Killing vector fields and holonomy algebras
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- by Carlos Currás-Bosch PDF
- Proc. Amer. Math. Soc. 90 (1984), 97-102 Request permission
Abstract:
We prove that for each Killing vector field $X$ on a complete Riemannian manifold, whose orthogonal distribution is involutive, the $(1,1)$ skew-symmetric operator ${A_X}$ associated to $X$ by ${A_X} = {L_X} - {\nabla _X}$ lies in the holonomy algebra at each point. By using the same techniques, we also study when that operator lies in the infinitesimal and local holonomy algebras respectively.References
- Dale Husemoller, Fibre bundles, 2nd ed., Graduate Texts in Mathematics, No. 20, Springer-Verlag, New York-Heidelberg, 1975. MR 0370578
- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
- Bertram Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold, Trans. Amer. Math. Soc. 80 (1955), 528–542. MR 84825, DOI 10.1090/S0002-9947-1955-0084825-8
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 90 (1984), 97-102
- MSC: Primary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-1984-0722424-6
- MathSciNet review: 722424