Affine transformations in bundles. (English. Russian original) Zbl 1509.53004

Sultanov, A. Ya.; Monakhova, O. A.

J. Math. Sci., New York 245, No. 5, 601-643 (2020); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 146, 48-88 (2018).

Summary: This paper is a review of results of studies of affine transformations in generalized spaces over real linear algebras over the past 15–20 years.

MSC:

53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
53B05	Linear and affine connections
53B15	Other connections

Keywords:

affine transformation; generalized space over an algebra; smooth manifold; connection; torsion

References:

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[135]	A. Ya. Sultanov, “On the dimension of the algebra of differentiations of a unital linear algebra,” in: Proc. Int. Conf. On Geometry and Analysis [in Russian], Penza (2003), pp. 110-112.
[136]	A. Ya. Sultanov, “On the Whitney sum of Weil bundles,” in: Proc. Int. Conf. on Geometry and Analysis Dedicated to the Memory of Professor N. V. Efimov, Abrau-Dyurso, Sept. 5-11, 2004 [in Russian], Rostov-on-Don (2004), p. 57.
[137]	A. Ya. Sultanov, “On differentiations of free Weil algebras and their representations on Weil bundles,” Tr. Geom. Semin. G. F. Lapteva, 128-137 (2004).
[138]	Sultanov, AY, Actions of automorphism groups of Weil algebras on Weil bundles, Uch. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki, 147, 1, 159-172 (2005) · Zbl 1206.58003
[139]	Sultanov, AY, Horizontal lifts of linear connections on second-order Weil bundles, Differ. Geom. Mnogoobr. Figur, 36, 133-140 (2005) · Zbl 1384.58005
[140]	A. Ya. Sultanov, “On differentiations of linear algebras,” in: Motions in Generalized Spaces [in Russian], Penza (2005), pp. 111-136.
[141]	A. Ya. Sultanov, “Holomorphic affine vector fields on Weil bundles,” Proc. Int. Conf. on Geometry and Analysis Dedicated to the Memory of Professor N. V. Efimov, Abrau-Dyurso, Sept. 5-11, 2006, Rostov-on-Don (2006), p. 90. · Zbl 1283.53017
[142]	Sultanov, AY, On dimensions of Lie algebras of holomorphic affine vector fields in affinely connected spaces over algebras, Differ. Geom. Mnogoobr. Figur, 37, 164-168 (2006) · Zbl 1270.53009
[143]	A. Ya. Sultanov, “On the Lie algebra of holomorphic affine vector fields on Weil bundles with connections of natural lifts,” in: Proc. Int. Conf. “Geometry on Odessa-2006” [in Russian], Odessa (2006), p. 152. · Zbl 1270.53009
[144]	A. Ya. Sultanov, “On Weil algebras,” in: Proc. Int. Conf. “Modern Problems in Geometry and Mechanics of Deformable Bodies” [in Russian], Cheboksary (2006), pp. 38-39.
[145]	Sultanov, AY, On Weil algebras with additional conditions, Vestn. Chuvash. Pedagog. Inst., 5, 172-174 (2006)
[146]	Sultanov, AY, On real dimensions of Lie algebras of holomorphic affine vector fields, Izv. Vyssh. Ucheb. Zaved. Mat., 4, 54-67 (2007) · Zbl 1226.17018
[147]	Sultanov, AY, On the real realization of a holomorphic linear connection over an algebra, Differ. Geom. Mnogoobr. Figur, 38, 136-139 (2007) · Zbl 1270.53041
[148]	Sultanov, AY; Moshin, AY, On the Whitney sum of Weil bundles, Tr. Inst. Sistem. Anal. Ross. Akad. Nauk, 31, 1, 215-223 (2007)
[149]	Sultanov, AY; Mukhin, AV, Decomposition of Weil bundles into Whitney sums, Tr. Inst. Sistem. Anal. Ross. Akad. Nauk, 31, 1, 224-229 (2007)
[150]	A. Ya. Sultanov, “Linear connections in the module of differentiations of an algebra,” Tr. Geom. Semin. G. F. Lapteva, 78-109 (2007).
[151]	Sultanov, AY, On Lie algebras of holomorphic affine vector fields on Weil bundles, Uch. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki, 151, 4, 171-177 (2009) · Zbl 1216.53021
[152]	Sultanov, AY, Derivations of linear algebras and linear connections, J. Math. Sci., 169, 3, 362-412 (2010) · Zbl 1218.53027
[153]	Sultanov, AY, On automorphism groups of special linear algebras, Izv. Penz. Gos. Pedagog. Inst., 18, 22, 70-74 (2010)
[154]	Sultanov, AY, On algebras of differentiations of linear algebras of maximal dimension, Izv. Penz. Gos. Pedagog. Inst., 18, 22, 75-77 (2010)
[155]	Sultanov, AY, On Lie algebras of holomorphic affine vector fields and holomorphic linear connections with symmetric Ricci fields, Vestn. Tatar. Gos. Guman.-Pedagog. Inst., 1, 23, 41-45 (2011)
[156]	A. Ya. Sultanov, “On affine differentiations in the module of differentiations of the algebra of polynomials,” in: Proc. VII Int. Symp. “Fundamental and Applied Scientific Problems” [in Russian], Vol. 1, Moscow (2012), pp. 16-22.
[157]	Sultanov, AY, Holomorphic affine vector fields on Weil bundles, Mat. Zametki, 91, 6, 896-899 (2012) · Zbl 1283.53017
[158]	Sultanov, AY; Morgun, MV, On Lie algebras of vector fields of real realizations of holomorphic linear connection, Izv. Vyssh. Ucheb. Zaved. Mat., 4, 59-65 (2008)
[159]	Sultanov, AY; Sultanova, GA, Estimate of dimensions of Lie algebras of infinitesimal affine transformations of the tangent bundles T(M) with connections of total lift, Uch. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki, 156, 2, 43-54 (2014) · Zbl 1353.58001
[160]	Sultanov, AY; Sultanova, NS, On affine automorphisms of locally trivial bundles, Differ. Geom. Mnogoobr. Figur, 34, 136-140 (2003) · Zbl 1158.53318
[161]	A. Ya. Sultanov and N. S. Sultanova, “Automorphism groups of Weil algebras and their actions on Weil bundles,” Actual Problems im Mathematics and Mechanics. Proc. N. I. Lobachevsky Math. Center [in Russian], 25, Kazan (2004), pp. 252-253.
[162]	Sultanova, GA, Some lifts of tensor fields of the type (1, r) from the base to the tangent bundle, Izv. Vyssh. Ucheb. Zaved. Povolzh. Region. Fiz.-Mat. Nauki, 1, 29, 54-64 (2014)
[163]	Sultanova, GA, On motion groups in tangent bundles with connections of total lifts over two-dimensional maximally mobile affinely connected spaces, Differ. Geom. Mnogoobr. Figur, 46, 153-161 (2015) · Zbl 1376.53037
[164]	Sultanova, GA, On the estimate of dimensions of Lie algebras of infinitesimal automorphisms of tangent bundles with connections of total lifts over non-projectively-Euclidean bases, Differ. Geom. Mnogoobr. Figur, 47, 146-153 (2016) · Zbl 1397.53028
[165]	Sultanova, GA, On dimensions of Lie algebras of automorphisms in tangent bundles with connections of total lifts over projectively-Euclidean bases, Dalnevost. Mat. Zh., 16, 1, 83-95 (2016) · Zbl 1375.53026
[166]	N. S. Sultanova, “Infinitesimal affine transformations of cotangent bundles with connections of horizontal lifts,” in: Motions in Generalized Spaces [in Russian], Penza (1999), pp. 150-156.
[167]	N. S. Sultanova, “On affine transformations of cotangent bundles over maximally mobile affinely connected spaces,” in: Motions in Generalized Spaces [in Russian], Penza (2000), pp. 162-167.
[168]	N. S. Sultanova, “Natural extensions of diffeomorphisms from a smooth manifold to its cotangent bundle,” in: Motions in Generalized Spaces [in Russian], Penza (2002), pp. 215-220.
[169]	Sultanova, NS, On the dimension of the total motion group of the cotangent bundle with the connection of horizontal lift, Differ. Geom. Mnogoobr. Figur, 33, 104-107 (2002) · Zbl 1063.53018
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[171]	Vishnevsky, VV, Integrable affine structures and their plural interpretation, Itogi Nauki Tekh. Sovr. Mat. Prilozh. Temat. Obzory, 5-64 (2002), Moscow: VINITI, Moscow
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