A co-dimensional 3 sub-Riemannian structure on Gromoll-Meyer exotic sphere

2016/08/08 by Wolfram Bauer, Bauer, Wolfram, Kenro Furutani +3
Mathematics · #53C17 #57S25 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C17 #msc:57S25

paper · pdf · doi:10.48550/arxiv.1608.02444

21 pages

arxiv created 2016/08/08 · arxiv updated 2016/08/09

Abstract

We construct a co-dimension 3 completely non-holonomic sub-bundle on the Gromoll-Meyer exotic 7 sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7 sphere (or more general on a 4n+3 dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides an alternated simple proof for the standard sphere \mathbbS⁷.

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• Pulling back the Gromoll-Meyer construction and models of exotic spheres
• An infinite family of Gromoll-Meyer spheres
• Framed bordism and Lagrangian embeddings of exotic spheres
• Curvature of an exotic 7-sphere
• Sub-Riemannian geodesics on nested principal bundles