Frame bundle

About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.

Heat kernel expansions in vector bundles

Abstract: Let M be a complete connected smooth (compact) Riemannian manifold of dimension n . Let Π : V → M be a smooth vector bundle over M . Let L = 1 2 Δ + b be a second order differential operator on M , where Δ is a Laplace-Type operator on the sections of the vector bundle V and b a smooth vector field on M . Let k t ( − , − ) be the heat kernel of V relative to L . In this paper we will derive an exact and an asymptotic expansion for k t ( x , y 0 ) where y 0 is the center of normal coordinates defined on M , x is a point in the normal neighborhood centered at y 0 . The leading coefficients of the expansion are then computed at x = y 0 in terms of the linear and quadratic Riemannian curvature invariants of the Riemannian manifold M , of the vector bundle V , and of the vector bundle section ϕ and its derivatives. We end by comparing our results with those of previous authors (I. Avramidi, P. Gilkey, and McKean–Singer).

The Atiyah bundle and connections on a principal bundle

Abstract: Let M be a C∞ manifold and G a Lie a group. Let EG be a C∞ principal G-bundle over M. There is a fiber bundle C(EG) over M whose smooth sections correspond to the connections on EG. The pull back of EG to C(EG) has a tautological connection. We investigate the curvature of this tautological connection.

A Fourier transform for Higgs bundles

Abstract: We define a Fourier-Mukai transform for Higgs bundles on smooth curves and study its properties. It is shown that the transform of a stable zero-degree Higgs bundle is an algebraic vector bundle on the cotangent bundle of the Jacobian of the curve. We show that this transformed bundle admits a natural extension to an algebraic vector bundle over a projective compactification of the base. The main result is that the original Higgs bundle can be reconstructed from this extension. We also compute certain invariants of the transformed bundle.

A note on the Whitehead torsion of a bundle modulo a subbundle

Abstract: The purpose of this note is to show how the sum theorem for Whitehead torsion due to K. W. Kwun and R. H. Szczarba, and generalized by L. C. Siebenmann, may be applied to compute the Whitehead torsion of the total space pair of a bundle pair in terms of the Whitehead torsion of the fiber pair and the Euler characteristic of the base. Let p:E-+B be a PL fiber bundle with fiber F and suppose p':E'->-B is a PL subbundle with fiber F'. If the inclusion F'c F is a homotopy equivalence, a theorem of Dold [1, Theorem 6.3] implies that the inclusion £<=£' ¡s also a homotopy equivalence. It is the object of this note to answer the natural question1 "How are the Whitehead torsions tíF, F')e Wh tt^F) and t(£, £')eWh «i(£) related?" Specifically we prove the Theorem. Let B and F be connected. Then r(£, £') = XiB)MiF, F') where %iB) is the Euler characteristic of B and j*: Wh 771(F)^-Wh 77,(£) is induced by the inclusion y.F^-E. To set the context precisely, we recall that a PL bundle pair with fiber (F, F') is a PL map p: (£, £')->-(£, B), denoted simply by B in the sequel, such that for some triangulation K of B and each simplex aeK, there is a PL homeomorphism ha:iaxF,axF')-^-ip~1ia),p~1ia)C\E') such that pha=P\ where p^.iaxF, axF')->a is projection on the first factor. A PL bundle p':E'-*B with fiber F' is a subbundle ,of the PL bundle p: E-*B with fiber F if £'<= E,p'=p | £', and />:(£, E')-*B is a PL bundle pair with fiber (£, £'). In this event for each xeB we denote the pair ip-1ix),p'-1ix))=ip~1ix),p~1ix)r\E') by iFx,F'x) and the inclusion (F„F;)c (£,£') by;,. Throughout the remainder of this note we work in the category of compact PL spaces and PL maps. Received by the editors June 1, 1971. AMS 1970 subject classifications. Primary 57C10; Secondary 57C50.

A Two Point Calibration on an SP(1) Bundle Over the Three-Sphere

Abstract: Gluck and Ziller proved that Hopf vector fields on S3 have minimum volume among all unit vector fields. Thinking of S3 as a Lie group, Hopf vector fields are exactly those with unit length which are left or right invariant, and TS3 is a trivial vector bundle with a connection induced by the adjoint representation. We prove the analogue of the stated result of Gluck and Ziller for the representation given by quaternionic multiplication. The resulting vector bundle over S3, with the Sasaki metric, has as well no parallel unit sections. We provide an application of a double point calibration, proving that the submanifolds determined by the left and right invariant sections minimize volume in their homology classes.