On conformally curved Riemann spaces Vn, n≥6, admitting a group of motions Gγ of order γ>n(n+1)/2-(3n-11)

Yosio MUTO

doi:10.2969/jmsj/00910038

On conformally curved Riemann spaces V_n, n≥6, admitting a group of motions G_γ of order γ>n(n+1)/2-(3n-11)

Yosio MUTO

著者情報

ジャーナルフリー

1957 年 9 巻 1 号 p. 38-61

DOI https://doi.org/10.2969/jmsj/00910038

詳細

Published: 1957 Received: 1956/04/12 Available on J-STAGE: 2006/08/29 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/08/29 訂正理由: - 訂正箇所: 論文タイトル訂正内容: Wrong : On conformally curved Riemann spaces V_n, n_??_6, admitting a group of motions G_r of order r>n(n+1)/2-(3n-11)
Right : On conformally curved Riemann spaces V_n, n≥6, admitting a group of motions G_γ of order γ>n(n+1)/2-(3n-11)

訂正日: 2006/08/29 訂正理由: - 訂正箇所: 論文抄録訂正内容: Wrong : The conformal curvature tensor C_λμvω of a Riemann space V_n, n_??_6, admitting a group of motions of order r>n(n+1)/2-(3n-11) is studied with the use of tensor calculus. The form of C_λμvω is obtained by virtue of the fact that the equations XC_λμvω=0 can contain at most a certain number of linearly independent equations. The C_λμvω is in general of the form C_λμvω=[Cδ_λωδ_μv-δ_λvδ_μω] -((n-1)/2)C[δ_λω(A_μA_v+B_μB_v)+δ_μv (A_λA_ω+B_λB_ω)-δ_λv (A_μA_ω+B_μB_ω)-δ_μω(A_λA_v+B_λB_v)]+((n-1) (n-2)/2)C [A_λA_ωB_μB_v+A_μA_ωB_λB_ω -A_λA_vB_μB_ω-A_μA_ωB_λB_v] with A_αA_α=B_αB_α=1, A_αB_α=0. But for n=6, 8 some other form is also possible.
Right : The conformal curvature tensor C_λμνω of a Riemann space V_n, n≥6, admitting a group of motions of order r>n(n+1)/2-(3n-11) is studied with the use of tensor calculus. The form of C_λμνω is obtained by virtue of the fact that the equations XC_λμνω=0 can contain at most a certain number of linearly independent equations. The C_λμνω is in general of the form
C_λμνω=C[δ_λωδ_μν-δ_λνδ_μω] -((n-1)/2)C[δ_λω(A_μA_ν+B_μB_ν)+δ_μν (A_λA_ω+B_λB_ω)-δ_λν (A_μA_ω+B_μB_ω)-δ_μω(A_λA_ν+B_λB_ν)]+((n-1) (n-2)/2)C [A_λA_ωB_μB_ν+A_μA_νB_λB_ω -A_λA_νB_μB_ω-A_μA_ωB_λB_ν]
with A_αA_α=B_αB_α=1, A_αB_α=0. But for n=6, 8 some other form is also possible.

訂正日: 2006/08/29 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) E. Cartan, Leçons sur la géométrie des espaces de Riemann, 2d ed., Paris, Gauthier-Villars, 1946.
2) I. P. Egorov, On a strengthening of Fubini's theorem on the order of the group of motions of a Riemannian space, Doklady Akad. Nauk SSSR. N. S., 66 (1947), pp. 793-796.
3) L. P. Eisenhart, Continuous groups of transformations, Princeton University Press, 1933.
4) M. Kurita, On the isometry of a homogeneous Riemann space, Tensor, N. S., 3 (1954), pp. 91-100.
5) D. Montgomery and H. Samelson, Transformation groups of spheres, Ann. of Math. (2), 44 (1943), pp. 454-470.
6) H. Wakakuwa, On n-dimensional Riemannian spaces admitting some groups of motions of order less than n(n-1)/2, Tôhoku Math. J. Sec. Series, 6 (1954), pp. 121-134.
7) H. C. Wang, On Finsler spaces with completely integrable equations of Killing, J. London Math. Soc., 22 (1947), pp. 5-9.
8) K. Yano, On n-dimensional Riemannian spaces admitting a group of motions of order n(n-1)/2+1, Trans. Amer. Math. Soc., 74 (1953), pp. 260-279.
9) K. Yano, Groups of transformations in generalized spaces, Tokyo, 1949.
10) G. Vranceanu, Groupes de mouvement des espaces à conexion, Acad. Repub. Pop. Române, Studii si Cercetari Matematice, 2 (1951), pp. 387-444.

Right : [1] E. Cartan, Leçons sur la géométrie des espaces de Riemann, 2d ed., Paris, Gauthier-Villars, 1946.
[2] I. P. Egorov, On a strengthening of Fubini's theorem on the order of the group of motions of a Riemannian space, Doklady Akad. Nauk SSSR. N. S., 66 (1947), pp. 793-796.
[3] L. P. Eisenhart, Continuous groups of transformations, Princeton University Press, 1933.
[4] M. Kurita, On the isometry of a homogeneous Riemann space, Tensor, N. S., 3 (1954), pp. 91-100.
[5] D. Montgomery and H. Samelson, Transformation groups of spheres, Ann. of Math. (2), 44 (1943), pp. 454-470.
[6] H. Wakakuwa, On n-dimensional Riemannian spaces admitting some groups of motions of order less than n(n-1)/2, Tôhoku Math. J. Sec. Series, 6 (1954), pp. 121-134.
[7] H. C. Wang, On Finsler spaces with completely integrable equations of Killing, J. London Math. Soc., 22 (1947), pp. 5-9.
[8] K. Yano, On n-dimensional Riemannian spaces admitting a group of motions of order n(n-1)/2+1, Trans. Amer. Math. Soc., 74 (1953), pp. 260-279.
[9] K. Yano, Groups of transformations in generalized spaces, Tokyo, 1949.
[10] G. Vranceanu, Groupes de mouvement des espaces à conexion, Acad. Repub. Pop. Române, Studii si Cercetari Matematice, 2 (1951), pp. 387-444.

訂正日: 2006/08/29 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

抄録

The conformal curvature tensor C_λμνω of a Riemann space V_n, n≥6, admitting a group of motions of order r>n(n+1)/2-(3n-11) is studied with the use of tensor calculus. The form of C_λμνω is obtained by virtue of the fact that the equations XC_λμνω=0 can contain at most a certain number of linearly independent equations. The C_λμνω is in general of the form
C_λμνω=C[δ_λωδ_μν-δ_λνδ_μω] -((n-1)/2)C[δ_λω(A_μA_ν+B_μB_ν)+δ_μν (A_λA_ω+B_λB_ω)-δ_λν (A_μA_ω+B_μB_ω)-δ_μω(A_λA_ν+B_λB_ν)]+((n-1) (n-2)/2)C [A_λA_ωB_μB_ν+A_μA_νB_λB_ω -A_λA_νB_μB_ω-A_μA_ωB_λB_ν]
with A_αA_α=B_αB_α=1, A_αB_α=0. But for n=6, 8 some other form is also possible.

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