Abstract
Let M be a differentiable manifold of class C ?. All tensor fields discussed below are assumed to be of class C ?. Let X be a vector field on M. If X vanishes at a point 0 ? M then X induces, in a natural way, an endomorphism a X of the tangent space V o at 0. In fact if y ? V 0 and Y is any vector field whose value at 0 is y, then define a x y = [X, Y] 0 . It is not hard to see that [X, Y] 0 does not depend on Y so long as the value of Y at 0 is y.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. vol. 75 (1953) pp. 428–443.
A. Borel and A. Lichnerowicz, Espaces riemannien et hermitiens symetrique, C. R. Acad. Sci. Paris vol. 234 (1952) pp. 2332–2334.
A. Borel Groupes d’holonomie des varietes riemanniennes, C. R. Acad. Sci. Paris vol. 234 (1952) pp. 1835–1837.
E. Cartan, Sur une classe remarquable d’espaces de Riemann, Bull. Soc. Math. France vol. 54 (1926) pp. 214–264; vol, 55 (1927) pp. 114–134.
E. Cartan, La geomUrie des groupes de transformations, J. Math. Pures Appl. vol. 6 (1927) pp. 1–119.
R. Hermann, Sur les isometries infinitesimales et le group d’holonomie d J un espace de Riemann, C. R. Acad. Sci. Paris vol. 239 (1954) pp. 1178–1180.
R. Hermann, Sur les automorphismes infinitesimaux d’une G-structure, C. R. Acad. Sci. Paris vol. 239 (1954) pp. 1760–1761.
A. Nijenhuis, On the holonomy group of linear connections, Indagationes Math. vol. 15 (1953) pp. 233–249; vol. 16 (1954) pp. 17–25.
K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. vol. 76 (1954) pp. 33–65.
K. Nomizu, Sur les transformations affine d’une varitti riemannienne, C. R. Acad. Sci. Paris vol. 237 (1953) pp. 1308–1310.
K. Nomizu, Applications de Vttude de transformation affine aux espaces homogeneous reimanniens, C. R. Acad. Sci. Paris vol. 237 (1953) pp. 1386–1387.
A. Lichnerowicz, Espaces homogenes kahleriens, Colloque International de Geometrie Differentielle, Strasbourg, 1953, pp. 171–184.
K. Yano and S. Bochner, Curvature and Bette numbers, Annals of Mathematics Studies, no. 32, Princeton, 1953.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag New York
About this chapter
Cite this chapter
Kostant, B. (2009). Holonomy and the Lie Algebra of Infinitesimal Motions of A Riemannian Manifold. In: Joseph, A., Kumar, S., Vergne, M. (eds) Collected Papers. Springer, New York, NY. https://doi.org/10.1007/b94535_1
Download citation
DOI: https://doi.org/10.1007/b94535_1
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-09582-0
Online ISBN: 978-0-387-09583-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)