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

1998 ◽  
Vol 126 (10) ◽  
pp. 2827-2834 ◽  
Author(s):  
Indranil Biswas
Keyword(s):  
Tangent Bundle ◽  
Vector Bundles ◽  
Projective Manifold

scholarly journals ON PROJECTIVE MANIFOLDS WITH PSEUDO-EFFECTIVE TANGENT BUNDLE

2021 ◽  
pp. 1-30
Author(s):  
Genki Hosono ◽  
Masataka Iwai ◽  
Shin-ichi Matsumura
Keyword(s):  
Minimal Surfaces ◽  
Tangent Bundle ◽  
Vector Bundles ◽  
Structure Theorem ◽  
Projective Manifold ◽  
Hermitian Metrics ◽  
Rationally Connected ◽  
Projective Manifolds ◽  
Positively Curved ◽  
Tangent Bundles

Abstract In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration $X \to Y$ to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

Parallel Translation in Vector Bundles with Abelian Structure Group and the Gauss-Bonnet Formula

1973 ◽  
Vol 25 (4) ◽  
pp. 765-771
Author(s):  
Hansklaus Rummler
Keyword(s):  
Riemannian Manifold ◽  
Simple Proof ◽  
General Result ◽  
Tangent Bundle ◽  
Vector Bundles ◽  
Structure Group ◽  
Parallel Translation ◽  
Result Theorem

Most proofs for the classical Gauss-Bonnet formula use special coordinates, or other non-trivial preparations. Here, a simple proof is given, based on the fact that the structure group SO(2) of the tangent bundle of an oriented 2-dimensional Riemannian manifold is abelian. Since only this hypothesis is used, we prove a slightly more general result (Theorem 1).

scholarly journals LIE ALGEBROIDS AS SPACES

2018 ◽  
Vol 19 (2) ◽  
pp. 487-535 ◽  
Author(s):  
Ryan Grady ◽  
Owen Gwilliam
Keyword(s):  
Tangent Bundle ◽  
Symplectic Structure ◽  
Vector Bundles ◽  
Natural Generalization ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Faithful Functor ◽  
Derived Geometry

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

HORIZONTAL LAPLACIAN ON TANGENT BUNDLE OF FINSLER MANIFOLD WITH g-NATURAL METRIC

2012 ◽  
Vol 09 (07) ◽  
pp. 1250061 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
AKBAR TAYEBI ◽  
CHUNPING ZHONG
Keyword(s):  
Tangent Bundle ◽  
Vector Fields ◽  
Vector Bundles ◽  
Killing Vector ◽  
Finsler Manifold ◽  
Order Differential Operator ◽  
Finsler Manifolds ◽  
Scalar And Vector Fields ◽  
Matsumoto Metric ◽  
Weitzenböck Formula

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 Weitzenböck 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.

scholarly journals Negative Vector Bundles and Complex Finsler Structures

1975 ◽  
Vol 57 ◽  
pp. 153-166 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Complex Manifold ◽  
Tangent Bundle ◽  
Vector Bundles ◽  
Finsler Structure

A complex Finsler structure F on a complex manifold M is a function on the tangent bundle T(M) with the following properties. (We denote a point of T(M) symbolically by (z, ζ), where z represents the base coordinate and ζ the fibre coordinate.)

scholarly journals Dirac Structures on Banach Lie Algebroids

2014 ◽  
Vol 22 (3) ◽  
pp. 219-228
Author(s):  
Vlad-Augustin Vulcu
Keyword(s):  
Vector Bundle ◽  
Tangent Bundle ◽  
Differential Calculus ◽  
Vector Bundles ◽  
Dirac Structure ◽  
Lie Algebroid ◽  
Symmetric Form ◽  
Courant Bracket ◽  
Dirac Structures ◽  
Original Definition

Abstract In the original definition due to A. Weinstein and T. Courant a Dirac structure is a subbundle of the big tangent bundle T M ⊕ T* M that is equal to its ortho-complement with respect to the so-called neutral metric on the big tangent bundle. In this paper, instead of the big tangent bundle we consider the vector bundle E ⊕ E*, where E is a Banach Lie algebroid and E* its dual. Recall that E* is not in general a Lie algebroid. We define a bilinear and symmetric form on the vector bundle E ⊕ E* and say that a subbundle of it is a Dirac structure if it is equal with its orthocomplement. Our main result is that any Dirac structure that is closed with respect to a type of Courant bracket, endowed with a natural anchor is a Lie algebroid. In the proof the differential calculus on a Lie algebroid is essentially involved. We work in the category of Banach vector bundles.

scholarly journals SEMISTABILITY OF RESTRICTED TANGENT BUNDLES AND A QUESTION OF I. BISWAS

2013 ◽  
Vol 24 (01) ◽  
pp. 1250122 ◽  
Author(s):  
PRISKA JAHNKE ◽  
IVO RADLOFF
Keyword(s):  
Riemann Surface ◽  
Abelian Variety ◽  
Tangent Bundle ◽  
Compact Riemann Surface ◽  
Projective Manifold ◽  
Holomorphic Map ◽  
Complex Projective ◽  
Tangent Bundles

Let M be a complex projective manifold with the property that for any compact Riemann surface C and holomorphic map f : C → M the pullback of the tangent bundle of M is semistable. We prove that in this case M is a curve or a finite étale quotient of an abelian variety answering a conjecture of Biswas.

scholarly journals Connections on Bundles

2012 ◽  
Vol 60 (2) ◽  
pp. 191-194
Author(s):  
Md. Showkat Ali ◽  
Md. Mirazul Islam ◽  
Farzana Nasrin ◽  
Md. Abu Hanif Sarkar ◽  
Tanzia Zerin Khan
Keyword(s):  
Tangent Bundle ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Affine Connection ◽  
Basic Theory

This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle TM, is called an affine connection on an m-dimensional smooth manifold M. By the general discussion of affine connection on vector bundles that necessarily exists on M which is compatible with tensors.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11492 Dhaka Univ. J. Sci. 60(2): 191-194, 2012 (July)

scholarly journals Vector bundles over (8k + 5)-dimensional manifolds

1989 ◽  
Vol 111 (1-2) ◽  
pp. 183-197 ◽  
Author(s):  
Tze-Beng Ng
Keyword(s):  
Tangent Bundle ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Suppose that M is a closed, connected and smooth manifold of dimension n = 8k + 5, with k ≧1. Let η be an n-plane bundle over M. Under suitable conditions on M, we derive necessary and sufficient conditions for the span of η to be ≧j, j = 5 or 6. We then apply the results to the tangent bundle of M. In particular, we prove a conjecture of E. Thomas, namely, if M is 3-connected mod 2, then span M ≧ 5 if, and only if, χ2(M) = 0. We prove that if also w8k(M) = 0, then span M≧6. We also derive some immersion theorems for M.

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