On n-dimensional projectively flat spaces admitting a group of affine motions Gr of order r>n^2-n

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The necessary and sufficient conditions that an n-dimensional manifold A_n with symmetric affine connection and with vanishing projective curvature admit a group of affine motions Gr of order r>n^2-n are obtained in the form of tensor equations for n≧3, by studying the integrability conditions of the differential equations of affine motions [numerical formula] We find that we have three types of spaces. The spaces of the first type, T1, admit a transitive group Gr with r=n^2, or an intransitive group Gr with r=n^2-1. The spaces of the second type, T2, admit a transitive group Gr, r=n^2-n+1, and the spaces of the third type, T3, also admit a transitive group Gr, r=n^2-n+1. The rank of the symmetric part of the Ricci tensor is one for T1 and T2, while it is two for T3. The skew symmetric part vanishes for T1 and T3, while it is of rank two for T2. The existence of such types are proved. As, according to G. Vranceanu, a projectively not flat space does not allow a group of affine motions of order more than n^2-2n+5, we find that we can omit the words "with vanishing projective curvature" for n≧5.
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