Geometrical methods in mathematical physics. Applications in quantum mechanics. Part~2
Lekcionnye kursy NOC, Geometrical methods in mathematical physics. Applications in quantum mechanics. Part 2 (2015), pp. 3-184.

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M. O. Katanaev. Geometrical methods in mathematical physics. Applications in quantum mechanics. Part~2. Lekcionnye kursy NOC, Geometrical methods in mathematical physics. Applications in quantum mechanics. Part 2 (2015), pp. 3-184. http://geodesic.mathdoc.fr/item/LKN_2015_a0/

[1] B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, Sovremennaya geometriya. Metody i prilozheniya, Nauka, M., 1998 | MR | Zbl

[2] S. P. Novikov, I. A. Taimanov, Sovremennye geometricheskie struktury i polya, MTsNMO, M., 2005

[3] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. 1, Interscience Publ., New York, 1963 ; S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. 2, Interscience Publ., New York, 1969 ; Sh. Kobayasi, K. Nomidzu, Osnovy differentsialnoi geometrii, T. 1, Nauka, M., 1981 ; Ш. Кобаяси, К. Номидзу, Основы дифференциальной геометрии, Т. 2, Наука, М., 1981 | MR | Zbl | MR | Zbl | Zbl | Zbl

[4] S. S. Chern, W. H. Chen, K. S. Lam, Lectures on Differential Geometry, World Sci., Singapore, 1999 | MR | Zbl

[5] V. I. Arnold, Obyknovennye diferentsialnye uravneniya, Nauka, M., 1975 | MR | Zbl

[6] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki, Editorial URSS, M., 2002

[7] V. S. Vladimirov, Uravneniya matematicheskoi fiziki, Nauka, M., 1988 | MR | Zbl

[8] I. M. Gelfand, Lektsii po lineinoi algebre, MTsNMO, M., 1998

[9] I. M. Gelfand, R. A. Minlos, Z. Ya. Shapiro, Predstavleniya gruppy vraschenii i gruppy Lorentsa, ikh primeneniya, Fizmatlit, M., 1958 | MR

[10] Yu. N. Drozhzhinov, B. I. Zavyalov, Vvedenie v teoriyu obobschennykh funktsii, Lekts. kursy NOTs, 5, MIAN, M., 2006 | DOI | DOI | Zbl

[11] V. V. Zharinov, Algebro-geometricheskie osnovy matematicheskoi fiziki, Lekts. kursy NOTs, 9, MIAN, M., 2008, 210 pp. | DOI | DOI | Zbl

[12] J. L. Kelley, General Topology, D. Van Nostrand Company, Toronto, 1957 ; Dzh. L. Kelli, Obschaya topologiya, Nauka, M., 1968 | MR | Zbl | MR | Zbl

[13] A. A. Kirillov, Elementy teorii predstavlenii, Nauka, M., 1978 | MR

[14] A. I. Kostrikin, Vvedenie v algebru. Chast I. Osnovy algebry, Nauka, M., 2000 | Zbl

[15] A. I. Kostrikin, Yu. I. Manin, Lineinaya algebra i geometriya, Nauka, M., 1986 | MR | Zbl

[16] M. M. Postnikov, Analiticheskaya geometriya, Nauka, M., 1979

[17] P. K. Rashevskii, Rimanova geometriya i tenzornyi analiz, Nauka, M., 1967 | MR | Zbl

[18] V. A. Rokhlin, D. B. Fuks, Nachalnyi kurs topologii. Geometricheskie glavy, Nauka, M., 1977 | MR | Zbl

[19] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1988 | MR | Zbl

[20] H. Whitney, “Differentiable manifolds”, Ann. of Math. (2), 37:3 (1936), 645–680 | DOI | MR | Zbl

[21] J. Munkres, “Obstructions to the smoothing of piecewise-differentiable homeomorphisms”, Ann. of Math. (2), 72:3 (1960), 521–554 | DOI | MR | Zbl

[22] J. Milnor, “On manifolds homeomorphic to the 7-sphere”, Ann. of Math. (2), 64:2 (1956), 394–405 | DOI | MR | Zbl

[23] S. K. Donaldson, P. B. Kronheimer, The Geometry of Four-Manifolds, The Clarendon Press, Oxford, 1990 | MR | Zbl

[24] M. A. Kervaire, “A manifold which does not admit any differentiable structure”, Comment. Math. Helv., 34 (1960), 257–270 | MR | Zbl

[25] T. Aubin, A Course in Differential Geometry, Grad. Stud. Math., 27, Amer. Math. Soc., Providence, RI, 2001 | MR | Zbl

[26] F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Grad. Texts in Math., 94, Springer-Verlag, Berlin, 1983 ; F. Uorner, Osnovy teorii gladkikh mnogoobrazii i grupp Li, Mir, M., 1987 | MR | Zbl | MR

[27] S. S. Chern, “Curves and surfaces in Euclidean space”, Studies in Global Geometry and Analysis, Math. Assoc. Amer., Englewood Cliffs, NJ, 1967, 16–56 | MR

[28] I. G. Petrovskii, Lektsii po teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1970 | MR | Zbl

[29] C. J. Isham, Modern Differential Geometry, World Sci. Lecture Notes Phys., 61, World Sci., Singapore, 1999 | MR | Zbl

[30] L. Schwartz, Analyse Mathématique, Cours I et II, Hermann, Paris, 1967 ; L. Shvarts, Analiz, T. 1, 2, Mir, M., 1972 | Zbl

[31] C. Chevalley, Theory of Lie Groups. I, Princeton Math. Ser., 8, Princeton Univ. Press, Princeton, NJ, 1946 ; K. Shevalle, Teoriya grupp Li, T. 1, IL, M., 1948 | Zbl

[32] C. Godbillon, Géométrie différetielle et mécanique analytique, Hermann, Paris, 1969 ; K. Godbiion, Differentsialnaya geometriya i analiticheskaya mekhanika, Mir, M., 1973 | MR | Zbl | Zbl

[33] G. de Rham, “Sur la théorie des formes différentielles harmoniques”, Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.), 22 (1946), 132–152 | MR | Zbl

[34] G. de Rham, Vari'etés différentiables. Formes, courants, formes harmoniques, Hermann, Paris, 1955 ; Zh. de Ram, Differentsiruemye mnogoobraziya, URSS, M., 2006 | Zbl

[35] J. A. Wolf, Spaces of Constant Curvature, University of California, Berkley, CA, 1972 ; D. Volf, Prostranstva postoyannoi krivizny, Nauka, M., 1982 | MR | MR | Zbl

[36] N. E. Steenrod, The Topology of Fiber Bundles, Princeton Math. Ser., 14, Princeton University Press, Princeton, NJ, 1951 ; N. Stinrod, Topologiya kosykh proizvedenii, IL, M., 1953 | MR | Zbl

[37] H. Weyl, “Gravitation und Elektrizität”, Berl. Ber., 1918 (1918), 465–480 ; G. Veil, “Gravitatsiya i elektrichesvo”: E. Kuranskii (red.), Albert Einshtein i teoriya gravitatsii, Mir, M., 1979, 513–527 | Zbl | MR

[38] T. Levi-Civita, “Nozione di parallelismo una varietá qualungue e consequente specificazione geometrica della curvatura Riemanniana”, Rend. Palermo, 42:1 (1916), 173–204 | DOI

[39] M. O. Katanaev, “Geometricheskaya teoriya defektov”, UFN, 175:7 (2005), 705–733 | DOI

[40] A. M. Gleason, “Groups without small subgroups”, Ann. of Math. (2), 56:2 (1952), 193–212 | DOI | MR | Zbl

[41] D. Montgomery, L. Zippin, “Small subgroups of finite-dimensional groups”, Ann. of Math. (2), 56:2 (1952), 213–241 | DOI | MR | Zbl

[42] L. S. Pontryagin, Nepreryvnye gruppy, Nauka, M., 1984 | MR | Zbl

[43] A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications, PWN – Polish Scientific Publishers, Warszawa, 1977 ; A. Barut, R. Ronchka, Teoriya predstavlenii grupp i ee prilozheniya, T. 1, 2, Mir, M., 1980 | MR | MR | MR

[44] M. Goto, F. D. Grosshans, Semisimple Lie Algebras, Lecture Notes in Pure Appl. Math., 38, Marcel Dekker, New York, 1978 | MR | Zbl

[45] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math., 34, Amer. Math. Soc., Providence, RI, 2001 ; S. Khelgason, Differentsialnaya geometriya, gruppy Li i simmetricheskie prostranstva, Faktorial Press, M., 2005 | MR | Zbl

[46] S. Helgason, Differential Geometry and Symmetric Spaces, Pure Appl. Math., 12, Academic Press, New York, 1962 ; S. Khelgason, Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1983 | MR | Zbl

[47] M. M. Postnikov, Lektsii po geometrii. Semestr V. Gruppy i algebry Li, Nauka, M., 1982 | MR | Zbl

[48] E. B. Dynkin, “Struktura poluprostykh algebr Li”, UMN, 2:4(20) (1947), 59–127 | MR

[49] E. B. Vinberg, A. L. Onischik, Seminar po gruppam Li i algebraicheskim gruppam, Nauka, M., 1988 | MR | Zbl

[50] J. F. Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, Chicago, 1996 | MR | Zbl

[51] I. D. Ado, “Predstavlenie algebr Li matritsami”, UMN, 2:6(22) (1947), 159–173 | MR

[52] E. Cartan, “Groupes simples clos et ouverts et géométrie riemanniene”, Journ. de Math. (9), 8 (1929), 1–33 | Zbl

[53] M. Hausner, J. T. Schwartz, Lie Groups; Lie Algebras, Gordon Breach Science Publ., New York, 1968 | MR | Zbl

[54] C. Kosniowski, A First Course in Algebraic Topology, Cambridge Univ. Press, Cambridge, 1980 ; Ch. Kosnevski, Nachalnyi kurs algebraicheskoi topologii, Sovremennaya matematika. Vvodnye kursy, Mir, M., 1983 | MR | Zbl | MR | Zbl

[55] J. F. Adams, “Vector fields on spheres”, Ann. of Math. (2), 75:3 (1962), 603–632 | DOI | MR | Zbl

[56] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966 ; E. Spener, Algebraicheskaya topologiya, Mir, M., 1971 | MR | Zbl | MR | Zbl

[57] R. Narasimhan, Analysis on Real and Complex Manifolds, North-Holland Publ., Amsterdam, 1971 | MR

[58] A. A. Bolibrukh, Fuksovy differentsialnye uravneniya i golomorfnye rassloeniya, MTsNMO, M., 2000

[59] K. Iwasawa, “On some types of topological groups”, Ann. of Math. (2), 50:3 (1949), 507–558 | DOI | MR | Zbl

[60] K. Nomizu, H. Ozeki, “On the degree of differentiability of curves used in the definition of the holonomy groups”, Bull. Amer. Math. Soc., 68 (1962), 74–75 | DOI | MR | Zbl

[61] W. Ambrose, I. M. Singer, “A theorem on holonomy”, Trans. Amer. Math. Soc., 75 (1953), 428–443 | DOI | MR | Zbl

[62] J. Hano, H. Ozeki, “On the holonomy groups of linear connexions”, Nagoya Math. J., 10 (1956), 97–100 | MR | Zbl

[63] K. Nomizu, “Un théorème sur les groupes d'holonomie”, Nagoya Math. J., 10 (1956), 101–103 | MR | Zbl

[64] H. Ozeki, “Infinitesimal holonomy groups of bundle connections”, Nagoya Math. J., 10 (1956), 105–123 | MR | Zbl

[65] H. Wang, “On invariant connections over a principal fibre bundle”, Nagoya Math. J., 13 (1958), 1–19 | MR | Zbl

[66] M. V. Berry, “Quantal phase factors accompanying adiabatic changes”, Proc. Roy. Soc. London Ser. A, 392:1802 (1984), 45–57 | DOI | MR | Zbl

[67] Y. Aharonov, D. Bohm, “Significance of electromagnetic potentials in the quantum theory”, Phys. Rev. (2), 115:3 (1959), 485–491 | DOI | MR | Zbl

[68] F. Wilczek, A. Zee, “Appearance of gauge structure in simple dynamical systems”, Phys. Rev. Lett., 52:24 (1984), 2111 | DOI | MR

[69] M. Born, V. Fock, “Beweis des Adiabatensatzes”, Z. Phys., 51 (1928), 165–180 | DOI | Zbl

[70] E. Schrödinger, “Quantizierung als Eigenwertproblem (Erste Mitteilung)”, Ann. Phys. Leipzig, 384:4 (1926), 361–376 | DOI

[71] E. Schrödinger, “Quantizierung als Eigenwertproblem (Zweite Mitteilung)”, Ann. Phys. Leipzig, 384:6 (1926), 489–527 | DOI | Zbl

[72] V. S. Vladimirov, I. V. Volovich, “Lokalnye i nelokalnye toki dlya nelineinykh uravnenii”, TMF, 62:1 (1985), 3–29 | MR | Zbl

[73] V. S. Vladimirov, I. V. Volovich, “Zakony sokhraneniya dlya nelineinykh uravnenii”, UMN, 40:4 (1985), 17–26 | MR | Zbl

[74] A. Messiah, Quantum Mechanics, Vol. II, North-Holland Publ., Amsterdam, 1962 ; A. Messia, Kvantovaya mekhanika, T. 2, Nauka, M., 1979 | MR | Zbl | MR

[75] V. A. Fok, Nachala kvantovoi mekhaniki, Nauka, M., 1976

[76] T. Bitter, D. Dubbers, “Manifestation of Berry's topological phase in neutron spin rotation”, Phys. Rev. Lett., 59 (1987), 251–254 | DOI

[77] F. G. Werner, D. R. Brill, “Significance of electromagnetic potentials on the quantum theory in the interpretation of electron interferometer fringe observations.”, Phys. Rev. Lett., 4:7 (1960), 344–347 | DOI

[78] R. G. Chambers, “Shift of an electron interference pattern by enclosed magnetic flux”, Phys. Rev. Lett., 5:1 (1960), 3–5 | DOI

[79] H. Boersch, H. Hamisch, D. Wohlleben, K. Grohmann, “Antiparallele Weißsche Bereiche als Biprisma für Elektroneninterferenzen”, Z. Phys., 159:4 (1960), 397–404 | DOI