Abstract:
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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. |