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The Vector Field Problem for Homogeneous Spaces

  • Parameswaran SankaranEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Let M be a smooth connected manifold of dimension \(n\ge 1\). A vector field on M is an association \(p\rightarrow v(p)\) of a tangent vector \(v(p)\in T_pM\) for each \(p\in M\) which varies continuously with p. In more technical language, it is a (continuous) cross section of the tangent bundle \(\tau (M)\). The vector field problem asks: Given M, what is the largest possible number r such that there exist vector fields \(v_1,\ldots , v_r\) which are everywhere linearly independent, that is, \(v_1(x),\ldots ,v_r(x)\in T_xM\) are linearly independent for every \(x\in M\). The number r is called the span of M, written \({\text {span}}(M)\). It is clear that \(0\le {\text {span}}(M)\le \dim (M)\). The vector field problem is an important and classical problem in differential topology. In this survey, we shall consider the vector field problem focussing mainly on the class of compact homogeneous spaces.

Keywords

Vector fields Span Stable span Parallelizability Stable parallelizability Homogeneous spaces 

1991 Mathematics Subject Classification

57R25 

Notes

Acknowledgements

I am grateful to Professor Peter Zvengrowski for sharing with me his insights into the vector field problem and for long years of collaboration. I am grateful to referees for their very thorough reading of the paper, for their comments and for pointing out numerous errors. One of them also pointed out that Theorem 1.19 and parts of Theorem 1.20 were proved in the paper of Staples [79]. Also, I thank Július Korbaš, Arghya Mondal, Avijit Nath and Peter Zvengrowski for their comments and for pointing out errors. I thank Mahender Singh for the invitation to participate in the Seventh East Asian Conference on Algebraic Topology held in December 2017 at IISER Mohali and for his interest in publishing these notes as part of the Conference proceedings.

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Authors and Affiliations

  1. 1.The Institute of Mathematical Sciences (HBNI)Taramani, ChennaiIndia

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