Abstract
In this paper we give a generalisation of Kostant’s Theorem about theA x -operator associated to a Killing vector fieldX on a compact Riemannian manifold. Kostant proved (see [6], [5] or [7]) that in a compact Riemannian manifold, the (1, 1) skew-symmetric operatorA x =L x −≡ x associated to a Killing vector fieldX lies in the holonomy algebra at each point. We prove that in a complete non-compact Riemannian manifold (M, g) theA x -operator associated to a Killing vector field, with finite global norm, lies in the holonomy algebra at each point. Finally we give examples of Killing vector fields with infinite global norms on non-flat manifolds such thatA x does not lie in the holonomy algebra at any point.
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References
A. Andreotti and E. Vesentini,Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Etudes Sci., Publ. Math.25 (1965), 313–362.
E. Calabi,Métriques Kählériennes et fibrés holomorphes, Ann. Sci. Ec. Norm. Sup 4e série12 (1979), 269–294.
E. Calabi,Isometric families of Kähler structures, The Chern Symposium (1979), Springer-Verlag, 1980, pp. 23–39.
J. Dodziuk,Vanishing theorems for square-integrable harmonic forms, inGeometry and Analysis, Papers dedicated to the memory of V. K. Patodi, Springer-Verlag, 1981, pp. 21–27.
S. Kobayashi and K. Nomizu,Foundations of Differential Geometry, Vol. I, Int. Publ., 1963.
B. Kostant,Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold, Trans. Am. Math. Soc.80 (1953), 528–542.
A. Lichnérowicz,Géométrie des groupes de transformations, Dunod, Paris, 1958.
S. Yorozu,Killing vector fields on complete Riemannian manifolds, Proc. Am. Math. Soc.84 (1982), 111–120.
S. Yorozu,Conformal and Killing vector fields on complete noncompact Riemannian manifolds, inAdvanced Studies in Pure Math., Vol. 3,Geometry of geodesics and related topics, North-Holland (to appear).
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Currás-Bosch, C. A x -operator on complete riemannian manifolds. Israel J. Math. 53, 315–320 (1986). https://doi.org/10.1007/BF02786564
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DOI: https://doi.org/10.1007/BF02786564