Bundles with spherical Euler class (original) (raw)
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Holonomy Groups of Normal Bundles, II
Journal of the London Mathematical Society, 1980
Let i: N -> M be an isometric immersion of a Riemannian manifold N into a Riemannian manifold M. We denote by D the induced connection on the normal bundle v of N in M. Then D is known to be a Riemannian connection on v (see, for instance, ).
Holonomy of homogeneous vector bundles and polar representations
Indiana University Mathematics Journal, 1995
Given a homogeneous vector bundle with a connection, we are concerned with the computation of its holonomy group using the homogeneous structure. In the case of the normal bundle of an irreducible homogeneous submanifold of euclidean space, we obtain that the Lie algebra of the normal holonomy group is algebraically generated by the projection to the normal space at a point of the corresponding Killing fields of the ambient space.
The Euler Class from a General Connection, Relative to a Metric
2021
We extend the well-known formula for the Euler class of a real oriented even-dimensional vector bundle in terms of the curvature of a metric connection to the case of a general linear connection provided a metric is present. We rewrite the classical Gauss-Bonnet theorem in dimension two in light of this formula. We also discuss a potential application to a conjecture of Chern, and make a brief digression to discuss m-quasi-Einstein manifolds.
A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30(9), (1991)], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation.
On the Algebraic Holonomy of Stable Principal Bundles
International Journal of Mathematics, 2011
Let EG be a stable principal G–bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over ℂ. Let H ⊂ G be a complex reductive subgroup which is not necessarily connected, and let EH ⊂ EG be a holomorphic reduction of structure group to H. We prove that EH is preserved by the Einstein–Hermitian connection on EG. Using this we show that if EH is a minimal reductive reduction (which means that there is no complex reductive proper subgroup of H to which EH admits a holomorphic reduction of structure group), then EH is unique in the following sense: For any other minimal reduction of structure group (H′, EH′) of EG to some reductive subgroup H′, there is some element g ∈ G such that H′ = g-1Hg and EH′ = EHg. As an application, we show the following: Let M be a simply connected, irreducible smooth complex projective variety of dimension n such that the Picard number of M is one. If the canonical line bundle KM is ample, then the a...
On the Geometry of Vector Bundles with Flat Connections
2019
Let E → M be an arbitrary vector bundle of rank k over a Riemannian manifold M equipped with a fiber metric and a compatible connection DE . R. Albuquerque constructed a general class of (twoweights) spherically symmetric metrics on E. In this paper, we give a characterization of locally symmetric spherically symmetric metrics on E in the case when DE is flat. We study also the Einstein property on E proving, among other results, that if k ≥ 2 and the base manifold is Einstein with positive constant scalar curvature, then there is a 1parameter family of Einstein spherically symmetric metrics on E, which are not Ricci-flat. Introduction and main results In the framework of Riemannian geometry, many special kinds of vector bundles have been considered and extensively studied, such as the cotangent bundle or the tangent bundle the literature of whose is very rich. Indeed, a wide range of interesting works have been published on the geometry of tangent bundles endowed with special types...
Structure groups and holonomy in infinite dimensions
2004
We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T. Robart in [Can. J. Math. 49 (4) (1997) 820–839], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose–Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.
Flat Bundles With Complex Analytic Holonomy
The Quarterly Journal of Mathematics, 2016
Let G be a connected complex Lie group. We show that any flat principal G-bundle over any finite CW-complex pulls back to a trivial bundle over some finite covering space of the base space if and only if each real characteristic class of positive degree of G vanishes. A third equivalent condition is that the derived group of the radical of G is simply connected. As a corollary, the same conditions are equivalent if G is a connected amenable Lie group. In particular, if G is a connected compact Lie group then any flat principal G-bundle over any finite CW-complex pulls back to a trivial bundle over some finite covering space of the base space.
The normal holonomy group of Khler submanifolds
Proceedings of the London Mathematical Society, 2004
We study the (restricted) holonomy group Hol(∇ ⊥) of the normal connection ∇ ⊥ (shortly, normal holonomy group) of a Kähler submanifold of a complex space form. We prove that if the normal holonomy group acts irreducibly on the normal space then it is linear isomorphic to the holonomy group of an irreducible Hermitian symmetric space. In particular, it is a compact group and the complex structure J belongs to its Lie algebra. We prove that the normal holonomy group acts irreducibly if the submanifold is full (i.e. it is not contained in a totally geodesic proper Kähler submanifold) and the second fundamental form at some point has no kernel. For example, a Kähler-Einstein submanifold of CP n has this property. We define a new invariant µ of a Kähler submanifold of a complex space form. For non-full submanifolds, the invariant µ measures the deviation of J from belonging to the normal holonomy algebra. For a Kähler-Einstein submanifold, the invariant µ is a rational function of the Einstein constant. By using the invariant µ , we prove that the normal holonomy group of a not necessary full Kähler-Einstein submanifold of CP n is compact and give a list of possible holonomy groups. The approach is based on a definition of the holonomy algebra hol(P) of arbitrary curvature tensor field P on a vector bundle with a connection and on a De Rham type decomposition theorem for hol(P) .
On compact Riemannian manifolds with noncompact holonomy groups
Journal of Differential Geometry
Solving a long standing problem in Riemannian geometry we construct a compact Riemannian manifold with a noncompact holonomy group. As the title indicates we then prove structure theorems for these manifolds. We employ an argument of Cheeger and Gromoll [1971] to show that the holonomy group of a compact Riemannian manifold is compact if and only if the image of the so called holonomy representation of its fundamental group is finite. Then we characterize these holonomy representations algebraically. As a consequence we prove that a finite cover of a compact Riemannian manifold M' n) with a noncompact holonomy group is the total space of a torus bundle over another compact Riemannian manifold ß' 6 ' with b < n-4.