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On the stratification of the orbit space for the action of automorphisms on connections

Seria
Rozprawy Matematyczne tom/nr w serii: 250 wydano: 1986
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Warianty tytułu
Abstrakty
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CONTENTS

Introduction..................................................................................................................................................5

§1. Basic notions and notation.....................................................................................................................7
  1.1. Automorphisms of principal bundles....................................................................................................7
  1.2. Connections and parallel translations.................................................................................................9
  1.3. Symmetries and connections.............................................................................................................11

§2. The action of the gauge group on connections....................................................................................14
  2.1. The gauge group...............................................................................................................................14
  2.2. The action of on $G^{k+1}$ on $C^k$................................................................................................17
  2.3. Weak and strong invariant metrics on $C^k$.....................................................................................20
  2.4. The equivariant embedding of $C^k$ into the space of $H^k$ Riemannian metrics on P...................23

§3. The Slice Theorem...............................................................................................................................30
  3.1. The Hodge-Kodaira-like decomposition for $T_e Φ_A$....................................................................30
  3.2. The orbits are submanifolds..............................................................................................................36
  3.3. The Slice Theorem............................................................................................................................38

§4. The geometric structure of $R^k = C^k/G^{k+1}$..................................................................................43
  4.1. Consequences of the Slice Theorem.................................................................................................44
  4.2. The Countability Theorem.................................................................................................................47
  4.3. Density theorems...............................................................................................................................49
  4.4. The stratification of $R^k$.................................................................................................................57
References.................................................................................................................................................61
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 250
Liczba stron
62
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCL
Daty
wydano
1986
Twórcy
  • Instytut Matematyczny PAN, Śniadeckich 8, 00-950 Warszawa, Polska
autor
  • Instytut Matematyki Politechniki Warszawskiej, Plac Jedności Robotniczej 1, 00-661 Warszawa, Polska
Bibliografia
  • [1] M. F. Atiyah, N. J. Hitchin, I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. (London) A 362 (1978), 425-461.
  • [2] M. F. Atiyah, J. D. S. Jones, Topological aspects of Yang-Mills theory, Commun. Math. Phys. 61 (1978), 97-118.
  • [3] K. Borsuk, Theory of retracts, PWN, Warszawa 1967.
  • [4] N. Bourbaki, Eléments de mathématique, Paris: Hermann, [a] Fasc. III, Topologie générale. Groupes topologiques (1960), [b] Fasc. XXXIII, Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 a 7) (1971), [c] Fasc. XXXVII, Groupes et algèbres de Lie (1972).
  • [5] J. P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Comp. Math. 30 (1975), 1-41.
  • [6] G. E. Bred on, Introduction to compact transformation groups, Academic Press, New York 1972.
  • [7] M. Cantor, Elliptic operators and the decomposition of tensor fields. Bull. Amer. Math. Soc. 5 (1981), 235-262.
  • [8] P. R. Chernoff, J. E. Marsden, Properties of infinite dimensional Hamiltonian systems. In: Lecture notes in mathematics. Vol. 425 Springer, Berlin-Heidelberg-New York 1974.
  • [9] Y. Choquet-Bruhat, D. Christodoulou, Elliptic systems in Hilbert spaces on manifolds which are euclidean at infinity, Acta Math. 146 (1981), 129-150.
  • [10] Y. Choquet-Bruhat, D. Christodoulou: Systèmes elliptiques sur une variété euclidienne à l'infini, C. R. Acad. Sci. Paris 290 Ser. A. (1980), 781-785.
  • [11] D. G. Ebin, The manifold of Riemannian metrics, Proc. Symp. Pure Math. Amer. Math. Soc. XV (1970), 11-40.
  • [12] D. G. Ebin, J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163.
  • [13] A. E. Fisher, The theory of superspace. In: Relativity (M. Carmeli; S. Fickler and L. Witten eds), Plenum Press, New York 1970.
  • [14] V. N. Gribov, Instability of non-abelian gauge theories and impossibility of choice of Coulomb gauge, SLAC Translation 176 (1977). See also: Quantization of non-abelian gauge theories, Nuclear Phys. B 139 (1978), 1-19.
  • [15] G.'t Hooft, Aspects of quark confinement, Phys. Scripta 24 (1981), 841-846.
  • [16] D. Husemoller, Fibre bundles, Mc Graw-Hill Book Co. New York 1966.
  • [17] J. Isenberg, J. Marsden, A slice theorem for the space of solutions of Einstein equations, Phys. Rep. 89 (1982), 179-222.
  • [18] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Vol. I, Interscience, New York 1963.
  • [19] W. Kondracki, J. S. Rogulski, [a] On conjugacy classes of closed subgroup, [b] On the notion of stratification, Preprint 281 PAN, Warszawa 1983.
  • [20] K. Kuratowski, Introduction to set theory and topology PWN, Warszawa 1972.
  • [21] P. K. Mitter, C. M. Viallet, On the bundle of connections and the gauge orbit manifold in Yang-Mills theory, Commun. Math. Phys. 79 (1981), 457-472.
  • [22] G. D. Mostov, Equivariant embeddings in euclidean spare, Ann. of Math. 65 (1957), 432-446.
  • [23] M. S. Narasimhan, T. R. Ramadas, Geometry of SU(2) gauge fields, Commun. Math. Phys. 67 (1979), 121-136.
  • [24] R. S. Palais, Embedding of compact differentiate transformation groups in orthogonal representations, J. Math. Mech. 6 (1957), 673-678.
  • [25] R. S. Palais, Foundations of global non linear analysis, Benjamin Company Inc, New York 1968.
  • [26] R. S. Palais, Seminar on the Atiyah-Singer index theorem, Princeton Univ. Press (1965).
  • [27] J. S. Rogulski, Operators with $H^k$ coefficients and generalized Hodge-de Rham decompositions, Demonstratio Mathematica vol. XVIII, No 1. (1985), 77-89.
  • [28] I. M. Singer, Some remarks on the Gribov ambiguity, Commun. Math. Phys. 60 (1978). 7-12.
  • [29] R. Thom, Les singularités des applications différentiables. Annales de l'Institut Fourier VI (1956), 43-87.
  • [30] H. Whitney, Elementary structures of real algebraic varieties, Ann. of Math. 66 (1957), 546-556.
Języki publikacji
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Identyfikator YADDA
bwmeta1.element.zamlynska-fda8e116-db0b-44a1-866f-e3751b3154c2
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ISBN
83-01-06782-9
ISSN
0012-3862
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DML-PL
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