 0 over compact Riemann surfaces, which are provided with irreducible meromophic tracefree connections. In the case of a logarithmic connection on the Riemann sphere, such a vector bundle will be trivial up to the isomonodromic deformation associated to a small move of the poles, according to a result of A. Bolibruch. In the general case of meromorphic connections over Riemann surfaces of arbitrary genus, we prove that the vector bundle will be semi-stable, up to a small isomonodromic deformation. More precisely, the vector bundle underlying the universal isomonodromic deformation is generically semi-stable along the deformation, and even maximally stable. For curves of genus g ≥ 2, this result is non-trivial even in the case of non-singular connections. The author was partially supported by ANR SYMPLEXE BLAN06-3-137237.     

Coupled connections on a compact Riemann surface

We consider holomorphic differential operators on a compact Riemann surface X whose symbol is an isomorphism. Such a differential operator of order n on a vector bundle E sends E to K^⊗n_X⊗E, where K_X is the holomorphic cotangent bundle. We classify all those holomorphic vector bundles E over X that admit such a differential operator. The space of all differential operators whose symbol is an isomorphism is in bijective correspondence with the collection of pairs consisting of a flat vector bundle E over X and a holomorphic subbundle of E satisfying a transversality condition with respect to the connection.     

Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree

For a projective manifold whose tangent bundle is of nonnegative degree, a vector bundle on it with a holomorphic connection actually admits a compatible flat holomorphic connection, if the manifold satisfies certain conditions. The conditions in question are on the Harder-Narasimhan filtration of the tangent bundle, and on the Neron-Severi group.

     

A remark on “Connections and Higgs fields on a principal bundle”

The authors show that a unipotent vector bundle on a non–Kähler compact complex manifold does not admit a flat holomorphic connection in general. It was also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.     

A Construction of Non-Split Supermanifolds

A construction is presented associating with any closed (1,1)-form on a complex manifold M a supermanifold, whose retract is the split supermanifold (M,), where is the sheaf of holomorphic forms on M. This supermanifold is non-split whenever is not -exact. In particular, any complex line bundle L over M determines the supermanifold associated with the curvature form of a metric on L; it is non-split whenever the refined Chern class of L is non-zero. The case of a flag manifold M is studied in more details.     

On the relative connections

Abstract

We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and Kähler.     

A modification of the Hodge star operator on manifolds with boundary

For smooth compact oriented Riemannian manifolds M of dimension 4k+2, k≥0, with or without boundary, and a vector bundle F on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of M twisted by F. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when k=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.     

Geometric interpretation of the Wagner curvature tensor in the case of a manifold with contact metric structure

Considering a manifold (φ, ξ, η, g, X, D) with contact metric structure, we introduce the concept of N-extended connection (connection on a vector bundle (D, π,X)), with N an endomorphism of the distribution D, and show that the curvature tensor of each N-extended connection for a suitably chosen endomorphism N coincides with the Wagner curvature tensor.