Bundles with spherical Euler class
نویسندگان
چکیده
In this note, we show that the holonomy group of a Riemannian connection on a k-dimensional Euclidean vector bundle is transitive on the unit sphere bundle whenever the Euler class χ is spherical. We extract several consequences from this, among them that this is always the case as long as χ does not vanish, and the base of the bundle is simply connected and rationally (k + 1)/2-connected.
منابع مشابه
On Transgression in Associated Bundles
We formulate and prove a formula for transgressing characteristic forms in general associated bundles following a method of Chern [Che91]. As applications, we derive D. Johnson’s explicit formula in [Joh07] for such general transgression and Chern’s first transgression formula in [Che45] for the Euler class.
متن کاملMathai-quillen Formalism
Characteristic classes play an essential role in the study of global properties of vector bundles. Particularly important is the Euler class of real orientable vector bundles. A de Rham representative of the Euler class (for tangent bundles) first appeared in Chern’s generalisation of the Gauss-Bonnet theorem to higher dimensions. The representative is the Pfaffian of the curvature, whose cohom...
متن کاملOn the Second Homology Group of the Discrete Group of Diffeomorphisms of the Circle
Let G be the discrete group of orientation preserving diffeomorphisms of the circle, and let H be the subgroup of diffeomorphisms which are the identity in a neighborhood of a given point. It is proved that there is a short exact sequence in the integral homology of groups O-Hz(H)Hz(G) SZ-0. The epimorphism E can be identified with the Euler class for transversally foliated circle bundles. The ...
متن کاملVanishing of Universal Characteristic Classes for Handlebody Groups and Boundary Bundles
Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral cohomology when restricted to the handlebody subgroup. This is a special case of a more general theorem valid in all dimensions: universal characteristic classes ma...
متن کاملRemarks on the Geometry of Almost Complex 6-manifolds
This article is mostly a writeup of two talks, the first [3] given in the Besse Seminar at the École Polytechnique in 1998 and the second [4] given at the 2000 International Congress on Differential Geometry in memory of Alfred Gray in Bilbao, Spain. It begins with a discussion of basic geometry of almost complex 6-manifolds. In particular, I define a 2-parameter family of intrinsic first-order...
متن کامل