On some properties of a kind of affinely connected manifolds admitting a group of affine motions, II
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概要
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In a previous paper [1] the present author showed that, if an n-dimensional manifold A_n with symmetric affine connection and n≧7 admits a group of affine motions of order r>n^2-2n, then its curvature tensor has the form R^λ_<.μνω>=A^λB_<μνω>+C^λD_<μνω>+δ^λ_μ(P_<νω>-P_<ων>)+δ^λ_νP_<μω>-δ^λ_ωP_<μν> In another paper [2] it was also shown that, if an A_n, n≧7, with symmetric affine. connection admits a group of affine motions of order r>n^2-2n, then the curvature tensor has the form (1) R^λ_<.μνω>=A^λB_<μνω>+δ^λ_μ(P_<νω>-P_<ων>)+δ^λ_νP_<μω>-δ^λ_ωP_<μν>, or it has the form (2) R^λ_<.μνω>=(A^λP_μ+C^λQ_μ) (P_νQ_ω-P_ωQ_ν)+δ^λ_μ(P_<νω>-P_<ων>)+δ^λ_νP_<μω>-δ^λ_ωP_<μν>, which is possible only for n=7. Besides, it was shown that, if the curvature tensor has the form (2), then the order r of the group of affine motions admitted must satisfy r≦n^2-3n+8. In the present paper an A_n, n≧7, having the curvature tensor of the form (1) where A^λB_<μνω>≠0 is studied with the results that, if it admits a group of affine motions of order r>n^2-2n, then the rank of the tensor B_<μνω> is 2 or 3 and, moreover, that it has the form B_<μνω>=P_μ(P_νQ_ω-P_ωQ_ν) or B_<μνω>=P_μ(P_νQ_ω-P_ωQ_ν) +Q_μ(R_νP_ω-R_ωP_ν)-R_μ(P_νQ_ω-P_ωQ_ν)+2R_μ(Q_νR_ω-Q_ωR_ν). It is also shown that the latter form is possible only for n=7 (or r≦n^2-3n+8). Then, a necessary and sufficient condition that an A_n, n≧7, with symmetric affine connection and with non-vanishing projective curvature tensor admit a group of affine motions of order r=n^2-2n+5 is obtained. It is that the curvature tensor have the form R^λ_<.μνω>=A^λP_μ(P_νQ_ω-P_ωQ_ν) where A^λ, P_μ and Q_ν are covariant constants. This is equivalent to the condition that the connection parameters Γ^λ_<μν> satisfy Γ^1_<22>=x^3, other Γ^λ_<μν>=0 when the coordinate system is suitably chosen. Such connection was already given by G. Vranceanu [3, 4] as an example, but now it is shown that no other connection is possible.
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