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Invariant Sub-Bundles of the Tangent Bundle of a Homogeneous Space

Published online by Cambridge University Press:  20 November 2018

Philippe Tondeur
Philippe Tondeur*
Affiliation:
Harvard University, Cambridge, Massachusetts
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Let M = G/H be the homogeneous space of a Lie group G and a closed subgroup H. Denote by p : G → G/H the canonical projection, e ∈ G the identity and x0 = p(e). Let W be a subspace of the tangent space Tx0(M).

Definition. A lift W* of W is a subspace of the Lie algebra of G satisfying ∩ W* = ﹛0﹜ and p*W* = W, where p* : → Tx0(M) denotes the tangent map of p at e.

Consider a G-invariant sub-bundle of the tangent bundle of M (4), i.e., a field of vector subspaces x ⊂ Tx(M) for every x ∈ M satisfying

1

Here μg : M → M denotes the diffeomorphism defined by g ∈ G and (μg)*x : Tx → Tμg(x) the induced tangent map at x.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 629 - 634
Copyright
Copyright © Canadian Mathematical Society 1966

References

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