Skip to main content
SpringerLink
Log in

Derivations of linear algebras and linear connections

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. R. J. Alonso, “Jet manifolds associated to a Weil bundle,” Arch. Math., 36, 195–199 (2000).

    MATH  MathSciNet  Google Scholar 

  2. A. V. Aminova, “Pseudo-Riemannian manifolds with general geodesics,” Usp. Mat. Nauk, 48, No. 2(290), 107–164 (1993).

    MathSciNet  Google Scholar 

  3. A. V. Aminova, “Projective-group properties of some Riemannian spaces, In: Proceedings of Geometric Workshop [in Russian], 6, All-Union Institute for Scientific and Technical Information, Moscow (1974), pp. 295–316.

  4. A. V. Aminova, “Projective and affine motion groups in spaces of general relativity theory,” In: Proceedings of a Geometric Workshop [in Russian], 6, All-Union Institute for Scientific and Technical Information, Moscow (1974), pp. 317–346.

  5. A. V. Aminova, “Almost projective motion groups of affine connection spaces,” Izv. Vuzov, Mat., No. 4, 71–75 (1979).

  6. M. F. Atiyah, “Complex analytic connections in fibre bundles,” Trans. Amer. Math. Soc., 85, No. 1, 181–207 (1957).

    MATH  MathSciNet  Google Scholar 

  7. D. E. Beklemishev, “Differential geometry of spaces with almost complex structure,” In: Progress in Science and Technology, Series on Problems of Geometry [in Russian], All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1965), pp. 165–212.

  8. R. Bishop and R. Crittenden, Geometry of Manifolds [Russian translation], Mir, Moscow (1967).

    MATH  Google Scholar 

  9. R. H. Bowman, “Concerning a problem of Yano and Kobayashi,” Tensor, 25, 105–112 (1972).

    MATH  MathSciNet  Google Scholar 

  10. R. H. Bowman, “Second order connections,” J. Different. Geom., 7, No. 3–4, 549–561 (1972).

    MATH  MathSciNet  Google Scholar 

  11. R. H. Bowman, “Second order connections. II,” J. Different. Geom., 8, No. 1, 75–84 (1973).

    MATH  MathSciNet  Google Scholar 

  12. N. Bourbaki, Algebra. Algebraic Structures, Linear and Multilinear Algebra [Russian translation], Fizmatgiz, Moscow (1962).

    Google Scholar 

  13. N. Bourbaki, Algebra. Modules, Rings, and Forms [Russian translation], Nauka, Moscow (1966).

    Google Scholar 

  14. F. Brickell and R. S. Clark, “Integrable almost tangent structures,” J. Different. Geom., 9, No. 4, 557–563 (1974).

    MATH  MathSciNet  Google Scholar 

  15. G. N. Bushueva and V. V. Shurygin, “On the higher order geometry of Weil bundles over smooth manifolds and over parameter-dependent manifolds,” Lobachevskii J. Math., 18, 53–105 (2005).

    MATH  MathSciNet  Google Scholar 

  16. E. Cartan, Affine, Projective, and Conformal Connection Spaces [Russian translation], Kazan’ University, Kazan’ (1962).

  17. E. Cartan, Geometry of Lie Groups and Symmetric Spaces [Russian translation], IL, Moscow (1949).

  18. S. S. Chen, Complex Manifolds [Russian translation], IL, Moscow (1961).

  19. L. A. Cordero, “Lifts of tensor fields to the frame bundle,” Rend. Circ. Mat. Palermo., 32, No. 2, 236–271 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  20. L. A. Cordero and M. De Leon, “Horizontal lift of a connection to the frame bundle,” Bull. Unione Mat. Ital., 6, No. 3, B, 223–240 (1984).

    Google Scholar 

  21. C. W. Curtis and I. Rayner, Representation Theory of Finite Groups and Associative Algebras [Russian translation], Nauka, Moscow (1969).

    Google Scholar 

  22. K. T. J. Dodson and M. Vasces-Abal, “Induced harmonic mappings of tangent bundles and frame bundles,” Mat. Zametki, 50, No. 3, 27–37 (1991).

    MATH  MathSciNet  Google Scholar 

  23. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry, Methods and Applications [in Russian], Nauka, Moscow (1986).

    MATH  Google Scholar 

  24. C. Duval and G. Valeat, “Quantum integrability of quadratic Killing tensors,” J. Math. Phys., 46, No. 05356, 05356-1–05356-22 (2005).

    Google Scholar 

  25. I. P. Egorov, “On order of motion groups of affine connection spaces,” Dokl. Akad. Nauk SSSR, 57, No. 2, 867–870 (1947).

    MATH  Google Scholar 

  26. I. P. Egorov, “On order of motion groups in affine connection spaces,” Dokl. Akad. Nauk SSSR, 61, No. 4, 605–608 (1948).

    MATH  Google Scholar 

  27. I. P. Egorov, “On motion groups of asymmetric affine connection spaces,” Dokl. Akad. Nauk SSSR, 64, No. 5, 621–624 (1949).

    MATH  Google Scholar 

  28. I. P. Egorov, “On motion groups of general asymmetric affine connection spaces,” Dokl. Akad. Nauk SSSR, 73, No. 2, 265–267 (1950).

    MATH  Google Scholar 

  29. I. P. Egorov, “Collineations of projective connection spaces,” Dokl. Akad. Nauk SSSR, 80, No. 5, 709–712 (1951).

    MATH  Google Scholar 

  30. I. P. Egorov, “Tensor characteristic of maximally movable A n of nonzero curvature,” Dokl. Akad. Nauk SSSR, 84, No. 2, 209–212 (1952).

    MATH  Google Scholar 

  31. I. P. Egorov, “Maximally movable L n of semisymmetric connection,” Dokl. Akad. Nauk SSSR, 84, No. 3, 433–435 (1952).

    MATH  Google Scholar 

  32. I. P. Egorov, “Motions in affine connection spaces,” Dokl. Akad. Nauk SSSR, 87, No. 5, 693–696 (1952).

    MATH  Google Scholar 

  33. I. P. Egorov, “On motions in affine connection spaces,” Dokl. Akad. Nauk SSSR, 89, No. 5, 781–784 (1953).

    MATH  Google Scholar 

  34. I. P. Egorov, Motions in Affine Connection Spaces [in Russian], Doctoral Dissertation in Physics and Mathematics, MGU, Moscow (1955).

    Google Scholar 

  35. I. P. Egorov, “Equiaffine spaces of third lacunarity,” Dokl. Akad. Nauk SSSR, 103, No. 1, 151–152 (1956).

    Google Scholar 

  36. I. P. Egorov, “Motions in generalized differential-geometric spaces,” In: Progress in Science and Technology, Series on Algebra, Topology, and Geometry [in Russian], All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1967), pp. 375–428.

  37. I. P. Egorov, “Automorphisms in generalized spaces,” In: Progress in Science and Technology, Series on Problems of Geometry [in Russian], 10, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1978), pp. 147–191.

  38. A. I. Egorov, N. S. Sinyukov, and A. Ya. Sultanov, “Scientific legacy of I. P. Egorov (25. 07. 1915–2. 10. 1990),” In: Progress in Science and Technology, Series on Contemporary Mathematics and Its Applications, Thematical Surveys [in Russian], 8, All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (1995), pp. 5–36.

  39. C. Ehresmann, “Les prolongments d’une variete differentiable. I. Calcul des jets, prolongment principal,” C. R. Acad. Sci., 233, No. 11, 598–600 (1951).

    MathSciNet  Google Scholar 

  40. C. Ehresmann, “Les prolongments d’une variete differentiable. II. L’espace des jets d’ordere r de V n dans V m .,” C. R. Acad. Sci., 233, No. 15, 777–779 (1951).

    MathSciNet  Google Scholar 

  41. C. Ehresmann, “Extension du calcul des jets aux jets non holonomes,” C. R. Acad. Sci., 239, No. 25, 1762–1764 (1954).

    MATH  MathSciNet  Google Scholar 

  42. L. P. Eisenhart, Continuous Transformation Groups [Russian translation], IL, Moscow (1947).

  43. L. P. Eisenhart, Riemannian Geometry [Russian translation], IL, Moscow(1948).

  44. L. E. Evtushik, “Lie derivative and differential equations of a geometric object field,” Dokl. Akad. Nauk SSSR, 132, No. 5, 998–1001 (1960).

    Google Scholar 

  45. L. E. Evtushik, “Differential connections and infinitesimal transformations of pseudogroup prolongations,” In: Proceedings of a Geometric Workshop [in Russian], 2, All-Union Institute for Scientific and Technical Information, Moscow (1969), pp. 119–150.

  46. L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, and A. P. Shirokov, “Differential-geometric structures on manifolds,” In: Progress in Science and Technology, Series on Problems of Geometry [in Russian], 9, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1979), pp. 1–247.

  47. A. T. Fomenko, Differential Geometry and Topology, Additional Chapters [in Russian], MGU, Moscow (1983).

    Google Scholar 

  48. D. Gromol, W. Klinkenberg, and W. Meyer, Riemannian Geometry in the Large [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  49. S. Helgason, Differential Geometry and Symmetric Spaces [Russian translation], Mir, Moscow (1964).

    MATH  Google Scholar 

  50. N. Jacobson, Lie Algebras [Russian translations], Mir, Moscow (1964).

    MATH  Google Scholar 

  51. S. Kobayashi, Transformation Groups in Differential Geometry [Russian translation], Nauka, Moscow (1986).

    MATH  Google Scholar 

  52. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry [Russian translation], Vol. 1, Nauka, Moscow (1981).

    Google Scholar 

  53. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry [Russian translation], Vol. 2, Nauka, Moscow (1981).

    Google Scholar 

  54. I. Kolář, “Affine structures on Weil bundles,” Nagoya Math. J., 158, 99–106 (2000).

    MATH  MathSciNet  Google Scholar 

  55. I. Kolář, P. W. Michor, and J. Slovak, Natural Operations in Differential Geometry, Springer (1993).

  56. G. I. Kruchkovich, “Hypercomplex structures on manifolds. I,” Trudy Sem. Vect. Tenz. Anal., No. 16, 174–201 (1972).

  57. G. I. Kruchkovich, “Hypercomplex structures on manifolds. II,” Trudy Sem. Vect. Tenz. Anal., No. 17, 213–227 (1974).

  58. A. G. Kurosh, Lectures on General Algebra [in Russian], FM, Moscow (1962).

    Google Scholar 

  59. B. L. Laptev, “Lie differentiations,” In: Progress in Science and Technology, Series on Algebra, Topology and Geometry [in Russian], All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1967), pp. 429–465.

  60. G. F. Laptev, “Basic higher-order infinitesimal structures,” In: Proceedings of a Geometric Workshop [in Russian], 1, All-Union Institute for Scientific and Technical Information, Moscow (1966), pp. 139–189.

  61. B. L. Laptev and A. Ya. Sultanov, “On the I. P. Egorov problem in motion theory,” Izv. Vuzov, Mat., No. 8, 34–39 (1986).

    Google Scholar 

  62. A. Lichnerowicz, Global Theory of Connections and Holonomy Groups [Russian translation], IL, Moscow (1960).

    Google Scholar 

  63. M. A. Malakhal’tsev, “Manifold structures over dual-number algebras on a torus,” In: Proceedings of a Geometric Workshop [in Russian], Kazan’ University, Kazan’ (1994), pp. 47–62.

  64. M. A. Malakhal’tsev, “Foliations with leaf structures,” In: Progress in Science and Technology, Series on Contemporary Mathematics and Its Applications, Thematical Surveys [in Russian], 73, All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (2002), pp. 65–102.

  65. Yu. I. Manin, Gauge Fields and Complex Geometry [in Russian], Nauka, Moscow (1984).

    MATH  Google Scholar 

  66. O. V. Manturov, “Riemannian spaces with orthogonal and symplectic motion groups and irreducible rotation group,” Dokl. Akad. Nauk SSSR, 141, No. 5, 1034–1037 (1961).

    MathSciNet  Google Scholar 

  67. O. V. Mel’nikov, V. N. Remeslennikov, V. A. Roman’kov, L. A. Skornyakov, and I. P. Shestakov, General Algebra [in Russian], Vol. 1, Nauka, Moscow (1990).

    MATH  Google Scholar 

  68. A. S. Mishchenko and A. T. Fomenko, Differential Geometry and Topology [in Russian], MGU, Moscow (1980).

    MATH  Google Scholar 

  69. Kam-Ping Mok, “Complete lifts of tensor fields and connections to the frame bundle,” Proc. London Math. Soc., 38, No. 1, 72–88 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  70. Kam-Ping Mok, “Lifts of vector fields to tensor bundles,” Geom. Dedic., 8, No. 1, 61–67 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  71. M. V. Morgun and A. Ya. Sultanov, “On Lie algebras of vector fields of holomorphic linear connection realizations,” Izv. Vuzov, Mat., No. 4, 59–65 (2008).

    Google Scholar 

  72. A. Morimoto, “Prolongation of connections to tangent bundles of higher order,” Nagoya Math. J., 40, 99–120 (1970).

    MATH  MathSciNet  Google Scholar 

  73. A. Morimoto, “Liftings of some types of tensor fields and connections to tangent bundles of p v-velocities,” Nagoya Math. J., 40, 13–31 (1970).

    MATH  MathSciNet  Google Scholar 

  74. A. Morimoto, “Prolongation of connections to tangent bundles of near points,” J. Different. Geom., 11, No. 4, 479–498 (1976).

    MATH  MathSciNet  Google Scholar 

  75. J. Munoz, J. Rodriguez, and F. Muriel, “Weil bundles and jet spaces,” Czech. Math. J., 50, No. 4, 721–748 ( 2000).

    Article  MATH  MathSciNet  Google Scholar 

  76. R. Narasimhan, Analysis on Real and Complex Manifolds [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  77. K. Nomizu, Lie Groups and Differential Geometry [Russian translation], IL, Moscow (1960).

  78. A. P. Norden, Affine Connection Spaces [in Russian], Nauka, Moscow (1976).

    MATH  Google Scholar 

  79. A. P. Norden, “On complex representation of tensors of Lorenz space,” Izv. Vuzov, Mat., No. 1, 156–163 (1959).

  80. A. P. Norden, “On a certain class of four-dimensional A-spaces,” Izv.Vuzov, Mat., No. 4, 145–157 (1960).

  81. A. P. Norden, “Biaxial geometry and its generalizations,” In: Proceedings of the 4th All-Union Mathematical Congress [in Russian], Vol. 2, Nauka, Leningrad (1964), pp. 236–243.

  82. A. P. Norden, “Congruences and dual numbers,” In: Proceedings of a Workshop of the Geometry Chair [in Russian], No. 3, Kazan’ University, Kazan’ (1968), pp. 82–98.

  83. A. P. Norden and G. G. Markov, “On holomorphically projective transformations,” Izv. Vuzov, Mat., No. 6, 82–87 (1975).

    Google Scholar 

  84. R. Pearce, Associative Algebras [Russian translation], Mir, Moscow (1986).

    Google Scholar 

  85. A. Z. Petrov, New Methods in General Relativity Theory [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  86. A. S. Podkovyrin, “Hypersurfaces of unitary space. I,” Izv. Vuzov, Mat., No. 8, 41–51 (1967).

  87. A. S. Podkovyrin, “Hypersurfaces of unitary space. II,” Izv. Vuzov, Mat., No. 9, 75–85 (1967).

  88. L. S. Pontryagin, Continuous Groups [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  89. M. M. Postnikov, Lectures on Geometry. Semester V. Lie Groups and Algebras [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  90. P. K. Rashevskii, “Symmetric affine connection spaces,” Trudy Sem. Vekt. Tenz. Anal., No. 8, 82–92 (1950).

  91. P. K. Rashevskii, Riemannian Geometry and Tensor Analysis [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  92. B. A. Rozenfel’d, Non-Euclidean Geometries [in Russian], Gostekhizdat, Moscow (1955).

    Google Scholar 

  93. A. A. Salimov, “A note on almost integrability of a structure,” Izv. Vuzov, Mat., No. 12, 70–71 (1985).

  94. B. N. Shapukov, “Structure of tensor bundles. I,” Izv. Vuzov, Mat., No. 5, 63–73 (1979).

  95. B. N. Shapukov, “Connections on a differentiable bundle,” In: Proceedings of a Geometric Workshop [in Russian], No. 12, Kazan’ University, Kazan’ (1980), pp. 97–111.

  96. B. N. Shapukov, “Structure of tensor bundles. II,” Izv. Vuzov, Mat., No. 9, 56–63 (1981).

  97. B. N. Shapukov, “Lie derivatives in fiber spaces,” In: Proceedings of a Geometric Workshop [in Russian], No. 13, Kazan’ University, Kazan’ (1081), pp. 90–101.

  98. B. N. Shapukov, “Automorphisms of fiber spaces,” In: Proceedings of a Geometric Workshop [in Russian], No. 14, Kazan’ University, Kazan’ (1982), pp. 97–108.

  99. B. N. Shapukov, “Lie derivatives in vector and tensor bundles,” In: Proceedings of a Geometric Workshop [in Russian], No. 15, Kazan’ University, Kazan’ (1983), pp. 84–93.

  100. B. N. Shapukov, “Connections on differentiable bundles,” In: Progress in Science and Technology, Series on Problems of Geometry [in Russian], 15, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1983), pp. 61–93.

  101. B. N. Shapukov, “Lie derivatives of fiber manifolds,” In: Progress in Science and Technology, Series on Contemporary Mathematics and Its Applications, Thematical Surveys [in Russian], 73, All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (2002), pp. 103–134.

  102. B. N. Shapukov, “Bundles of a non-Euclidean 3-space of hyperbolic type generated by an antiquaternion algebra,” Uch. Zap. Kazan. Univ. Ser. Fiz.-Mat. N.,, 147, Book 1, 181–191 (2005).

    Google Scholar 

  103. A. P. Shirokov, “Structures on differentiable manifolds,” In: Progress in Science and Technology, Series on Algebra, Topology, and Geometry [in Russian], All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1969), pp. 127–188.

  104. A. P. Shirokov, “Structures on differentiable manifolds,” In: Progress in Science and Technology, Series on Algebra, Topology and Geometry, 11, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1974), pp. 153–208.

  105. A. P. Shirokov, “A note on structures in tangent bundles,” In: Proceedings of a Geometric Workshop [in Russian], 5, All-Union Institute for Scientific and Technical Information, Moscow (1974), pp. 311–318.

  106. A. P. Shirokov, “On tangent bundles of a one-dimensional space over an algebra,” In: Proceedings of a Geometric Workshop [in Russian], No. 10, Kazan’ University, Kazan’ (1978), pp. 113–120.

  107. A. P. Shirokov, “Geometry of tangent bundles and spaces over algebras,” In: Progress in Science and Technology, Series on Problems of Geometry [in Russian], 12, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1981), pp. 61–96.

  108. P. A. Shirokov, “Constant fields of vectors and second-order tensors in Riemannian spaces,” Izv. Fiz.-Mat. Obshch. KGU. Ser. 2, 25, 86–114 (1925).

    Google Scholar 

  109. P. A. Shirokov, Tensor Calculus [in Russian], Kazan’ University, Kazan’ (1961).

  110. P. A. Shirokov, Selected Works in Geometry [in Russian], Kazan’ University, Kazan’ (1966).

  111. V. V. Shurygin, “Jet bundles as manifolds over algebras,” In: Progress in Science and Technology, Series on Problems of Geometry [in Russian], 19, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1987), pp. 3–22.

  112. V. V. Shurygin, “Structural equations of an A-affine frame bundle,” Izv. Vuzov, Mat., No. 12, 78-80 (1989).

  113. V. V. Shurygin, “Higher-order connections and lifts of geometric object fields,” Izv. Vuzov, Mat., No. 5, 96–104 (1992).

  114. V. V. Shurygin, “Manifolds over algebras and their applications in the geometry of jet bundles,” Usp. Mat. Nauk, 48, No. 2(290), 75–106 (1993).

    MathSciNet  Google Scholar 

  115. V. V. Shurygin, “On category of manifolds over algebras,” In: Proceedings of a Geometric Workshop [in Russian], No. 22, Kazan’ University, Kazan’ (1994), pp. 107–122.

  116. V. V. Shurygin, “Smooth manifolds over local algebras and Weil manifolds,” In: Progress in Science and Technology, Series on Contemporary Mathematics and Its Applications, Thematical Surveys [in Russian], 73, All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (2002), pp. 162-236.

  117. V. V. Shurygin, “On the structure of complete manifolds over local Weil algebras,” Izv. Vuzov, Mat., No. 11, 88–97 (2003).

  118. N. S. Sinyukov, Geodesic Mappings of Riemannian Spaces [in Russian], Nauka, Moscow (1979).

    MATH  Google Scholar 

  119. N. S. Sinyukov, A. Ya. Sultanov, and A. P. Shirokov, “Ivan Petrovich Egorov (on his 70th Anniversary),” Usp. Mat. Nauk, 45, No. 4(274), 177–178 (1990).

    MathSciNet  Google Scholar 

  120. L. B. Smolyakova, “On holonomy representations of manifolds modeled by modules over a Weil algebra,” In: Proceedings of a Geometric Workshop [in Russian], No. 24, Kazan’ University, Kazan’ (2003), pp. 129–138.

  121. L. B. Smolyakova, “Structure of complete radiant manifolds modeled by modules over a Weil algebra,” Izv. Vuzov, Mat., No. 5, 76–83 (2004).

  122. L. B. Smolyakova and V. V. Shurygin, “An analog of the Vaisman-Molino cohomology for manifolds modeled on some types of modules over Weil algebras and its application,” Lobachevskii J. Math., 9, 55–75 (2001).

    MATH  MathSciNet  Google Scholar 

  123. A. Ya. Sultanov, “Prolongations of Riemannian metrics from a differential manifold to its linear frame bundle,” Izv. Vuzov, Mat., No. 6, 93–102 (1992).

  124. A. Ya. Sultanov, “Infinitesimal transformations of a linear frame bundle with complete lift connection,” In: Proceedings of a Geometric Workshop [in Russian], No. 22, Kazan’ University, Kazan’ (1994), pp. 78–88.

  125. A. Ya. Sultanov, “On certain geometric structures in jet bundles of differential mappings,” Izv. Vuzov, Mat., No. 7, 51–64 (1995).

  126. A. Ya. Sultanov, “Infinitesimal transformations of a linear frame bundle with complete lift connection,” Izv. Vuzov, Mat., No. 2, 53–58 (1996).

  127. A. Ya. Sultanov, “Decomposition of a jet bundle of differentiable mappings into a Whitney sum of tangent bundles,” In: Proceedings of a Geometric Workshop [in Russian], No. 23, Kazan’ University, Kazan’ (1997), pp. 139–148.

  128. A. Ya. Sultanov, “Prolongations of tensor fields and connections to Weil bundles,” Izv. Vuzov, Mat., No. 9, 64–72 (1999).

  129. A. Ya. Sultanov, “On linear connections in Weil bundles,” In: Invariant Lifts of Study on Manifolds of Structures of Geometry, Analysis, and Mathematical Physics, Materials of the International Conference Devoted to the 90th Anniversary of the birth of G. F. Laptev [in Russian], MGU, Moscow (1999), pp. 48–60.

  130. A. Ya. Sultanov, “On maximal dimension of affine transformation groups of a Weil bundle with synectic connection,” In: Function Theory, Its Applications, and Related Problems, Materials of the Workshop-Conference Devoted to the 130th Anniversary of the birth of D. F. Egorov [in Russian], Kazan’ (1999), pp. 215–216.

  131. A. Ya. Sultanov, “Infinitesimal affine transformations in first-order Weil bundles with horizontal lift connection,” In: Motions in Generalized Spaces [in Russian], Penza (1999), pp. 142–149.

  132. A. Ya. Sultanov, “On linear connections over algebras,” In: Works of Participants of the International School-Workshop in Geometry and Analysis Devoted to the Memory of N. V. Efimov, Abrau-Dyurso, September 5–11 [in Russian], Rostov-on-Don (2000), p. 68.

  133. A. Ya. Sultanov, “On maximal dimension of intransitive motion groups of affine connection spaces,” In: Motions in Generalized Spaces [in Russian], Penza (2000), pp. 79–80.

  134. A. Ya. Sultanov, “On decomposition of infinitesimal affine transformations of Whitney sums of tangent bundles,” In: Proceedings of the N. I. Lobachevskii Mathematical Center, Topical Problems of Mathematics and Mechanics [in Russian], 5, Kazan’ (2000), pp. 200–201.

  135. A. Ya. Sultanov, “On intransitive motion groups of affine connection spaces,” In: Differential Geometry of Figure Manifolds [in Russian], No. 32, Kaliningrad (2001), pp. 101–104.

  136. A. Ya. Sultanov, “On affine transformations of Weil bundles with synectic connection,” In: Materials of the 4th International Conference in Geometry and Topology [in Russian], Cherkassy (2001), pp. 99–100.

  137. A. Ya. Sultanov, “Derivations of linear algebras of special type,” In: Motions in Generalized Spaces [in Russian], Penza (2002), pp. 206–215.

  138. A. Ya. Sultanov, “On motion groups of affine connection spaces with torsion,” In: Differential Geometry of Figure Manifolds [in Russian], No. 33, Kaliningrad (2002), pp. 100–103.

  139. A. Ya. Sultanov, “Affine transformations of linear connection manifolds and automorphisms of linear algebras,” Izv. Vuzov, Mat., No. 11, 77–81 (2003).

  140. A. Ya. Sultanov, “On infinitesimal automorphisms of bundles,” In Materials of the 5th International Conference in Geometry and Topology Devoted to the Memory of A. V. Pogorelov (1919–2002) [in Russian], Cherkassy (2003), pp. 137–138.

  141. A. Ya. Sultanov, “On the dimension of a derivation algebra of a linear algebra with unit,” In: International Conference in Analysis and Geometry, A Collection of Works [in Russian], Penza (2003), pp. 110–112.

  142. A. Ya. Sultanov, “On Whitney sum of Weil bundles,” In: Works of Participants of the International School-Workshop in Geometry and Analysis Devoted to the Memory of N. V. Efimov, Abrau-Dyurso, September 5–11, 2004 [in Russian], Rostov-on-Don (2004), p. 57.

  143. A. Ya. Sultanov, “Actions of Weil algebra automorphisms on Weil bundles,” Uch. Zap. Kazan. Univ., 147, Ser. Fiz.-Mat. N., Book 1, 159–172 (2005).

    Google Scholar 

  144. A. Ya. Sultanov, “On derivations of freeWeil algebras and their representations on Weil bundles,” A Collection of Works of the G. F. Laptev International Geometric Workshop [in Russian], Penza (2004), pp. 128–137.

  145. A. Ya. Sultanov, “Horizontal lifts of linear connections on second-order Weil bundles,” In: Differential Geometry of Figure Manifolds [in Russian], No. 36, Kaliningrad (2005), pp. 133–140.

  146. A. Ya. Sultanov, “On derivations of linear algebras,” In: Motions in Generalized Spaces [in Russian], Penza (2005), pp. 111–136.

  147. A. Ya. Sultanov, “Holomorphic affine vector fields on Weil bundles,” In: Works of Participants of the International School-Workshop in Geometry and Analysis Devoted to the Memory of N. V. Efimov, Abrau-Dyurso, September 5–11, 2006 [in Russian], Rostov-on-Don (2006), p. 90.

  148. A. Ya. Sultanov, “On dimensions of Lie algebras of holomorphic affine vector fields of affine connection spaces over an algebra,” In: Differential Geometry of Figure Manifolds [in Russian], No. 37, Kaliningrad (2006), pp. 164–168.

  149. A. Ya. Sultanov, “On Lie algebra of holomorphic affine vector fields of aWeil bundle with natural lift connection,” In: Abstracts of Reports of International Conference ‘Geometry in Odessa-2006’ [in Russian], Odessa (2006), p. 152.

  150. A. Ya. Sultanov, “On Weil algebras,” In: Modern Problems of Geometry and Mechanics of Deformable Bodies, Abstracts of Reports of a Regional Scientific Conference, October 19–20, 2006 [in Russian], Cheboksary (2006), pp. 38–39.

  151. A. Ya. Sultanov, “OnWeil algebras with additional conditions,” Vestn. N. Ya. Yakovlev Chuvash. Gos. Ped. Univ., No. 5, 172–174 (2006).

  152. A. Ya. Sultanov, “On real dimensions of Lie algebras of holomorphic affine vector fields,” Izv. Vuzov, Mat., No. 4, 54–67 (2007).

  153. A. Ya. Sultanov, “On real realization of a holomorphic linear connection over an algebra,” In: Differential Geometry of Figure Manifolds [in Russian], No. 38, Kaliningrad (2007), pp. 136–139.

  154. A. Ya. Sultanov and N. S. Sultanova, “Automorphism group of Weil algebras and their actions on Weil bundles,” In: Materials of the International Scientific Conference ‘Topical problems of Mathematics and Mechanics Devoted to the 200th Anniversary of Kazan’ State University and the 70th Anniversary of the N. G. Chebotarev Scientific-Research Institute of Kazan’ State University. Proceedings of the N. I. Lobachevskii Mathematical Center [in Russian], 25, Kazan’ (2004), pp. 252–253.

  155. N. V. Talantova and A. P. Shirokov, “A note on a certain metric in a tangent bundle,” Izv. Vuzov, Mat., No. 6, 143–146 (1975).

    Google Scholar 

  156. V. V. Trofimov and A. T. Fomenko, “Riemannian geometry,” In: Progress in Science and Technology, Series on Contemporary Mathematics and Its Applications, Thematical Surveys [in Russian], 76, All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (2002), pp. 3–262.

  157. V. V. Vagner, “Theory of composite manifolds,” Trudy Sem. Vekt. Tenz. Anal., No. 8, 11–72 (1950).

  158. V. V. Vagner, “Algebraic theory of differential groups,” Dokl. Akad. Nauk SSSR, 80, No. 6, 845–848 (1951).

    MATH  MathSciNet  Google Scholar 

  159. V. V. Vagner, “Algebraic theory of higher-order tangent spaces,” Trudy Sem. Vekt. Tenz. Anal., No. 10, 31–88 (1956).

  160. A. M. Vasil’ev, Theory of Differential-Geometric Structures [in Russian], MGU, Moscow (1987).

    MATH  Google Scholar 

  161. A. M. Vasil’ev, “Invariant affine connections in homogeneous spaces,” Dokl. Akad. Nauk SSSR, 131, No. 3, 489–492 (1960).

    Google Scholar 

  162. E. B. Vinberg, “On invariant linear connections,” Dokl. Akad. Nauk SSSR, 128, No. 4, 653–654 (1959).

    MATH  MathSciNet  Google Scholar 

  163. E. B. Vinberg, “Invariant linear connections in homogeneous space,” Trudy Mosk. Mat. Obshch., 9, 191–210 (1960).

    MathSciNet  Google Scholar 

  164. V. V. Vishnevskii, Spaces over Algebras Defined by Affinors [in Russian], Doctorial Dissertation in Physics and Mathematics, Kazan’ University, Kazan’ (1972).

  165. V. V. Vishnevskii, “On real realizations of tensor operations in spaces over algebras,” Izv. Vuzov, Mat., No. 5, 62–65 (1974).

  166. V. V. Vishnevskii, “Manifolds over plural numbers and semitangent structures,” In: Progress in Science and Technology, Series on Problems of Geometry [in Russian], 20, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1988), pp. 35–75.

  167. V. V. Vishnevskii, “Lifts of differential-geometric structures to higher-order semitangent bundles,” Izv. Vuzov, Mat., No. 5, 16–24 (1995).

    Google Scholar 

  168. V. V. Vishnevskii, “Integrable affine structures and their plural interpretations,” In: Progress in Science and Technology, Series on Contemporary Mathematics and Its Applications, Thematical Surveys [in Russian], 73, All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (2002), pp. 5–64.

  169. V. V. Vishnevskii, “A manifold with a pair of commutative semitangent structures,” Izv. Vuzov, Mat., No. 6, 1–8 (1998).

  170. V. V. Vishnevskii and T. A. Panteleeva, “Holomorphic prolongations of objects to second-order semitangent bundle,” Izv. Vuzov, Mat., No. 9, 3–10 (1985).

    Google Scholar 

  171. V. V. Vishnevskii, A. P. Shirokov, and V. V. Shurygin, Spaces over Algebras [in Russian], Kazan’ University, Kazan’ (1984).

  172. A. Weil, “Theorie des points proches sur les varietetes differentiables,” Colloq. Internat. Centre Nat. Rech. Sci., 52, Strasbourg (1953).

  173. R.Wells, Differential Calculus on Complex Manifolds [Russian translation], Mir, Moscow (1976).

    Google Scholar 

  174. K. Yano, The Theory of Lie Derivatives and Its Applications, North-Holland, Amsterdam (1957).

    MATH  Google Scholar 

  175. K. Yano and S. Ishihara, “Fibered spaces and projectable tensor fields,” Perspect. Geom. Relat., Indiana Univ. Press, Bloomington–London (1966).

  176. K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, New York (1973).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Penza State Pedagogical University, Penza, Russia

    A. Ya. Sultanov

Authors
  1. A. Ya. Sultanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Ya. Sultanov.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 123, Geometry, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sultanov, A.Y. Derivations of linear algebras and linear connections. J Math Sci 169, 362–412 (2010). https://doi.org/10.1007/s10958-010-0053-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-010-0053-4

Keywords

Search

Navigation