Existence of Hermitian n-Symmetric Spaces and of Non-Commutative Naturally Reductive Spaces.

J.Alfredo Jiménez

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In this note it is shown that almost Hermitian locally homogeneous manifolds are determined, up to local isometries, by an integer $k_{H}$ , the covariant derivatives of the curvature tensor up to order $k_{H} + 2$ and the covariant derivatives of the complex structure up to the second order calculated at some point. An example of a Hermitian locally homogeneous manifold which is not locally isometric to any Hermitian globally homogeneous manifold is given.

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