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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

1. Chevalley, C., Theory of Lie groups, vol. 1 (Princeton, 1946).Google Scholar
2. Frölicher, A., Zur Differentialgeometrie der komplexen Strukturen, Math. Ann., 129 (1955), 50–95.Google Scholar
3. Helgason, S., Differential geometry and symmetric spaces (New York, 1962).Google Scholar
4. Lang, S., Introduction to differentiate manifolds (New York, 1962).Google Scholar
5. Nomizu, K., Invariant affine connections on homogeneous spaces, Amer. J. Math., 76 (1954), 33–65.Google Scholar
6. Tondeur, Ph., Ein invariantes Vektorraumfeld auf einem reduktiven, lokal-symmetrischen homogenen Raum ist involutorisch, Math. Z., 85 (1964), 382–384.Google Scholar
7. Tondeur, Ph., Invariant sub-bundles of the tangent bundle of a reductive homogeneous space, Math. Z. 89 (1965), 420–421.Google Scholar
8. Tondeur, Ph., Champs invariants de p-plans sur un espace homogène, C. R. Acad. Sci. Paris, 259 (1964), 4473–4475.Google Scholar