Killing vector fields on semiriemannian manifolds

概要

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It is well known that a Killing vector field on a riemannian compact manifold is holonomic (Kostant (4)). In other words, the Ax operatpr (Ax=Lx-▽x=-▽x) lies in the holomony algebra of the manifold. The covariant derivative of Ax gives us a curvature transformation. This fact and the Ambrose-Singer theorem show that the Ax operator lies infinitesimally in the holonomy algebra h. (i.e. [?]Y, ▽rAx=Rxy∈h) (*) The subjebt of our study is the holonomicity of a Killing vector field on a semiriemannian compact manifold. We remarl the validity of (*) on semiriemannian manifolds. In order to simplify is study, we constrain it to Lorentz locally strictly weakly irreducible manifolds (1.SWI). We remark that Beger (1) showed that the holonomy algebra of a Lorentz manifold which is irreducible and non locally symmetric is the whole po(n, 1). Therefore, we can leave out this case. Stictly weakly irreducible manifolds, defined by H. Wu (5, 6) in 1963 are the cornerstones of this study. Among there we have found examples of compact manifolds with a non holonimoic Killing vector field.
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