Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T23:41:04.686Z Has data issue: false hasContentIssue false

On Holonomy and Homogeneous Spaces

Published online by Cambridge University Press:  22 January 2016

Bertram Kostant*
Affiliation:
Berkeley, California
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In general a homogeneous space admits many invariant affine connections. Among these are certain connections which appear in many ways to be more natural than the others. We refer to the connections which K. Nomizu in [4] calls canonical affine connections of the first kind. When G is a compact connected Lie group and K a closed subgroup we called an invariant Riemannian metric on G/K, natural (in [2]) when it induced such a connection.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1957

References

[1] Kostant, B., Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold, Trans. Amer. Math. Soc, vol. 80 (1955), pp. 528542.Google Scholar
[2] Kostant, B., On differential geometry and homogeneous spaces, I. Proc. Nat. Acad. Sci., vol. 42 (1956), pp. 258261.Google Scholar
[3] Kostant, B., On differential geometry and homogeneous spaces, II. Proc. Nat. Acad. Sci., vol. 42 (1956), pp. 354357.Google Scholar
[4] Nomizu, K., Invariant affine connections on homogeneous spaces, Amer. Jour. Math., vol. 76 (1954), pp. 3365.Google Scholar
[5] Lichnerowicz, A., Espaces homogènes riemannien et reductibilité, Comptes Rendus, vol. 242 (1956), pp. 14101413.Google Scholar
[6] Lichnerowicz, A., Sur la reductibilité âes espaces homogènes riemanniens. Comptes Rendus, vol. 243 (1956), pp. 640642.Google Scholar
[7] Matsushima, Y. and Hano, J., Some studies on Kaehlerian homogeneous spaces, Nagoya Math. J., vol. 11 (1957), pp. 7792.Google Scholar
[8] Lichnerowicz, A., Transformations affine et holonomie, Comptes Rendus, Vol. 244 (1957), pp. 18681870.Google Scholar