We define holomorphic connection on a parabolic vector bundle over a Riemann surface and prove that a parabolic vector bundle admits a holomorphic connection if and only if each direct summand of it is of parabolic degree zero. This is a generalization to the parabolic context of a well-known result of Weil which says that a holomorphic vector bundle on a Riemann surface admits a holomorphic co...

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 connection, a tensor, and a metric on the base space which together satisfy a ...

Any singularity free vector field X defined on an open set in a three-dimensional Euclidean space with curl X = 0 admits a complex line bundle Fa with a fibre-wise defined symplectic structure, a principal bundle Pa and a Heisenberg group bundle Ga . For the non-vanishing constant vector field X the geometry of Pa defines for each frequency a Schrödinger representation of any fibre of the Heise...

Let X be a smooth projective curve of genus g ≥ 2 defined over an arbitrary algebraically closed field K. For any vector bundle E on X, call μ(E) := deg(E)/rank(E) the slope of E. A vector bundle E on X is said to be spanned if the natural map H(X,E)⊗OX → E is surjective. Such bundles are important tools for the projective geometry ofX because they are exactly the vector bundles associated to a...

where Dvai is the directional derivative of the function ai in the direction v. Consequently we know when a vector field does not change along a curve γ: Dγ̇X = 0. Covariant derivatives generalize the directional derivatives allowing us to differentiate vector fields on arbitrary manifolds and, more generally, sections of arbitrary vector bundles. Definition 1.1 (Covariant derivative of sections...

This paper has three objectives. First to recall the link between the classical Legendre-Fenschel transformation and a useful isomorphism between 1-jets of functions on a vector bundle and on its dual. As a particular consequence we obtain the classical isomorphism between the cotangent bundle of the tangent bundle T ∗TM and the tangent bundle of the cotangent bundle TT ∗M of any manifold M. Se...

generators. Our goal is to make this description more concrete. Note that G(k, n) = U(n)/(U(k)×U(n− k)) = St(k, n)/U(k) is the quotient by U(k) of the complex Stiefel manifold St(k, n) of partial orthonormal k-frames in C. Thus any finite-dimensional representation ρ : U(k) −→ GL(W ) of U(k) gives rise to the natural vector bundle St(k, n)×ρ W −→ G(k, n), where St(k, n) −→ G(k, n) is viewed as ...

For any complex vector bundle Ek of rank k over a manifold Mm with Chern classes ci ∈ H2i(Mm, Z) and any non-negative integers l1, · · · , lk we show the existence of a positive number p(m, k) and the existence of a complex vector bundle Êk over Mm whose Chern classes are p(m, k) · li · ci ∈ H2i(Mm, Z). We also discuss a version of this statement for holomorphic vector bundles over projective a...

The existence problem for vector bundles on a smooth compact complex surface consists in determining which topological complex vector bundles admit holomorphic structures. For projective surfaces, Schwarzenberger proved that a topological complex vector bundle admits a holomorphic (algebraic) structure if and only if its first Chern class belongs to the Neron-Severi group of the surface. In con...

The existence problem for vector bundles on a smooth compact complex surface consists in determining which topological complex vector bundles admit holomorphic structures. For projective surfaces, Schwarzenberger proved that a topological complex vector bundle admits a holomorphic (algebraic) structure if and only if its first Chern class belongs to the Neron-Severi group of the surface. In con...