REFERENCES
B. N. Apanasov, Geometry of Discrete Groups and Manifolds [in Russian], Nauka, Moscow (1991).
M. F. Atiyah, "Complex analytic connections in fibre bundles," Trans. Amer. Math. Soc, 85, 181-207 (1957).
I. V. Bel'ko, "Atiyah-Molino class of a fiber Lie algebroid," Dokl. Akad. Nauk Belarusi, 37, No. 5, 16-18 (1993).
I. N. Bernshtein and B. I. Rozenfel'd, "Homogeneous spaces of infinite-dimensional Lie algebras and characteristic classes of foliations," Usp. Mat. Nauk, 28, No. 4, 103-138 (1973).
R. A. Blumenthal and J. J. Hebda, "Ehresmann connections for foliations," Indiana Math. J., 33, No. 4, 597-612 (1984).
R. A. Blumenthal and J. J. Hebda, "Complementary distributions which preserve the leaf geometry and applications to totally geodesic foliations," Quart. J. Math., 35, No. 4, 597-612 (1984).
R. A. Blumenthal and J. J. Hebda, "An analogue of the holonomy bundle for a foliated manifold," Tôhoku Math. J., 40, No. 2, 189-197 (1988).
A. V. Boyarshinova, "On the space of essential infinitesimal deformations," Izv. Vuzov, Mat., No. 8, 3-12 (1997).
A. V. Boyarshinova, "On the space of essential infinitesimal deformations of a manifold over the algebra of k-plural numbers," In: Tr. Geom. Sem., No. 23, Kazan Univ., Kazan (1997), pp. 7-21.
N. Bourbaki, Algebra, Chapter X. Homological Algebra [Russian translation], Nauka, Moscow (1987).
F. Brickell and R. S. Clark, "Integrable almost tangent structures," J. Differ. Geom., 9, No. 4, 557-563 (1974).
M. Crampin, "Defining Euler-Lagrange fields in terms of almost tangent structures," Phys. Lett., 97A, 466-468 (1983).
M. Crampin and G. Thompson, "Affine bundles and integrable almost tangent structures," Math. Proc. Cambridge Phil. Soc., 98, 61-71 (1985).
P. Dazord and G. Hector, "Integration symplectique des variét´es de Poisson totalement asphériques," In: Symplectic Geometry, Groupoids, and Integrable Systems. Séminaire Sud Rhodanien à Berkeley, 1989, Springer Math. SKIP 20 (1991), pp. 38-72.
S. De Filippo, G. Landi, G. Marmo, and G. Vilasi, "Tensor fields defining a tangent bundle structure," Ann. Inst. H. Poincaré, 50, No. 2, 205-218 (1989).
A. Dold, Lectures on Algebraic Topology [Russian translation], Mir, Moscow (1976).
M. Doupovec and I. Kolář, "Natural affinors on time-dependent Weil bundles," Arch. Math., 27b, 205-209 (1991).
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Methods and Applications [in Russian], Nauka, Moscow (1979).
T. V. Duc, "Structures presque-transverse," J. Differ. Geom., 14, No. 2, 215-219 (1979).
D. J. Eck, "Product-preserving functors on smooth manifolds," J. Pure Appl. Algebra, 42, 133-140 (1986).
C. Ehresmann, "Les connexions in.nitésimales dans un espace fibré différentiable," In: Colloq. de Topologie, Bruxelles, 1950, Paris (1951), pp. 29-55.
C. Ehresmann, "Les prolongements d'une variété différentiable, I. Calcul des jets, prolongement principal," C. R. Acad. Sci., 233, No. 11, 598-600 (1951).
C. Ehresmann, "Les prolongements d'une variété différentiable, II. L'espace des jets d'ordre r de Vn dans Vm," C. R. Acad. Sci., 233, No. 15, 777-779 (1951).
C. Ehresmann, "Extension du calcul des jets aux jets non holonomes," C. R. Acad. Sci., 239, No. 25, 1762-1764 (1954).
C. Ehresmann, "Sur les connexions d'ordre supérior," In: Atti del V Congresso dell'Unione Math. Italiana, Roma (1955), pp. 326-328.
C. Ehresmann, "Applications de la notion de jet non holonome," C. R. Acad. Sci., 240, No. 4, 397-399 (1955).
T. I. Gaisin, "The Spencer complex for manifolds over algebras," In: Tr. Geom. Sem., No. 23, Kazan Univ., Kazan (1997), pp. 33-41.
A. El Kasimi-Alaoui, "Sur la cohomologie feuilletée," Compos. Math., 49, 195-215 (1983).
A. El Kasimi-Alaoui and A. Tihami, "Cohomologie bigradu´ee de certains feuilletages," Bull. Soc. Math. Belg., 38, Ser. B, 144-156 (1986).
L. E. Evtushik, "Differential connections and infinitesimal transformations of a prolonged pseudogroup," In: Tr. Geom. Sem., Vol. 2, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1966), pp. 119-150.
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 in Geometry [in Russian], Vol. 9, All-Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1979).
D. Fried, W. Goldman, and M. W. Hirsch, "Affine manifolds with nilpotent holonomy," Comment. Math. Helv., 56, No. 4, 487-523 (1981).
J. Gancarzewicz, W. Mikulski, and Z. Pogoda, "Lifts of some tensor fields and connections to product preserving functors," Nagoya Math. J., 135, 1-41 (1994).
V. Guillemin and S. Sternberg, "Deformation theory of pseudogroup structures," Mem. Amer. Math. Soc., No. 64 (1966).
J. L. Heitsch, "A cohomology for foliated manifolds," Comment. Math. Helv., 50, 197-218 (1975).
R. Hermann, "On the differential geometry of foliations," Ann. Math. Helv., 72, 445-457 (1960).
T. Inaba, The tangentially affine structures of Lagrangian foliations and the tangentially projective structures of Legendrian foliations, Preprint.
G. Kainz and P. Michor, "Natural transformations in differential geometry," Czech. Math. J., 37, 584-607 (1987).
P. W. Ketchum, "Analytic functions of hypercomplex variables," Trans. Amer. Math. Soc., 30, 641-667 (1928).
F. W. Kamber and P. Tondeur, "Foliated bundles and characteristic classes," In: Lect. Notes Math., Vol. 493, Springer (1975).
S. Kobayashi, "Manifolds over function algebras and mapping spaces," Tôhoku Math. J., 41, No. 2, 263-282 (1989).
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry [Russian translation], Vol. 1, Nauka, Moscow (1981).
K. Kodaira, Complex Manifolds and Deformations of Complex Structures, Springer (1986).
I. Kolář, "On the torsion of spaces with connection," Czech. Math. J., 21, 124-136 (1971).
I. Kolář, "Covariant approach to natural transformations of Weil functors," Comment. Math. Univ. Carolinae, 27, 723-729 (1986).
I. Kolář, "An infinite dimensional motivation in higher order geometry," In: Proc. Conf. Diff. Geom. Appl. Aug. 28-Sept. 1, 1995, Brno, Czech Republic, Brno (1996), pp. 151-159.
I. Kolář, P. W. Michor, and J. Slovák, Natural Operations in Differential Geometry, Springer (1993).
A. Kriegl and P. W. Michor, "Product preserving functors of infinite dimensional manifolds," Arch. Math., 32, 289-306 (1996).
G. I. Kruchkovich, "Hypercomplex structures on manifolds, I," In: Tr. Sem. Vekt. Tens. Anal., No. 16, Moscow Univ., Moscow (1972), pp. 174-201.
G. I. Kruchkovich, "Hypercomplex structures on manifolds, II," In: Tr. Sem. Vekt. Tens. Anal., No. 17, Moscow Univ., Moscow (1974), pp. 218-227.
G. F. Laptev, "Main infinitesimal structures of higher order on a smooth manifolds," In: Tr. Geom. Sem., Vol. 1, All-Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1966), pp. 139-189.
M. de León, I. Méndes, and M. Salgado, "Integrable p-almost tangent manifolds and tangent bundles of p 1-velocities," Acta Math. Hung., 58, Nos. 1-2, 45-54 (1991).
M. V. Losik, "On the reduction theorem for connections of higher order," In: Differential Geometry [in Russian], No. 5, Saratov Univ., Saratov (1980), pp. 53-64.
O. O. Luciano, "Categories of multiplicative functors and Weil's infinitely near points," Nagoya Math. J., 109, 67-108 (1988).
M. A. Malakhal'tsev, "An analog of Dolbo cohomologies for manifolds over the algebra of dual numbers," Izv. Vuzov, Mat., No. 11, 82-84 (1990).
M. A. Malakhal'tsev, "Structures of a manifold over the algebra of dual numbers on the torus," In: Tr. Geom. Sem., No. 22, Kazan Univ., Kazan (1994,), pp. 47-62.
M. A. Malakhal'tsev, "(X,G)-foliations," Izv. Vuzov, Mat., No. 7, 55-65 (1996).
M. A. Malakhal'tsev, "Godbillon-Vei classes of a one-dimensional manifold over a local algebra," In: Tr. Geom. Sem., No. 23, Kazan Univ., Kazan (1997), pp. 65-76.
Yu. I. Manin, Gauge Fields and Complex Geometry [in Russian], Nauka, Moscow (1984).
W. M. Mikulski, "Natural transformations of Weil functors into bundle functors," Rend. Circ. Mat. Palermo, Sér. 2, 22, 177-191 (1989).
W. M. Mikulski, "Product preserving bundle functors on fibered manifolds," Arch. Math., 32, 307-316 (1996).
A. S. Mishchenko, Vector Bundles and Their Applications [in Russian], Nauka, Moscow (1984).
P. Molino, "Propriétés cohomologiques et propriétés topologiques des feuilletages à connexion transverse projectable," Topology, 12, 317-325 (1973).
P. Molino, "Théorie des G-structure: le problème d'equivalence," In: Lect. Notes Math., Vol. 588, Springer (1977).
P. Molino, "Actions de groupes de Lie et presque-connexions," In: Lect. Notes Math., Vol. 484, Springer (1975), pp. 153-161.
P. Molino, Riemannian Foliations, Birkhäuser (1988).
A. Morimoto, "Liftings of some types of tensor fields and connections to tangent bundles of P r-velocities," Nagoya Math. J., 40, 13-31 (1970).
A. Morimoto, "Prolongation of connections to bundles of infinitely near points," J. Differ. Geom., 11, No. 4, 479-498 (1976).
A. P. Norden, "On the parallel translation of dual vectors," Uch. Zap. Kazan Univ., 110, No. 3, 95-103 (1950).
A. P. Norden, "Bi-axial geometry and its generalization," In: Proc. 4th All-Russian Math. Session [in Russian], Vol. 2, Leningrad (1964), pp. 236-243.
A. P. Norden, "On the structure of a connection on the line manifold of a non-Euclidean space," Izv. Vuzov, Mat., No. 12, 84-94 (1972).
E. Okassa, "Prolongements des champs de vecteur à des variétés de points proches," C. R. Acad. Sci., Sér. 1, 300, No. 6, 173-176 (1985).
E. Okassa, "Relévements des structures symplectiques et pseudo-Riemanniennes à des variétés de points proches," Nagoya Math. J., No. 115, 63-71 (1989).
N. M. Ostianu, "Stepwise fiber spaces," In: Tr. Geom. Sem., Vol. 5, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1974), pp. 259-309.
M. H. Papatriantafillou, "Partitions of unit on A-manifolds," Int. J. Math., 9, No. 7, 877-883 (1998).
M. H. Papatriantafillou, "Connections on A-bundles," In: Proc. Symp. New Developments in Differential Geometry, Budapest 1996, Kluwer Academic, Dordrecht (1999) pp. 307-316.
L.-N. Patterson, "Connexions and prolongations," Can. J. Math., 27, No. 4, 766-791 (1975).
P. Pierce, Associative Algebras [Russian translation], Mir, Moscow (1986).
Z. Pogoda, "Horizontal lifts and foliations," Rend. Circ. Mat. Palermo, Ser. 2, 38, No. 21, 279-289 (1989).
P. K. Rashevskii, "Scalar field in a fiber space," In: Tr. Sem. Vekt. Tens. Anal., No. 6, Moscow Univ., Moscow (1948), pp. 225-248.
B. L. Reinhart, Differential Geometry of Foliations. The Fundamental Integrability Problem, Springer (1983).
G. Scheffers, "Verallgemeinerung der Grundlagen der gewöhnlichen komplexen Funktionen," Berichte Sächs. Akad. Wiss., 45, 828-842 (1893).
B. N. Shapukov, "Connections on vector bundles," In: Progress in Science and Technology, Series on Problems in Geometry [in Russian], Vol. 15, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1985), pp. 61-95.
A. P. Shirokov, "On the theory of spaces defined by commutative algebras," Uch. Zap. Kazan Univ., 125, No. 1, 165-182 (1965).
A. P. Shirokov, "On a certain type of G-structure definned by algebras," In: Tr. Geom. Sem., Vol. 1, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1966), pp. 425-456.
A. P. Shirokov, "On the problem on pure tensors and invariant subspaces in manifolds with almost algebraic structure," In: Proc. Dept. Geom. [in Russian], No. 2, Kazan Univ., Kazan (1966), pp. 81-89.
A. P. Shirokov, "A note on structures in tangent bundles," In: Tr. Geom. Sem., Vol. 5, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1974), pp. 311-318.
A. P. Shirokov, "Geometry of tangent bundles and spaces over algebras," In: Progress in Science and Technology, Series on Problems in Geometry [in Russian], Vol. 12, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1981), pp. 61-95.
V. V. Shurygin, "On the theory of differential groups of higher order," Izv. Vuzov, Mat., No. 11, 77-81 (1984).
V. V. Shurygin, "Equations of parallel translation of an A-jet," Izv. Vuzov, Mat., No. 9, 78-80 (1987).
V. V. Shurygin, "Jet bundles as manifolds over algebras," In: Progress in Science and Technology, Series on Problems in Geometry [in Russian], Vol. 19, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1987), pp. 3-22.
V. V. Shurygin, "Structural equations of the equation of the bundle of A-affine frames," Izv. Vuzov, Mat., No. 12, 78-80 (1989).
V. V. Shurygin, "Higher-order connections and lifts of fields of geometric objects," Izv. Vuzov, Mat., No. 5, 96-104 (1992).
V. V. Shurygin, "Manifolds over algebras and their applications in the geometry of jet bundles," Usp. Mat. Nauk, 48, No. 2, 75-106 (1993).
V. V. Shurygin, "On the category of manifolds over algebras," In: Tr. Geom. Sem., No. 22, Kazan Univ., Kazan (1994), pp. 107-122.
V. V. Shurygin, "On a certain geometric property of an object of torsion of a higher-order connection," Izv. Vuzov, Mat., No. 2, 71-74 (1994).
V. V. Shurygin, "Ehresmann connection for the canonical foliation over a local algebra," Mat. Zametki, 59, No. 2, 303-310 (1996).
V. V. Shurygin, "On cohomologies of manifolds over local algebras," Izv. Vuzov, Mat., No. 9, 74-88 (1996).
V. V. Shurygin, "Atiah-Molino classes of a smooth manifold over a local algebra A as obstructions to the prolongation of transversal connections to A-smooth connections, In: Tr. Geom. Sem., No. 23, Kazan Univ., Kazan (1997), pp. 199-210.
V. V. Shurygin, "Smooth connections and horizontal distributions on manifolds over local algebras," In: Proc. Conf. Diff. Geom. Appl. Aug. 28-Sept. 1, 1995. Brno, Czech Republic, Brno (1996), pp. 309-319.
V. V. Shurygin, "Structure of smooth mappings over Weil algebras and the category of manifolds over algebras," Lobachevskii J. Math., 5 (1999); www.ksn.ru/tat en/science/ljm
E. P. Shustova, "On the interrelation of geometries of the tangent bundle of the third order and the Whitney sum," In: Tr. Geom. Sem., No. 23, Kazan Univ., Kazan (1997), pp. 211-221.
J. Slovàk, "Prolongations of connections and sprays with respect to Weil functors," Rend. Circ. Mat. Palermo, 36, No. 14, 143-156 (1987).
E. Study, Geometrie der Dymamen, Leipzig (1902).
A. Ya. Sultanov, "Decomposition of the jet bundle of differentiable mappings into the Whitney sum of tangent bundles," In: Tr. Geom. Sem., No. 23, Kazan Univ., Kazan (1997), pp. 139-148.
A. Ya. Sultanov, "Prolongation of tensor fields and connections to Weil bundles," Izv. Vuzov, Mat., No. 9, 81-90 (1999).
G. Thompson and U. Schwardmann, "Almost tangent and cotangent structures in the large," Trans. Amer. Math. Soc., 327, No. 1, 313-327 (1991).
W. P. Thurston, The Geometry and Topology of 3-Manifolds, Princeton Univ. Lecture Notes (1978/1979).
I. Vaisman, "Variétés riemanniennes feuilletées," Czech. Math. J., 21, No. 1, 46-75 (1971).
I. Vaisman, "d f-Cohomology of Lagrangian foliations," Monat. Math., 106, No. 3, 221-244 (1988).
V. V. Vagner, "Theory of differential objects and foundations of differential geometry," In: O. Veblen and J. Whitehead, Foundations of Differential Geometry [Russian translation], IL, Moscow (1949).
V. V. Vagner, "Theory of a composition manifold," In: Tr. Sem. Vekt. Tens. Anal., No. 8, Moscow Univ., Moscow (1950), pp. 11-72.
V. V. Vagner, "Algebraic theory of differential groups," Dokl. Akad. Nauk SSSR, 80, No. 6, 845-848 (1951).
V. V. Vagner, "Algebraic theory of tangent spaces of higher order," In: Tr. Sem. Vekt. Tens. Anal., No. 10, Moscow Univ., Moscow (1956), pp. 31-88.
J. Vanžura, "On the geometry and topology of manifolds over algebras," Weiterbildungszentr. Math. Kybern. Rechentechn., Sect. Math., 28, 133-136 (1978).
O. Veblen and J. Whitehead, Foundations of Differential Geometry [Russian translation], IL, Moscow (1949). 293
V. V. Vishnevskii, "On a certain property of analytic functions over algebras and its applications to the study of complex structures in Riemannian spaces," In: Proc. Dept. Geom. [in Russian], No. 2, Kazan Univ., Kazan (1966), pp. 5-12.
V. V. Vishnevskii, "Polynomial algebras and affinor structures," In: Proc. Dept. Geom. [in Russian], No. 6, Kazan Univ., Kazan (1971), pp. 22-35.
V. V. Vishnevskii, "On real realizations of tensor operations in spaces over algebras," Izv. Vuzov, Mat., No. 5, 62-65 (1974).
V. V. Vishnevskii, "Manifolds over plural numbers and semi-tangent structures," In: Progress in Science and Technology, Series on Problems of Geometry [in Russian], Vol. 20, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1988), pp. 35-75.
V. V. Vishnevskii, "Lifts of differential-geometric structures to semi-tangent bundles of higher order," Izv. Vuzov, Mat., No. 5, 16-24 (1995).
V. V. Vishnevskii, A. P. Shirokov, and V. V. Shurygin, Spaces over Algebras [in Russian], Kazan Univ., Kazan (1984).
V. S. Vladimirov and I. V. Volovich, "Superanalysis. I. Differential calculus," Teor. Mat. Fiz., 59, No. 1, 3-27 (1984).
R. D. Wagner, "The generalized Laplace equations in a function theory for commutative algebras," Duke Math. J., 15, 455-461 (1948).
A. Weil, "Théorie des points proches sur les variététes différentiables," In: Colloq. Int. Centre nat. Rech. Sci., 52, Strasbourg (1953), pp. 111-117.
R. Wells, Differential Calculus on Complex Manifolds [Russian translation], Mir, Moscow (1976).
H. Winkelnkemper, "The graph of a foliation," Ann. Glob. Anal. Geom., 1, 51-75 (1983).
R. Woblak, "Normal bundles of foliations of order r," Demonstr. Math., 18, No. 4, 977-994 (1985).
R. Woblak, "Graphs, Ehresmann connections, and vanishing cycles," In: Proc. Conf. Diff. Geom. Appl. Aug. 28-Sept. 1, 1995, Brno, Czech Republic, Brno (1996), pp. 345-352.
P. C. Yuen, "Prolongements de G-structures aux espaces de prolongement," C. R. Acad. Sci., 270, No. 3, A538-A540 (1970).
P. C. Yuen, "Sur la notion d'une G-structure géométrique et les A-prolongements de G-structures," C. R. Acad. Sci., 270, No. 24, A1589-A1592 (1970).
N. I. Zhukova, "Graph of a foliation with an Ehresmann connection and stability of leaves," Izv. Vuzov, Mat., No. 2, 79-81 (1994).
Rights and permissions
About this article
Cite this article
Shurygin, V.V. Smooth Manifolds over Local Algebras and Weil Bundles. Journal of Mathematical Sciences 108, 249–294 (2002). https://doi.org/10.1023/A:1012848404391
Issue Date:
DOI: https://doi.org/10.1023/A:1012848404391