Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, lie groups, and grassmanians are all presented here. Definition 568 the curvature form of the connection. Curvature of vector bundles associated to holomorphic brations. Curvature of higher direct image sheaves geiger, thomas and schumacher, georg, 2017. Stable vector bundles and curvature in the survey in geometry. M but also a module over the ring of all complexvalued functions c. Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology.
Curvatures of direct image sheaves of vector bundles and applications liu, kefeng and yang, xiaokui, journal of differential geometry. Conditions for nonnegative curvature on vector bundles and sphere bundles tapp, kristopher, duke mathematical journal, 2003. A c1complex vector bundle consists of a family fe xg x2m of complex vector spaces parametrized by m, together with a c1manifold structure of e x2me x such that 1. Vector bundles and connections the exposition of vector. Differential geometric invariants for timereversal symmetric bloch. In particular, one can compute using the standard representations of chern classes in relationship to curvature that if e is an hermtian. Curvature of vector bundles associated to holomorphic. The curvature of a tensor product of vector bundles. This is then clearly an even locally trivial holomorphic bration and there is a naturally. M, in other words, it assigns smoothly to each tangent space tpm a kdimensional subspace. Covariant derivatives and curvature on general vector bundles. The only two vector bundles with base space a circle and onedimensional. A real vector bundle of rank r over a manifold m consists of. Chern classes a connection and curvature in a complex.
We will consider rst the simplest theory of vector bundles where the vector space is a onedimensional complex vector space line bundles. Kefeng liu, xiaokui yang submitted on 11 mar 20 v1, last revised 12 oct 20 this version, v2. Lectures on fibre bundles and differential geometry. Bundles, connections, metrics, and curvature are the lingua franca of modern differential geometry and theoretical physics. Hermitian vector bundles and dirac operators we rst describe our objects of interest. In order to construct good moduli spaces for vector bundles over algebraic curves, mumford introduced the concept of a stable vector bundle. The notion of equivariant curvature can be formulated also at level of vector bundles. Curvatures of direct image sheaves of vector bundles and applications i authors. In this section we explain how hermitian holomorphic line bundles carry a natural connection and hence one can talk of the curvature of such a bundle. But there are many other interesting vector bundles. Recall that a k ahler structure on mconsists of a hermitian metric gon tm.
The focus is thus quite di erent from that of several popular treatments of di erential geometry for undergraduates, e. My goal here is to make a treatment which is readily understandable by anyone with a year of graduate. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. This approach can be generalized to higher order skewsymmetric curvature operator. In this paper, we study the curvature estimate of the hermitianyangmills flow on holomorphic vector bundles. Curvature can, in fact, be understood as a measure of the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Symmetries and nonnegative curvature of vector bundles. Standard references for vector bundles and connections are ghv and kn, where. Introduction this1 is a pdf with a treatment of principal bundles. A principal bundles, vector bundles and connections. Curvature of vector bundles 535 another example of the situation in theorem 1.
The main purpose of this paper is to study the surfaces in em with the mean curvature vector parallel in the normal bundle and to prove two classification theorems. In the figure we assume an infinitesimal curve c with tangent v, an orthonormal frame, a spin connection, and a unitary gauge. Wellknown examples of vector bundles are the tangent and cotangent bundles, tmand t m, as well as the tensor bundle tm x2m p. It is a real vector bundle of rank n dimm which in general is not trivial. Vector bundles with semidefinite curvature and cohomology. Moreover, in section 3, we obtain further vanishing theorems of cohomology, under assumptions on the curvature.
Extensions of vector bundles on the farguesfontaine curve 5 claim 1. Vector bundles over grassmannians and the spectral. Curvatures of direct image sheaves of vector bundles and. Let e m be a vector bundle, either real or complex.
The notion of a riemannian metric generalizes to arbitrary vector bundles. If e b is a vector bundle, then one can also think of. Find materials for this course in the pages linked along the left. Let mbe a compact, complex manifold of complex dimension m. The most important example is the tangent bundle, a real vector bundle. Ramanan no part of this book may be reproduced in any form by print, micro. E b, onto b, is called a vector bundle of rank k over b if the following holds. This concept has been generalized to vector bundles and, more generally, coherent sheaves. On the differential geometry of homogeneous vector bundles authors. Chern classes a connection and curvature in a complex vector bundle. In this chapter, we extend some of the onedimensional notions of chern and ricci forms to vector bundles. Vector bundles william schulz department of mathematics and statistics northern arizona university, flagsta. Introduction hermitian vector bundles and dirac operators.
Roughly speaking, we prove this by stratifying home. Let be a curve in rn and let rdt be the derivative induced along by the connection. Our proof is independent from the injectivity of the baumconnes assembly map for the fundamental group of m and relies on the construction of a certain infinite dimensional flat vector bundle. Note that e is a sub bundle of the trivial bundle 1m over gr2m. A fiber metric on the vector bundle e m assigns smoothly. F 2 which factor through a proper sub bundle of f 2 is strictly smaller than the dimension of home. In one simple case, we show that the curvature of the evolved hermitian metric is uniformly bounded away from the analytic subvariety determined by the hardernarasimhanseshadri filtration of the holomorphic vector bundle. Yang mills hermitian vector bundles, that is holomorphic vector bundles that admit an yang mills hermitianhermitian yang mills metric have special properties not obviously related to stability. In this case x is also a section of the vector bundle. E is not only a module over the ring of all realvaluedfunctions c. Curvature and torsion of linear transports along paths in, respectively, vector bundles and the tangent bundle to a differentiable manifold are defined and certain their properties are derived.
A note on curvature estimate of the hermitianyangmills. Our proof is independent from the injectivity of the baumconnes assembly map for the fundamental group of m and relies on the construction of a certain infinite dimensional flat vector bundle out. The topology of fiber bundles lecture notes ralph l. Pdf linear transports along paths in vector bundles. On parallel transport and curvature graduate project. On the differential geometry of homogeneous vector bundles. A bundle over a manifold is trivial if it is simply the cartesian product of. However, since his not commutative, one has to be careful when writing down formulas. Vector bundles, connections and curvature 3 there is an obvious projection t. Pdf surfaces with mean curvature vector parallel in the.
Hermitian einstein metrics on vector bundles and stability. Hot network questions was it illegal for nancy pelosi to. Using this technique, we will generalize to vector bundles some of the results of the first two sections of this paper. The case of quaternionic vector bundles can be handled in the same way. Dependence of riemann curvature tensor on the values of the vector fields in a neighborhood. Moreover, given such a x, we seek its curvature form, and, in the case where p is the principal tangent bundle, we ask for the complex. Vector bundles, connections, and curvature tony perkins. Pdf document information annals of mathematics fine hall washington road princeton university princeton, nj 08544, usa phone. Covariant derivatives and curvature on general vector bundles 3 the connection coe. Suppose e m is an ndimensional complex vector bundle. For any hermitian manifold, the curvature tensor is an endtm cvalued two form. Lecture notes geometry of manifolds mathematics mit. Curvature of vector bundles and subharmonicity of bergman.
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