Showing papers on "Frame bundle published in 1969"
TL;DR: In this article, the action of the Pontryagin squares in the cohomology of the Thorn space of a vector bundle is investigated. But the results of the normal bundle of a manifold imbedded in Euclidean space are not applicable to the case of an orientable manifold.
Abstract: The object of this note is to determine the action of the Pontryagin squares in the cohomology of the Thorn space of a vector bundle. This computation is then applied to the case of the normal bundle of a manifold imbedded in Euclidean space to give simplified proofs of some theorems of Mahowald. The first of Mahowald's theorems [3] was inspired by some 1940 results of Whitney [9], who showed that in certain cases the Euler class (with twisted integer coefficients) of the normal bundle of a nonorientable surface imbedded in Euclidean 4-space could be nonzero. This contrasts with the well-known theorem that the Euler class of the normal bundle of an orientable manifold in Euclidean space is always zero.
18 citations
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TL;DR: The tangent bundle is regarded as an almost product manifold and connections adapted to this structure are introduced in this article, where the theory of submersions is applied to this manifold.
Abstract: The tangent bundle is regarded as an almost product manifold. Connections adapted to this structure are introduced. Use is made of the theory of submersions,
7 citations
3 citations