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Transformations in hypercomplex Riemannian spaces

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Abstract

It is well known that an integrable regular H-structure induces on a real manifold Mn the structure of a hypercomplex analytic manifold (h-manifold)\(\mathop M\limits^* _m \). We prove that the Lie derivative of a pure tensor T on Mn is an h-derivative of Lie providing T is h-analytic. With the h-derivative of Lie there is associated on\(\mathop M\limits^* _m \) the hypercomplex derivative of Lie. This enables us to associate to the motions and affine collineations in the Riemannian space\(\mathop V\limits^* _m \) corresponding transformations in a real space Vn.

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Literature cited

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    G. I. Kruchkovich, “Hypercomplex structures on manifolds, I,” in: Proceedings of the Vector and Tensor Analysis Seminar, Vol. 16 [in Russian] (1972).

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    K. Yano and S. Bochner, Curvature and Betti Numbers, Ann. of Math. Studies, No. 32, Princeton University Press, Princeton, N.J. (1953).

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    V. V. Navrozov, “Fundamental concepts of hypercomplex Riemannian geometry,” Trudy MIRÉA, Matematika, No. 67, 50–64 (1973).

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    G. I. Kruchkovich, “Hypercomplex geodesies and their real realizations,” Trudy MIRÉA, Matematika, No. 67, 3–11 (1973).

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Additional information

Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 603–612, April, 1974.

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Navrozov, V.V. Transformations in hypercomplex Riemannian spaces. Mathematical Notes of the Academy of Sciences of the USSR 15, 356–361 (1974). https://doi.org/10.1007/BF01095128

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Keywords

  • Manifold
  • Real Space
  • Riemannian Space
  • Analytic Manifold
  • Real Manifold