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Deformation theory of subspaces in a Riemann space


Author: V. Hlavatý
Journal: Proc. Amer. Math. Soc. 1 (1950), 600-617
MSC: Primary 53.0X
DOI: https://doi.org/10.1090/S0002-9939-1950-0038120-4
MathSciNet review: 0038120
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Abstract: This paper deals with the infinitesimal transformation (2,1) of a family ($ {V_m}$) of subspaces $ {V_m}$ in a Riemann space. In §§1-4 the transformation (2,1) is applied on internal objects of a $ {V_m}$, while in §§5 and 6 the characteristic mixed tensors $ K_{{a_r}...{a_1}}^v$ (cf. the equation (1,2;4)) of a $ {V_m}$ are investigated with respect to (2,1). Finally, some applications of the theory are given in §§7 and 8. In particular the statement expressed by the equation (8,10)b is the generalization of the well known Levi-Civita result for $ m = 1$, while the statement expressed by the equation (8,10)c generalizes the classical result (for $ m = 1,\;n = 2$) by Jacobi.


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  • [1] J. A. Schouten, On infinitesimal deformations of $ {V_m}$ in $ {V_n}$, Proc. K. Akademie van Wetenschappen vol. 31 (1928) pp. 208-218.
  • [2] E. Bortolotti, Scostamento geodetico $ e$ sue generalizzazioni, Giornale di matematiche vol. 66 (1928) pp. 153-191.
  • [3] A. G. Walker, On small deformation of sub-spaces of a flat space, Proceedings of the Edinburgh Mathematical Society vol. 3 (1932) pp. 77-86.
  • [4] J. A. Schouten and E. R. van Kampen, Beiträge zur theorie der deformation, Prace Matematyczno-Fizyczne Warszawa vol. 41 (1933) pp. 1-19.
  • [5] H. A. Hayden, Infinitesimal deformations of sub-spaces in a general metrical space, Proc. London Math. Soc. vol. 37 (1934) pp. 416-440.
  • [6] -, Infinitesimal deformation of an $ {L_m}$ in an $ {L_n}$, Proc. London Math. Soc. vol. 41 (1936) pp. 332-336.
  • [7] J. A. Schouten and D. J. Struik, Einführung in die neueren Methoden der Differentialgeometrie, vol. 1, Groningen, 1935, p. 151.
  • [8] E. T. Davies, On the deformation of a subspace, J. London Math. Soc. vol. 11 (1936) pp. 295-301.
  • [9] -, On the second and third fundamental forms of a subspace, J. London Math. Soc. vol. 12 (1937) pp. 290-295.
  • [10] P. Dienes and E. T. Davies, On the infinitesimal deformations of tensor submanifolds, Journal de Mathématiques vol. 16 (1937) pp. 111-150.
  • [11] Kentaro Yano, Sur la déformation infinitésimale des sous-espaces dans un espace affine, Proc. Japan Acad. 21 (1945), 248–260 (1949) (French). MR 0033146
  • [12] Kentaro Yano, Sur la déformation infinitésimale tangentielle d’un sous-espace, Proc. Japan Acad. 21 (1945), 261–268 (1949) (French). MR 0033147

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DOI: https://doi.org/10.1090/S0002-9939-1950-0038120-4
Article copyright: © Copyright 1950 American Mathematical Society

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