Download - ΥΠΟΜΝΗΜΑ ΕΠΙΣΤΗΜΟΝΙΚΗΣ ∆ΡΑΣΤΗΡΙΟΤΗΤΑΣusers.uoa.gr/~evassil/bio2017B.pdf · µινάϱιο αυτό έγινε συστηµατική παρουσίαση

Transcript
  • .

    2017

  • 2005,

    ( 2006).

    () -

    *

  • 1 1

    1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.5 . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.6 . . . . . . . . . . . . . . . . . . . . . . . 2

    1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.9 . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    2 4

    2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    2.5 . . . . . . . . . . . . . . . . . . . . . . . . 9

    2.6 . . . . . . . . . . . . . . . . . . . . . . . 9

    2.7 . . . . . . . . . . . . . . . . . . . . . . . 10

    2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    2.9 . . . . . . . . . . . . . . . . . . . . . . . . . 11

    2.10 . . . . . . . . . . . . . . . . . . . . . 12

    3 13

    3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    3.2 . . . . . . . . . . . . 13

    3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    3.5 . . . . . . . . . . . . . . . . . . . . . . . 16

    3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    4 17

    5 23

    5.1 . . . . . . . . . . . . . . . . . . . . . . . . . 23

    5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

    5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

    5.5 . . . . . . . . . . . . . . . . . 67

    i

  • 1.

    1.1.

    : 9 1945, .

    : . .

    : 24, 164 51. . 210-9941927

    : [email protected]

    URL: http://users.uoa.gr/ evassil

    Research Gate: Efstathios Vassiliou

    : , -

    () ( 2011).

    * : -

    . , 20067

    Chern-

    Weil" .

    1.2.

    ( , 1979)

    ( , 1973)

    ( , 1968).

    1.3.

    1976 () -

    ( 103, . /27-2-76)

    1979 ( 1979)

    ( 145, . ..../8-8-79)

    1979 ( 241, . ..../11-12-79)

    1982 , . 1268/1982 ( 187,

    . ..../26-8-82)

    1984 ( )

    ( 23, . ..../14-2-84)

    1987 ( 92, . ..../6-6-88)

    2011 .

    1

    http://users.uoa.gr/~evassilhttps://www.researchgate.net/profile/Efstathios_Vassiliou

  • 1.4.

    . ,

    ( Banach, Frechet) Lie,

    ( Lie), (

    ). -

    Mathematical Reviews: 18B40, 18F15, 18F20, 53C05, 53C30, 55N30, 57R22,

    58A20, 58A30, 58B10, 58B20, 58B25.

    1.5.

    ( )

    1963 , 18 & 6/11.

    .

    1968 , 7 & 10/12.

    1967, -

    , . . ,

    . -

    1973 .

    ( 1974 - -

    1975), 1976 -

    , 2011.

    1.6.

    1972-73

    Leeds .

    (. G. R. Allan), (A. West

    S. Carter) (E. Stout).

    - 1976 -

    1977 , Guest Staff Member,

    Aarhus . -

    : (J. Dupont),

    (Singularities) (H. A. Salomonsen)

    ( J. Dupont I. Madsen).

    1979 - 1980 ,

    , . -

    P. Libermann ( Paris VII),

    Lie A. Lichnerowicz ( College de Fra-

    nce) - , Pham Mau

    Quan G. Pichon ( Poincare).

    1991-92 ,

    , Cambridge () Visiting Scholar.

    2

  • 1.7.

    1. Corso Estivo di Matematica,

    (CNRS) Scuola Normale Superiore ( 1971).

    2. Representations of Lie Groups and Harmonic Analysis,

    ( 1977).

    3. Summer Seminar on Complex Analysis, (ICTP)

    ( 1980).

    4. Summer Workshop on Fibre Bundles and Geometry, ( 1982).

    1.8.

    1. .., (1963-1967).

    2. (British Counsil), -

    Leeds.

    3. , -

    Aarhus.

    4. 4 ,

    1.7,

    .

    1.9.

    :

    American Mathematical Society,

    London Mathematical Society,

    Societe Mathematique de France,

    Balkan Society of Geometers.

    , , (..).

    1.10.

    , , .

    3

  • 2.

    2.1.

    :

    : (Affine) , , -

    , (

    Lie), ( Riemann

    ), - ( ).

    : ( ), -

    .

    2.2.

    ,

    , , ,

    :

    1976-77 ( ): Gauss Hopf-Rinow. -

    D. Gromoll W. Klingen-

    berg W. Meyer: Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics

    #55, Springer ( . ).

    1980-81 (): Lie-Banach. Lie-

    Banach N.Bourbaki: Groupes et Algebres de Lie, Chapitres 23

    J. Marsden: Hamiltonian Mechanics, Infinite Dimensional Lie

    Groups, Geodesics Flows and Hydrodynamics (Berkley, 1968).

    1981-82 (): .

    , -

    N. Bourbaki: Varietes Differentielles et

    Analytiques (fascicule de resultats, 17) S. Lang: Introduction to

    Differentiable Manifolds.

    1981-82 ( ): . -

    W. Drechsler M. E. Mayer: Fibre Bundle Techniques

    in Gauge Theories, Lecture Notes in Physics #67, Springer, D. Bleecker: Gauge

    Theory and Variational Principles. 4 -

    .

    1983-84 (): . -

    ( ) S. G. Kren

    N. Jackin, [.

    3.6 ( # 2)].

    1984-85 ( ): Floquet-Liapunov.

    Floquet-Liapunov

    ( )

    4

  • , -

    (. ). ,

    , .

    1987-88 ( ): .

    (-

    ) , .

    1988-89 (): .

    -

    .

    1992-93 (): Lie.

    K. Mackenzie: Lie Groupoids and Lie Algebroids in Dif-

    ferential Geometry, LMS Lecture Notes Series 124 (1987).

    (

    ) . .

    1994-95 (): Banach Frechet.

    ( J. A. Leslie, H. Omori, M. E. Verona),

    Banach Lie-Banach.

    ( ) . -

    ,

    .

    1996-97 ( ): Sikorski. -

    . . .

    , R.

    Sikorski , -

    . (

    ) . .

    1997-98 ( ): LaTEX AMS-LaTEX .

    -

    .

    2000-01 (): . . -

    . -

    . . (. Geometry of Vector Sheaves,

    Vols. I-II, Kluwer, 1999), -

    .

    .

    2001-02 ( ): . -

    . . ,

    J. Leslie Frechet,

    P. Michor A. Kriegl,

    The convenient setting for global Analysis, AMS, 1997.

    5

  • 2002-03 ( ): . -

    , . , .

    , . , . , . . .

    2007-08 ( ): * . , - Kobayashi ( . ),

    ( . -

    ), A- ( . -

    ).

    2008-09 ( ): * . Chern-Weil ( . ).

    -

    , :

    1. N. Karkanias (The City University, ):

    ( 1983).

    2. A. Asada (Shinshu University, ): Non-abelian de Rham Theory (

    1984).

    3. Pham Mau Quan (Universite Paris XIII, ): 1) Infinitesimal Automorphisms

    and Characterizations of the Cotangent Bundles. 2) Stability of Periodic Orbits and

    Poincares Isometric Problem ( 1985).

    4. A. ASADA: Characteristic Classes in Non-commutative Geometry ( 1993).

    5. A. Asada: Non-commutative Version of Monodromy Theory ( 1994).

    6. M. Anastasiei (University Al I. Cuza , Iasi - ): Introduction to Lagrange

    Spaces ( 1995).

    7. K. Buchner (Technical University, Munchen - ): Differential Spaces: a Ge-

    neralization of Differential Manifolds ( 1996).

    8. W. Mozgawa (University Marie Curie, Lublin - ): Foliation theory and

    projective structures ( 1998).

    9. C.T.J. Dodson (University of Manchester, UK): Information geometric Riemannian

    neighbourhoods of randomness and independence ( 2005).

    2.3.

    , , :

    Colloquium on Differential Geometry, Janos Bolyai Mathematical Society, Budapest

    (3 7 1979). : Conjugate connections and differential

    equations.

    6

  • CSSR - GDR - Polish Conference on Differential Geometry and its Applications,

    Nove Mesto na Morave (8 12 1980). -

    : Linear connections on bundles over S1.

    7 , , (19 23

    1983). : On applications of related connections.

    3rd International Congress of Differential Geometry, (16

    1 1991). : Some applications of conjugate conne-

    ctions.

    1 , (27 28 1994).

    : .

    24th National Conference of Geometry and Topology, Timisoara (5 9 1994).

    : On a type of total differential equations in Frechet spaces (

    . ).

    2 , , (1 3 -

    1995). : .

    25th National Conference of Geometry and Topology, Univ. Al. I. Cusa, Iasi (18

    23 1995). : From principal connections to connections

    on principal sheaves.

    Workshop on Differential Geometry, Global Analysis and Lie Algebras, -

    (13 16 1995). : On the geometry

    of the sheaf of frames of a vector sheaf.

    4th International Congress of Geometry, (26 1

    1996). : On flat principal homogeneous bundles.

    Conference on Differential Geometry (Satelite conference of the 2nd European Con-

    gress of Mathematics), Budapest (27 30 1996). : Prin-

    cipal sheaves and connections.

    2nd Conference of Balkan Society of Geometers, (24

    27 1998). : Vector sheaves associated with principal she-

    aves.

    4 , (28 30 1999). -

    : On the geometry of associated sheaves.

    Colloquium on Differential Geometry, Debrecen (25 30 2000). -

    : Cohomological and geometrical aspects of principal and vector sheaves.

    4th Conference of Balkan Society of Geometers, * (24 27 2007). : Cohomological classification of Frechet bund-

    les.

    7

  • 8 , * (24 27 2007). :

    ( . ).

    11 , * ( 2013). : Grassmann sheaves and the classification of vector sheaves (

    . ).

    30 (invited talk) 5th International

    Workshop in Differential Geometry, Timisoara (18 22 2001),

    11 2001.

    , 3 -

    , (30 31 1997) (-

    : . , . ).

    2.4.

    , -

    :

    . , 1981.

    Connections, Total Equations and Floquet-type Theorems. Differential Geometry

    Day, University of Leeds, 1992.

    . -

    , 1995.

    On the Geometry of Principal Sheaves. Maria Curie - Skodowska University, Lublin,

    1997.

    Connections on Principal Sheaves: an abstract approach to gauge theory. Univer-

    sity of Timisoara, 1997.

    , :

    . -

    . ,

    Banach (-

    1973).

    , . Colloquium -

    ,

    ( 1982).

    Floquet . 2 -, -

    ( 1987).

    8

  • . * , 2330 2006.

    : .

    * , ( 2009). , , -

    ( 2010).

    N. Bourbaki: (;) . * , , 2530 2009.

    .* - , ( 2015).

    2.5.

    :

    1. . : (1995).

    2. . : Lie (2000).

    2.6.

    :

    1. . : Banach (1995).

    2. . : de Rham Thom

    (1996).

    3. . : Lie (1997).

    4. . (1998).

    5. . : (2002).

    6. . : * (2008).

    7. . : * Mobius (2009).

    8. . : * (2010 ).

    9. . : * - (2014).

    9

  • 2.7.

    ,

    :

    Balkan Journal of Geometry and its Applications,

    Bulletin of the Greek Mathematical Society,

    International Journal of Pure and Applied Mathematical Sciences,

    Global Journal of Mathematics and Mathematical Sciences.

    ,

    (Reviewer) Mathematical Reviews Zentralblatt fur

    Mathematik.

    (refereeing) : ,

    Portugaliae Math., Bulletin Greek Math. Soc., Balkan J. Geom. Appl. Int. J.

    Math. Math. Sci.

    2.8.

    -

    :

    , . -. 70/4/2554 (1996).

    , ,. . 70/4/3410 (1997).

    , ( ), . . 70/4/3410 (1998).

    , . . 70/4/3410(1999).

    ( -), . . 70/4/3410 (2001).

    ( -), . . 70/4/3410 (2002).

    ( -), . . 70/4/3410 (2004).

    Frechet , . . 70/4/3410 (2006).

    10

  • 2.9.

    :

    :

    1. . ( .. 22-3-1985 / : . . )

    2. . ( .. 7-7-1987 / : . . )

    3. . ( .. 10-7-1987 / : . . . )

    4. . ( 28-3-1991 / : . . . ).

    5. . ( .. 30-3-2004 / : . . . ).

    :

    1. .

    .

    , ( : . /

    .. 14-5-1984).

    2. . ( .. 5-3-1985).

    3. .

    .

    - ( : . / ..

    4-10-1990).

    4. .

    (-

    : . / .. 24-6-1992).

    5.

    . ( .. 7-4-1993).

    6. .

    ( : . ,

    . , . / .. 27-6 5-7-1995).

    7. .

    . ( .. 20-11-1996).

    8. .

    -

    ( : . , . , . / .. 17-7-1997).

    9. .

    , Riemann

    ( : . , . / .. 27-5-2003).

    10.

    ( : . / . .. 11-6-2008).

    .

    ( ..

    . / 20-10-1983 / : . . ).

    . ( .. . 3-6-2003 /

    : . . ).

    11

  • :

    1. . ( .. 30-4-1996 / : . . ).

    2. . ( .. 30-4-1996 / : . . . ).

    3. . ( .. 4-2-1997 / : . . . -

    ).

    4. . ( .. 3-7-1997 / : . ).

    5. . ( .. 22-5-2002 / : . . . -

    ).

    6. . ( .. 1-7-2002 / : . . . ).

    7. . ( .. 24-6-2003 / : . . -

    ).

    8. . ( .. 24-6-2003 / : . . . ).

    9. * . ( . . / . ).10. * . ( . . / . -

    ).

    9. * . ( . . / . . -).

    10. . ( . . 22-3-2005 / ).

    11. * . ( . . / . . . ).12. * . ( . . 22-6-2005/ . . . ).12. * . ( . . 20-6-2006/ . . . ).13. * . ( . . 22-6-2006/ . . . ).

    2.10.

    (), , , , , .

    ,

    .. (1986 - 1987 / .. 28-3-1986),

    ( ) Ismene Fitch(Ismene Fitch Scholarship Board) British Council ( 1988),

    ......( 20-9-1994 15-11-1997, ),

    ... (1996,1997), (...) . (Warwick Univer-

    sity).

    12

  • 3.

    ( 4) (

    , 5), ,

    ,

    .

    3.1.

    [1] . ( -

    ), 1973, . 138.

    [2] Banach . -

    ( ), 1978, . xvi+394.

    3.2.

    [3] Christoffel symbols and connection forms on infinite-dimensional fibre bundles.

    Bull. Soc. Math. Grece 15 (1974), 115122 [MR. 58 #31161; Zbl. 309.53021].

    [4] On the holonomy theorem. Prakt. Akad. Athenon 51 (1976), 463467 [MR. 81c:

    58011; Zbl. 368.53020].

    [5] Connections on 1-jet principal fibre bundles. Rend. Sem. Mat. Univ. Padova 56

    (1976), 2533 [MR. 58 #13117; Zbl. 372.53007].

    [6] Affine transformations of banachable bundles of frames. Mathematica Balkanica

    6:46 (1978), 291295 [MR. 81c:58012; Zbl. 413.58003].

    [7] (f, , h)-related connections and Liapunoffs theorem. Rend. Circ. Mat. Palermo,Ser. II, 27 (1978), 337341 [MR. 81d:58013; Zbl. 429.53022].

    [8] On the canonical connection of the 1-jet bundle of a principal bundle. Bull. Soc.

    R. S. Liege 47 (1978), 511 [MR. 58 #13116; Zbl. 385.53031].

    [9] On the infinite-dimensional holonomy theorem. Bull. Soc. R. S. Liege 47(1978),

    223228 [MR. 80m:58006; Zbl. 357.58007].

    [10] On affine transformations of banachable bundles. Colloq. Math. 44 (1981), 117

    123 [MR. 84i:58013; Zbl. 477.58008].

    [11] Conjugate connections and ordinary differential equations. Colloq. Math. Soc. Ja-

    nos Bolyai 31 (1982), 184804 [MR. 85a:58010; Zbl. 515.53030].

    [12] Sur les connexions plates d un fibre banachique. C. R. Acad. Sci. Paris Ser. I Ma-

    th. 295 (1982), 353356 [MR. 83k:58010; Zbl. 511.58009].

    [13] Transformations of linear connections. Period. Math. Hungar. 13 (1982), 289308

    [MR. 84i:58013; Zbl. 525.53044].

    13

  • [14] Flat bundles and holonomy homomorphisms. Manuscripta Math. 42 (1983), 161

    170 [MR. 84e:57024; Zbl. 519.58010].

    [15] Transformations of linear connections II. Period. Math. Hungar. 17 (1986), 111

    [MR. 87f:58015; Zbl. 617.58003].

    [16] Total differential equations and the structure of fibre bundles. Bull. Greek Ma-

    th. Soc. 27 (1986), 149159 [MR. 89g:58012; Zbl. 671.58002].

    [17] Floquet-type connections on principal bundles. Tensor (N.S.) 43 (1986), 189195

    [MR. 88j:58012; Zbl. 649.53008].

    [18] On affine extensions of the holonomy homomorphisms of flat principal bundles.

    Colloq. Math. 59 (1990), 263268 [MR. 92a:53033; Zbl. 791.53033].

    [19] Characterizations of flat bundles via total differentiation. Tensor (N.S.) 52 (1993),

    16 [MR. 94m:53035; Zbl. 804.53034].

    [20] On a class of principal bundles over symplectic bases on Euclidean spaces. De-

    monstratio Math. 26 (1993), 7592 [MR. 94h:53045; Zbl. 798.53025].

    [21] On a type of total differential equations in Frechet spaces (with G. Galanis). Proc.

    24th Conf. Geom. Topol., Timisoara (1994), 119128 [Zbl. 866.58008]

    [22] From principal connections to connections on principal sheaves. Anal. Stiint. U-

    niv. Al. I. Cuza Iasi 42, Suppl., (1996), 149160 [MR. 98j:53029; Zbl. 883.53029].

    [23] Transformations of sheaf connections. Balkan J. Geom. Appl. 1 (1996), 117133

    [MR. 97k:58003; Zbl. 970.16098].

    [24] On the triviality of homogeneous principal bundles. Proc. 4th Intern. Congress

    of Geometry, Thessaloniki (1966), N. Artemiadis N. Stefanidis Eds., Giachoudis

    Giapoulis Publ., 417423 [MR. 98i:53032; Zbl. 899.53018].

    [25] A generalized frame bundle for certain Frechet vector bundles and linear con-

    nections (with G. Galanis). Tokyo J. Math. 20 (1997), 129137 [MR. 98e:58020; Zbl.

    894.58006].

    [26] Connections on principal sheaves. New Developments in Differential Geometry,

    Budapest 1996, (J. Szenthe ed.) 459484, Kluwer Acad. Publ., Dordrect, 1998 [MR.

    2000c:58002; Zbl. 940.53020].

    [27] On Mallios A-connections as connections on principal sheaves. Note Mat. 14 (1994),237249 (1997) [MR. 98e:58009; Zbl. 883.53028].

    [28] A FloquetLiapunov theorem in Frechet spaces (with G. Galanis). Ann. Scuola

    Norm. Sup. Pisa 27 (1998), 427436 [MR. 2000a:34106; Zbl. 932.34061].

    [29] Topological algebras and abstract differential geometry. J. Math. Sci. (continua-

    tion of J. Soviet Math., New York), 95 (1999), 26692680 [MR. 2000h:58014; Zbl.

    936.53022].

    14

  • [30] Vector sheaves associated with principal sheaves. 2nd Confe-

    rence of Balkan Geometers (Thessaloniki 1998) Proceedings

    of the Workshop on Global Analysis, Differential Geometry and Lie Algebras (Thes-

    saloniki 2001), 197206, BSG Proceedings 10, Geometry Balkan Press, Bucharest,

    2004.

    [31] On the geometry of associated sheaves. Bull. Greek Math. Soc. 44 (2000), 157

    170 [MR. 2002f:18024; Zbl. pre01828021].

    [32] On certain Flat Frechet principal bundles and their holonomy homomorphisms

    (with G. Galanis). Algebras Groups Geom. 17 (2000), 105122 [MR. 2001c:58007;

    Zbl. 1011.58002].

    [33] Connections on A-frame bundles (with M. Papatriantafillou). Sci. Math. Jpn. 54

    (2001), 2938 [MR. 2002k:58012; Zbl. 1021.58004].

    [34] On associated Frechet vector bundles (with G. Galanis). Algebras Groups Geom

    19(2002), 277288 [MR. 2003i:58011].

    [35] Universal connections on groupoids (with A. Nikolopoulos). Int. J. Math. Math.

    Sci. 23 (2003), 14651480 [MR. 2004e:58033; Zbl. 1023.58011].

    [36] On the cohomology and geometry of principal sheaves. Demonstratio Math. 36

    (2003), 289306 [MR. 2004g:18011; Zbl. pre02012276].

    [37] Remarks on the cohomological classification of certain Frechet fiber bundles.

    Balkan J. Geom. Appl. 9 (2004), 2331.

    [38] A generalized second order frame bundle for Frechet manifolds (with C.T.J. Dod-

    son and G. Galanis). J. Geom. Phys. 55 (2004), 291305.

    * [39] Isomorphism classes for Banach vector bundle structures of second order (wi-th C. T. J. Dodson and G. Galanis). Math. Proc. Cambridge Philos. Soc. 141 (2006),

    489486.

    * [40] Grassmann sheaves and the classification of vector sheaves (with M. Papatrian-tafillou). Demonstratio Math. 46 (2013), 427436.

    * [41] Local connection forms revisited Rend. Circ. Mat. Palermo, Ser. II, 62 (2013),393408

    3.3.

    [42] Geometry of Principal Sheaves. Mathematics and Its Applications Vol. 578, Sprin-

    ger, pp. xvi + 444 (2005).

    * [43] Geometry in a Frechet Context: A Projective Limit Approach (with C.T.J. Dod-son and G. Galanis). London Mathematical Society Lecture Note Series Vol. 428,

    Cambridge University Press, pp. xii + 302 (2015).

    15

  • 3.4. ( )

    [44] On related connections and the 1-jet principal fibre bundle. Preprint Ser. 1977/

    1978, No 7, Matematisk Institut, Aarhus Univ. pp. 17 [Zbl. 363.53014].

    . , Zentralblatt

    B. N. Shapukov (. 4).

    [45] Some applications of conjugate connections. Proceeding 3rd International Con-

    gress of Geometry, Thessaloniki (1991), N. Stefanidis Ed., 434442, Aristotle Uni-

    versity of Thessaloniki [MR. 93f:58014].

    [46] On the geometry of the sheaf of frames of a vector sheaf. Proceedings of the W-

    orkshop on Global Analysis, Differential Geometry and Lie Algebras (Thessaloniki

    1995), 141146, BSG Proceedings 1, Geometry Balkan Press, Bucharest, 1997 [MR.

    99j:58004].

    . [27].

    [47] Flat principal sheaves. Semin. de Mecanica, Univ. Timisoara, 55 (1997), 134 [Zbl.

    906.53018].

    . -

    Timisoara .

    A. Mallios E. Rosinger (. . 3.7).

    [42].

    3.5.

    [48] I (-

    ). 1982, . vi + 210.

    [49] (). 1984,

    . viii + 164 [ : S. G. Kren N. I.kin,Linenye Differenialnye Uravneni na Mnogoobrazih, Iz. Voro-ne. Univ. (1980)].

    [50] , :

    Floquet-Liapunov (). 1986, .

    ix + 113.

    *[51] (), , 2007, .xii + 291 [ , ,

    , , 1989].

    3.6.

    [52] , . 1985, . 88.

    [53] :

    Lie ( . ), 1988, . 158.

    16

  • *[54] ( . ). 2007,. 223. [ , ,

    , 1994, . 206.]

    *[55] , 2010( ), . 203.

    *[56] , 2015 ( , , ), . 168.

    4.

    :

    -

    * M.C. Abbati - A. Mania A geometrical setting for geometric phases oncomplex Grassmann manifolds. J. Geom. Phys.

    53 (2007), 779797.

    [9]

    * M. Aghasi - A.R. Bahari Second order structures for sprays and con- [38], [39]C.T.J. Dodson nections on Frechet manifolds.

    G. Galanis - A. Suri arXiv:math/0810.5260

    * M. Aghasi - A.R. Bahari Higher order Hessian structures on Frechet [37]A. Suri manifolds. Differential Geometry - Dynamical

    systems 13 (2011), 118.

    * M. Aghasi - C.T.J. Dodson Infinite-dimensional second order ordinary dif- [15], [28]G. Galanis - A. Suri ferrential equations on T 2M . Nonlinear Analysis

    67 (2007), 28292838.

    [39]

    * M. Aghasi - C.T.J. Dodson Conjugate connections and differential equations [13], [28]G. Galanis - A. Suri in infinite=dimensional manifolds. Differential

    geometry, 227236, World Sci. Publ., Hacken-

    sack, N. J. 2009

    * M. Aghasi - A. Suri Splitting theorems for the double tangent bundlesof Frechet manifolds. Balkan J. Geom. Appl. 15

    (2010), 113.

    [15], [38]

    A.C. Albu - D. Opris Continuous connections on topological fibrations.

    Colloq. Math. Soc. Janos Bolyai 31, 51-71,

    North-Holland (1982).

    [11]

    * A.C. Anyaegbunam Geometric algebra via sheaf theory: A view tow-ards symplectic geometry. Ph. D Thesis, Univer-

    sity of Pretoria, 2010 (Repository.up.ac.za)

    [42]

    A. Asada Non-abelian de Rham theories. Colloq. Math.

    Soc. Janos Bolyai 46 (1984), 83-115.

    [16]

    17

    http://www.arXiv.org/abs/0810.5261

  • A. Asada Non abelian Poincare Lemma. Symp. Diff. Geo-

    metry, Peniscola, 1985, Lecture Notes in Mathe-

    matics #1209 (1986), 37-65.

    [16]

    Integrable forms on iterated loop spaces and h-

    igher dimensional non-abelian de Rham theory.

    Differential Geometry, Peniscola, 1988, Lecture

    Notes in Mathematics #1410.

    Chirability of Non-abelian Cohomology and 3-

    dimensional Non-abelian de Rham Set (manu-

    script).

    A. Asada Curvature forms with singularities and non-

    integral characteristic classes. J. Fac. Sci. Sh-

    inshu Univ. 20 (1985), 145-169.

    [12]

    Non-abelian de Rham theory. Proc. Intl. Conf. on

    Prospects of Math. Sci., Tokyo, 1987, World Sci.

    Pub., 13-40 (1988).

    Differential Geometry of Loop Spaces, Loop Gau-

    ge Theory and Non-abelian de Rham theory.

    Proc. Geometry of Manifolds 1. Topological a-

    spects of modern physics: Some recent topics,

    176208, Kyusyu Univ., 1987.

    Four lectures on the geometry of loop group bund-

    les and non-abelian de Rham theory. Lecture

    Notes Chalmers Univ. of Technology / Univ. of

    Goteborg, 1990.

    A. Asada Non-commutative geometry of GLp-bundles. Col-loq. Math. Soc. J. Bolyai 66, New developments

    in Differential Geometry, 2549, Kluwer, 1995.

    [19]

    * D. Beltida - J.E. Gale Infinitesimal aspects of idempotents in Banachalgebras. arXiv:math/1611.01470

    [3]

    . -

    . , -

    , 1995, . v+195.

    [10], [11],

    [14]

    * M. Callies - H. Schumacher The Yang-Mills module space on Riemann surfa- [14]K. Strokorb ces. crcg/Yang-Mills Project

    * J.S. Cook - R. Fulp Holonomy in Rogers supermanifolds with appli-cations to super Yang-Mills theory. International

    Journal of Geometric Methods in Modern phy-

    sics 8 (2011), 429458

    [9]

    18

    http://www.arXiv.org/pdf/1611.01470http://www.crcg.de/wiki/image/b/bo/YangMillsProject.pdf

  • * K. Drachal Remarks on the behaviour of higher-order deri-vations on the gluing of differential spaces. Che-

    choslovak Math. Journal 65 (140) (2015), 1137

    1154.

    [29]

    * C.T.J. Dodson A review of some recent work on hypercyclicity.Balkan J. Geom. Appl. 19 (2012)

    [25]

    C.T.J. Dodson G. Galanis Second order tangent bundles of infinite-

    dimensional manifolds. J. Geom. Phys. 52

    (2004), 127136

    [8]

    * K. Eftekharinasab Geometry of boundeb Frechet manifolds. RockyMountain J. Math. 2016 (to appear)

    [13]

    M. Fragoulopoulou Q-algebras. How close are they to Banach alge-

    bras? Proc. General Topological Algebras, Tartu

    1999, Esthonian Math. Soc., Mathematics Stu-

    dies 1 (2001).

    [33]

    Topological Algebras with Involution. Math.

    Studies 200, North-Holland 2005.

    * M. Fragoulopoulou M. Papatriantafillou Smooth manifolds vs diffrential triads. RevueRoumaine (2015).

    [42]

    G. Galanis The bundle of 1-jets of the sections of a Frechet

    principal bundle. Proc. 4th Intern. Congr. Geo-

    metry, Thessaloniki, 1996, 155162.

    [8]

    G. Galanis Universal connections in Frechet principal bund-

    les. Period. Math. Hungar. 54 (2007), 113.

    G. Galanis Projective limits of Banach-Lie groups. Period.

    Math. Hungarica 32 (1996), 179191.

    [15]

    G. Galanis On a type of Frechet bundles over Banach bases.

    Period. Math. Hungarica 35 (1995), 1530.

    [2], [7],

    [10], [14]

    G. Galanis Differential and geometric structure for the tan-

    gent bundle of a projective limit manifold. Rend.

    Sem. Math. Padova 112 (2004), 104115.

    [28]

    * G. Galanis - P. Palamides Nonlinear differential equations in Frechet spa-ces and cointinuum cros-sections. Anal. Stiint.

    Univ. Al. I. Cuza (Iasi), 51 (2005) Matematica,

    41-54.

    [28]

    * A. Gerstenberger A version of scale calculus and the associatedFredholm theory. arXiv:math/1602.07108

    [43]

    * H. Ghahremani-Gol - A. Razavi Ricci flow and the manifold of Riemannian metri-cs. Balcan J. Geom. Appl. 18 (2013), 2030.

    [38]

    * G. Giachetta - L. Mangiaroti Geometric and Algebraic Topological Methods in [9]S. Sardanashvily Quantum Mechanics. World Scientific, 2005.

    19

    http://www.arXiv.org/pdf/1602.07108

  • * G. Giachetta - L. Mangiaroti Geometric Formulation of classical and Quantum [9]S. Sardanashvily Mechanics. World Scientific, 2011.

    * W. E. Gryc On the holonomy of the Coulomb connection o-ver manifolds with boundary. J. Math. Phys. 49

    (2008).

    [9]

    D. Lappas On flat connections induced over covering maps.

    arXiv:math/0302074

    [14]

    J. P. Magnot Structure groups and holonomy in infinite dimen-

    sions. Bull. Sci. Math. 128 (2004) 513-529.

    [9]

    A. Mallios Topological Algebras. Selected Topics. Math.

    Studies 124, North-Holland, 1986.

    [9]

    A. Mallios On an abstract form of Floquets theorem. Ab-

    stracts AMS

    [17]

    A. Mallios Continuous vector bundles over topological al-

    gebras II. J. Math. Anal. Appl. 132 (1988), 401

    423

    [14]

    A. Mallios On an abstract formulation of differential geo-

    metry with physical applications. Proc. 3rd Pa-

    nhellenic Congress of Geometry, 1731, Univ.

    Athens 1997, (E. Vassiiou - M. Papatriantafil-

    lou - D. Lappas Eds.)

    [22], [26],

    [27], [29],

    [33]

    A. Mallios On an Axiomatic Treatment of Differential Geo-

    metry via Vector Sheaves. Applications. Math.

    Japonica 48 (1998), 93180.

    [22], [26],

    [27]

    A. Mallios Geometry of Vector Sheaves, Vol. II. Kluwer A-

    cad. Publ., 1998.

    [22], [23],

    [26], [27]

    * A. Mallios Modern Differenrial Geometry in Gauge TheoriesVol. I. Birkhauser, 2006.

    [42]

    On Utiyamas theme through A-invariancce.Complex Anal. Oper. Theory 6 (2012), 775780

    * A. Mallios Modern Differenrial Geometry in Gauge TheoriesVol. II. Birkhauser, 2010.

    [26], [27],

    [42]

    * A. Mallios P. Ntumba Fundamentals for symplectic A-manifolds.Rend. Circ. Mat. Palermo, Ser. II, 58 (2009),

    169198

    [42]

    A. Mallios I. Raptis Finitary, causal and quantal vacuum Einstei-

    ns gravity. Inter. J. Theorer. Phys. 42 (2003),

    14791619.

    [22], [26],

    [27], [31]

    * A. Mallios I. Raptis Smooth singularities exposed: Chimerasof the differential spacetime manifold.

    arXiv:gr-qc/0411121

    [31], [42]

    20

    http://www.arXiv.org/abs/math/0302074http://www.arxiv.org/abs/gr-qc/0411121

  • A. Mallios E. E. Rosinger Space-time foam dense singularities and de

    Rham cohomology. Acta Appl. Math. 67 (2001),

    5989.

    [26], [27],

    [29], [47]

    A. Mallios E. E. Rosinger Dense singularities and de Rham cohomology.

    Topological Algebras with applications in Dif-

    ferential Geometry and Mathematical Physics.

    Fest Colloquium in honor of A. Mallios, 5471,

    Univ. Athens 2002 (P. Strantzalos M. Fragou-

    lopoulou Eds.)

    [26], [27],

    [47]

    * A. Mallios E. Zafiris Differential Sheaves and Connections. WorldScientific, 2016.

    [42]

    R. Miron The Geometry of higher-order Lagrange Spaces.

    Applications to Mechanics and Physics. Kluwer

    Acad. Publ., 1996.

    [22]

    R. Miron - M. Anastasiei Vector bundles and Lagrange Spaces with ap-

    plications to Relativity. Balkan Soc. of Geome-

    ters, Monographs and Textbooks Nr. 1, Geome-

    try Balkan Press, 1997.

    [22]

    . Lie.

    , , -

    2000, . v+138.

    [14], [16]

    A. Nikolopoulos On differential equations in Lie algebroids. Bull.

    Greek Math. Soc. 44 (2000), 117127.

    [7], [16],

    [17]

    A. Nikolopoulos On the triviality of Lie groupoids. BSG Procee-

    dings (in press).

    [16]

    A. Nikolopoulos -universal connections on Lie groupoids. (pre-print)

    [5], [9]

    . A-. -

    , , -

    1986, . xii+270.

    [2]

    M. Papatriantafillou The category of differential triads. Bull. Greek

    Math. Soc. 44 (2002), 129141.

    [26], [27],

    [29]

    M. Papatriantafillou Hermitian structures and compatible connections

    on A-bundles. BSG Proceedings 4, 6575, Geo-

    metry Balkan Press, 2000.

    [33]

    * M. Papatriantafillou Pre-Lie groups in abstract differential geometry.Mediterr. J. Math. 12 (2015), 315328

    [27], [29],

    [42]

    * I. Raptis Finitary-algebraic resolutions of the innerSchwarzschild singularity. Inter. J. Theoret.

    Phys. 45 (1) 2004.

    [22], [27],

    [42]

    * I. Raptis Finitary topos for locally finite causal and quan- [22], [27]tal vacuum Einstein gravity. Inter. J. Theoret.

    Phys. 46 (3) 2005.

    [31], [42]

    21

  • * I. Raptis Third quantization of vacuum Einstein gravity [22], [27]and free Yang-Mills theories. Inter. J. Theoret.

    phys. 46 (5) 2006.

    [31], [42]

    * I. Raptis "Iconoclastic", Categorical quantum gravity.arXiv:gr-qc/0509089

    [27]

    * I. Raptis Categorical quantum gravity. Inter. J. Theoret.Phys. 45 (8) 2006, 14951523.

    [27]

    * G. Rezaie - R. Malekzahed Sprays on Frechet modelled manifolds. Inter-national Mathematical Forum 5 (2010) No 59,

    29012909

    [39]

    * G. Rezaie - R. Malekzahed Ordinary differential equations on trivial vectorbundles and a splitting of double tangent bund-

    le. Differential Geometry - Dynamical Systems,

    Vol. 12 (2010), 179186, Balkan Society of Ge-

    ometers, Geometry Balkan Press 2010.

    [15]

    E. E. Rosinger Parametric Lie Group Actions and Global Genera-

    lized Solutions of Nonlinear PDEs and an answ-

    er to Hilberts fifth problem. Kluwer Acad. Publ.,

    Dordrect, 1998.

    [26], [27],

    [29], [33]

    B.N. Shapukov Connections on differential fibre bundles. J. So-

    viet Math. 29 (1985), 1550-1571.

    [44]

    * V.V. Shurygin On the structure of complete varieties over Weilalgebras. Transl. Russian Math (Iz. VUZ) 2003,

    No 11, 8493 (2004).

    [33]

    * A. Suri Isomorphism classes of higher order tangentbundles. arXiv:math/1412.7321

    [15], [39]

    * A. Suri Geometry of the double tangent bundles of Bana-ch manifolds. J. Geom. Phys. 74 (2013), 9110.

    [15]

    .

    . , -

    , 1990.

    [2]

    * A. Vondra From semi-sprays to connections, from geometryof regular O.D.E in mechanics to geometry of ho-

    rizontal Pfaffian P.D.E on fibered manifolds (and

    vice versa). Proceedings of the Seminar on Dif-

    ferential Geometry, Mathematical Publications

    Vol. 2, Silesian University in Oprava, Oprava

    2000, 1752004.

    [11]

    22

    http://www.arXiv.org/abs/gr-qc/0509089http://www.arxiv.org/abs/1412.7321.pdf

  • 5.

    ,

    :

    (5.1): [1] [20], [24], [41], [44], [45], -

    Banach

    ,

    .

    (5.2): [21], [25], [28], [32], [34], [37], [38], [43],

    Frechet,

    Banach.

    .

    (5.3): [22], [23], [26], [27], [29] [31], [36], [40],

    [42], [46], [47] , ,

    .

    (5.4): [33], [35], [39] ,

    . A- -

    , Lie

    2 ( ).

    , 3.5 3.6,

    5.5.

    5.1.

    [1]

    ( )

    Banach.

    , ( ) -

    , , Ch. Ehresmann.

    ,

    ( )

    , 1960.

    () 1-, -

    , -

    .

    (related conjugate) ,

    , .

    B G. , , ,

    h : 1(B) G( B ).

    23

  • ,

    (. [3], [7], [14]).

    [2] Banach

    .

    Banach (. () Banach)

    () , Banach.

    , (, .., -

    , Christoffel, .).

    [1].

    , ( ,

    )

    (. [4], [9]). -

    , [1],

    (f, , h) (. [11]).

    . , -

    . R

    ,

    S1 . ,

    , ( ) -

    Floquet-Liapunov.

    Floquet-Liapunov, -

    (. [12], [16], [17], [18].

    (jets).

    , ,

    (.

    [5], [8]).

    [3] Christoffel symbols and connection forms on infinite-dimensional fibre bundles

    = (E,B, ) Banach F, ( ) K : TE E , , () B., Christoffel

    i : Ui L2(BF,F), C = {Ui} B B . ,

    1(P (E), L(F)) (E) = (P (E), GL(F), B, ). i

    1(Ui, L(F)),i I.

    , , i i.

    24

  • (E), - Banach . , (Banach) ,

    .

    i i, , (., , [Kobayashi Nomizu: Foundations of Differential Geometry I ).

    [4] On the holonomy theorem

    Ambrose - Singer

    Banach.

    (P,G,B, ) B . , p (.

    op)

    (. ) p P , P [p] h Lie q(Xq, Yq), q P [p] Xq, Yq . :

    op G h Lie-Banach g G , Lie-Banach op ( p) h.

    () [9].

    . [9].

    [5] Connections on 1-jet principal fibre bundles

    P. Garca ([Rend. Sem. Mat. Univ. Padova 47 (1976)]) , = (E,G,X, ) , G E (1-jets) J1E , J1() = (J1E,G, (J1E)/G, ) . , J1() , : G- p : J1E E (: p(j1xs) = s(x)) , , ,

    = . (1)

    :

    1) .

    2) (1) .

    , ( ) |(E) , (1). :

    1- J1E ( G), (1) .

    (1) (, idG, G)-, G : X (J

    1E)/G .

    /

    , - (. [P. Garca - A. Perez-Rendon: Commun. Math. Phys. 13 (1969)]).

    25

  • [6] Affine transformations of banachable bundles of frames

    affine

    ( ).

    = (E,B, ) = (E, B, ) Banach, F F , L(F,F ). (. ) - (E) (. (E)) (f, , h) (E) (E), , :

    f = ,

    f =

    (f |1((0))

    ),

    ( ) : [0, 1] B. Lie-Banach (.

    )

    (. = h ), (. ).

    f = (f, idB) , f (E), f affine f = . , :

    f affine,

    f = (f |1((0))

    ),

    f = (f |1((0))

    ),

    , , .

    [7] (f, , h)-related connections and Liapunoffs theorem

    = (P,G,B, ) = (P , G, B, ) Banach, , (f, , h) . (f, , h)- ( ) , f , .

    , (f, idG,idB)- , B G. , :

    = (P,G,B, ) = (P , G,B, ) , . (f, idG, idB)- (: f = ) , - hi : U G ( C = {Ui | i I} B), :

    gij = hi gij h1j , on Ui Uj

    i = Ad(h1i )

    i + h

    1i dhi, on Ui, (2)

    i, j I.

    26

  • gij (. gij ) (.

    ) C,i (.

    i) (.

    ), C, hi : Ui G f ( ). h1i (b) :=hi(b)

    1, (2)

    i,b(v) = Ad(hi(b)1) i,b(v) + Tb(hi(b)1 hi)(v),

    b Ui, v TbB (a G) G a.

    (E) = (E,B, ) F,

    i,b(v) = hi(b)1 i,b(v) hi(b) + hi(b)

    1 (Dhi(b)

    )(v),

    D Frechet. , F = Rn,

    i = Q1i (

    i Qi DQi),

    n n , ( Liapunov) i

    i. ,

    Liapunov.

    () , i hi, , - , [20].

    [8] On the canonical connection of the 1-jet bundle of a principal bundle

    [5]. J1() , .

    J1().

    () :

    f , () J1f J1() p J1f = f p. , J1f (. (J1f) =), :

    = f ,

    (J1f) = f,

    , G- p : J1E E (. [5]).

    , , :

    f (: f = ) (J1f) = f J1f |(E).

    -

    .

    27

  • . [16], , -

    , [49], [50].

    [9] On the infinite-dimensional holonomytheorem

    [4]. -

    , -

    ,

    Banach, I. Gelfand

    Banach. Gelfand

    ( Frechet)

    , ,

    , (..

    Frechet).

    -

    Ambrose-Singer ([Trans. Amer. Math. Soc. 75 (1953)]),

    . , , -

    , (Reduction

    Theorem).

    -

    (. .. [S. Kobayashi K. Nomizu: Foundations of Differential Geometry I ).

    J. P. Penot [Bull. Soc. Math. France

    98 (1970), p. 67] . -

    (2002) J.-P. Magnot ([Structure groups and holonomy in infinite dimen-

    sions], . 3.7),

    -

    Omori-Milnor (regular ) ,

    .

    Ambrose-Singer.

    [10] On affine transformations of banachable bundles

    affine [6].

    = (E,B, ) Banach F, f = (f, idB) . [6], f affine , ( ) .

    ( [6]) f ( f ) (: f = ). , :

    GL(F)-B- affine

    .

    affine

    28

  • .

    , [6] [10] affine -

    ,

    .

    , , -

    ., , -

    exp ( ) (. S. Kobayashi K. Nomizu: Foundations of Differential Geometry I, Interscience, 1963]). -

    , . ,

    affine -

    ( ) .

    .

    [11] Conjugate connections and ordinary differential equations

    = (P,G,B, ) = (P , G, B, ) - (f, , h) . :

    (f, , h)-, f = ( Lie-Banach ), hi : Ui G

    ( C = {Ui}iI B), {i} {

    i}

    i = Ad(h1i ) (h

    i) + h1i dhi, (3)

    i I.

    -

    (f, , h) hi (. G-B- [N.Bourbaki: Varietes differentielles et analytiques, fasc. res., no 6.4.4]).

    G = G, B = B = idG, h = idB, (2) [7].

    :

    ) L = (R GL(F), GL(F),R, pr1). , L

    dx/dt = A(t) x,

    A : R L(F) . , L , . , -

    L (. [7]).

    ) = (R F,R, pr1). ,

    Christoffel.

    ,

    ( L ) ,

    29

  • . ,

    Christoffel .

    Floquet-Liapunov,

    (. [16], [17]).

    [12] Sur les connexions plates d un fibre banachique

    ( ) , -

    . :

    1. (.

    [14]).

    2. -

    ,

    (. [17]).

    3. ( )

    (. [16]

    , [14]).

    4. Floquet-Liapunov -

    (S1 GL(F), GL(F), S1, pr1) (. [15]).

    .

    [13] Transformations of linear connections

    [6] [10], affine

    .

    (

    ), ,

    , -

    .

    Christoffel -

    ( [3]) -

    Christoffel (

    ).

    ,

    -

    , -

    ( ).

    , affine

    [6] [10], [11]

    [S. Kobayashi K. Nomizu: Foundations of Differential

    Geometry I ( ), [T. V. Duc: Kodai Math.

    Sem. Rep. 25 (1975)] ( ).

    [14] Flat bundles and the holonomy homomorphisms

    = (P,G,B, ) Banach (: ) .

    30

  • (. [1], [2]) (, ) h : 1(B) G, .

    B G, . (, ) . (, ) (, ) , G-B- f f = , . . H(B,G) , .

    , hi : 1(B) G (i = 1, 2) h2 = g h1 g

    1, g G. , , S(B,G) .

    ,

    H(B,G) = S(B,G).

    , -

    , (.,

    .., [F. Kamber - P. Tondeur: Amer. J. Math. 89 (1967)], [N. Steenrod: The topology of

    fibre bundles] [M. F. Atiyah: Trans. Amer. Math. Soc. 85 (1987), 181207]).

    G , S(B,G) 1(B) G, F,

    1(B) GL(F).

    [15] Transformations of linear connections II

    [13] -

    .

    ,

    ,

    .

    LS = (S1 F, S1, pr1), F Banach,

    . ,

    , LS (), Christoffel .

    [11] (R Rn,R, pr1).

    -

    B. : BF TB (:

    ), (direct) B - , (, idB)- (B F, B, pr1). (opposite) ( torsion free) B .

    /

    B, , , ,

    31

  • (. [B. Bishop R. Crittenden: Geometry of manifolds]).

    , Lie-Banach ,

    -

    , .. : [S. Helgason: Differential geome-

    try, Lie groups and symmetric spaces], [W. Graeub: Liesche Gruppen und affin zusamme-

    nhagende Mannigfaltigkeiten, Acta Math. 106 (1961)] [B. Maissen: Lie Gruppen mit

    Banachraumen als Parameterraume, Acta Math. 108 (1962)].

    [16] Total differential equations and the structure of fibre bundles

    Floquet-Liapunov, / .

    [2], [11] [15],

    .

    [14], (, ), = (P,G,B, ), B , h : 1(B) G.

    , () dx x1 = , 1- B Lie G, - # : 1(B) G. U B f C(U,G), () Df df f1 f

    Df(b) = df(b)f(b)1 dbf Tf(b)f(b)1 Tbf, b U,

    g G g G. ,

    . (

    2 ):

    o o = (BG,G,B, pr1) dx x1 = o , o = s

    o ( s o),

    #o = g ho g

    1, . #o ho .

    [14]

    (triviality) :

    : = (P,G,B, ). F : B G (B B)

    F (b []) = F (b) h([]), (4)

    b B [] 1(B).

    , = (P,GL(F), S1, ), S1 = R, :

    : :

    i) H : R GL(F) H|Z = h.

    ii) (, ) (. [14]) (1, 1), 1

    (S1GL(F), S1, pr1) 1 1 .

    32

  • 1 B(t) = (exp 1)t(t),

    exp : R S1, 1 1, R.

    .

    , [17], 1 = (S1 GL(F), S1, pr1)

    ,

    . ,

    Floquet-Liapunov.

    , , [12].

    [17] Floquet-type connections on principal bundles

    -

    , [12]

    [15] .

    1 = (S1GL(F), GL(F), S1, pr1) (

    Banach) .

    1.

    [14],

    [16] :

    ( ) = (P,GL(F), S1, ). :

    i) h(1).ii) () (, ) (1, 1), 1 .

    , B = S1, h : Z GL(F). [16].

    [18] On affine extensions of the holonomy homomorphisms of flat principal bundles

    B B . h : B B affine, , dh = (dh|T(0)B), : I B, (.

    )

    (. = h ), (.). , :

    = (P,G,B, ) B , , :

    i) (, ) (. [14]) (o, o) o = (B

    G,G,B) o o, so

    33

  • (: s o) :

    (so) = 0. (5)

    ii) affine H : B G, (direct) G (. 2o [15]) B,

    H(b []) = H(b) h([]), (6)

    b B [] 1(B).

    Floquet-Liapunov

    , [49] [50] (.

    3.5 5.5.

    (5)

    ,

    .

    [19] Characterizations of flat bundles via total differentiation

    , -

    , -

    ,

    .

    , = (P,G,B, ) :

    1. P . , :

    , . , , (f, k), f : P G f(po) = e (: po ) k :G G Lie,

    f(p g) = k(g) f(p) g (p, g) P G.

    , Df f .

    , :

    11 (f, k) .

    2. .

    :

    - C = {Ui X | i I} fi :

    1(Ui) Ui G,

    fi(p g) = fi(p) g; p 1(Ui), g G,

    fj(p) = Cji fi(p); p 1(Ui Uj),

    Cji : 1(Ui Uj) G . ,

    Dfi 1(Ui).

    34

  • (f, k), , .

    [20] On a class of principal bundles over symplectic bases on Euclidean spaces

    (M,) . (M,) ( ) () = (P,G,M, ) , - .

    , , -

    .

    , , -

    .

    [. (gauges)] . , -

    (: gauge potentials, ) -

    (

    hi : Ui G [7] [11]).

    J. Werth ([Int. J. Theor. Phys. 12 (1975)]),

    G .

    ( ). , ,

    M ( Werth),

    .

    [7] [11], [1] (

    Werth).

    [24] On the triviality of homogeneous principal bundles

    -

    ( [16], [17], [18])

    .

    ,

    (P,U,M, ), U - M = G/H, G Lie H . M (reductive) , . m Lie g G :

    g = h + m Ad(H)(m) = m,

    h Lie H.

    , :

    35

  • ( ) , - h : H U , H : G H

    H(Ho) = e (Ho e H),

    H(g g ho) = H(g) H(g); g G, g G[e], ho Ho,

    G[e] ( e G) G G/H.

    , -

    (G, (P,U,M, ), ), M . (P,U,M, ), Lie G, . P : G P P M : GM M

    P = M (idG ),

    P (g, p) u = g P (p u); p P, g G, u U,

    U P .

    [41] Local connection forms revisited

    ()

    (. ..

    [7], [11], [20]).

    , (-

    ) () (adjoint representation)

    . , -

    ,

    Banach.

    (13) [7]

    .

    , P (P,G,B, ) Q = (Q,H,B, ), , ( 1) -

    (f, , idB). ( 2) -

    , : G H , (f, , idB), (f, , idB)-. , = idG, [7], - : G H , , (f, , idB).

    3, ,

    (f, , idB), P Q, (f, , idB)-. , Banach, 6.2 [Kobayashi Nomizu:

    Foundations of Differential Geometry I].

    36

  • ( ).

    :

    1) -

    . ,

    , , -

    ,

    , .

    2) Banach.

    -

    ( ) ,

    ,

    Frechet.

    (.

    5, 5.2), [43]).

    3) . , , (-

    ) (sheafification),

    . -

    (. 5, 5.3),

    [42]).

    [44] On related connections and the 1-jet principal fibre bundle

    . (f, idG, h)-

    [7] [11].

    J1().

    J1(), [10].

    [45] Some applications of conjugate connections

    , .

    -

    , ,

    .

    , -

    ,

    :

    1. , ,

    ()

    , (. [20]).

    2. affine -

    ,

    (. [6], [13]).

    37

  • 3. ( ) -

    ,

    . Lie (.

    [15]).

    4. () .

    , -

    , .

    ,

    -

    ( ) FloquetLiapunov

    (. [16], [17]).

    5.2.

    [21] On a type of total differential equations in Frechet spaces

    [16], [17] [18]

    , Banach.

    Frechet,

    .

    Lie-Frechet,

    , , ,

    Frechet (. [28] ). Lie-Frechet

    Lie-Frechet

    Lie-Banach.

    , {Gi, gij}i,jN Lie-Banach, -

    , G := lim

    Gi Lie-Frechet :

    1) Lie G G Lie Gi Gi, . G = lim

    Gi.

    2) expG : G G, limexpGi ,

    , , C0(R,R+) C0(M,G), M C- G Lie .

    Lie-Frechet -

    (. [J. Grabowski: Ann. Global Anal. Geom. 11 (1993), 213220] [G.

    Galanis: Period. Math. Hungarica 32 (1996), 179191]). ,

    nested Lie-Banach H. Omori

    ([Lecture Notes in Mathematics #427 (1974)]), -

    ,

    .

    38

  • , , -

    1- Banach M Lie G, . 1(M,G), = lim

    i, i

    1(M,Gi).

    , (. [16])

    Dx = dx x1 = , 1(M,G), , , , . d+ 12 [,] = 0.

    i

    1(M,Gi) .

    , :

    1(M,G) (. = 0, M ) affine # : 1(M) G.

    ( [18].)

    [25] A generalized frame bundle for certain Frechet vector bundles and linear con-

    nections

    , E - P (E) E. Banach (.

    [3], [13]).

    -

    Frechet F. , , ,

    GL(F) P (E), ,

    , E Frechet. , .

    Banach, -

    . - (. ..

    - (jets) J(E) Banach E, - ( ) F. Takens [J. Differential Geom. 14(1979),

    543562]).

    , {(Ei, B, i)}iN Banach Fi, , (

    ) E = lim

    Ei Frechet F = lim

    Fi.

    : E = lim

    Ei F = limFi

    Frechet, Banach

    {Ei, ij} {Fi, ij},

    Hi(E,F) := {(f1, ..., fi)|fk L(Ek,Fk) : jk fj = fk jk ; i j k}; i N,

    H(E,F) := {(fi)iN|fi L(Ei,Fi) : ji fj = fi ji ; j i}.

    39

  • Hi(E,F) Banach H(E,F) = limHi(E,F). ,

    Hoi (F) = Hi(F,F) i

    j=1

    Lis(Fj); i N,

    Ho(F) = H(F,F)

    j=1

    Lis(Fj),

    Lie-Banach Hi(F) = Hi(F,F), Ho(F) = lim

    Hoi (F).

    E = lim

    Ei,

    P(Ei) :=

    bB

    Hoi (F, Eb),

    Hoi (F) . :

    P(E) = lim

    P(Ei) Ho(F), (associated) E.

    ,

    . , E = lim

    i P(E). ,

    P(E) E i Ei. :

    E P(E) = lim

    i.

    [28] A Floquet-Liapunov theorem in Frechet spaces

    [21],

    Frechet ,

    , .

    GL(F), F Frechet, .

    Frechet F -

    Banach Ei, GL(F) Ho(F) L(F) H(F) := H(F,F) ( [25]), :

    .

    x= A(t) x , (7)

    () A : I =[0, 1] L(F) A = A, A :I H(F) : H(F) L(F) ((fi)) = lim

    fi. (7) .

    40

  • (7)

    .

    xi= Ai(t) xi ; i N, (8)

    Banach Ei.

    Floquet-Liapunov (7), A - ( , , 1).

    :

    Floquet-Liapunov Frechet: (7) -

    , :

    1. (7)

    .

    y= B y ; B , (9)

    y = ( Q)(t) x, Q : R Ho(F).2. B H(F) Exp(B) = #(1).3. # : Z Ho(F) (7) - F : R Ho(F).

    # ( - ) #(n) = (#i (n))iN,

    #i : Z GL(Ei) -

    (8), -

    i, . #i (n) = i(n), n Z. ,

    Exp : H(F) Ho(F),

    Exp = lim

    (exp1 expk),

    exp1 expk : Hk(F) Hok(F); k N,

    . , B B , Banach, (B) = B.

    C, :

    (7) A : I L(C), () .

    , , Floquet-

    Liapunov, . ,

    .

    -

    , Frechet,

    [16], [17] (. [21]).

    41

  • [32] On certain flat Frechet principal bundles and their holonomy homomorphisms

    Frechet,

    Banach (.

    [21], [25]).

    , -

    , 1(B) G = lim

    Gi ( B ). -

    Banach , -

    [14].

    -

    E P(E) ( - [25]) . , -

    1(B) Ho(F), 1(B) GL(F).

    B G ( B Ho(F), - ), . ,

    Frechet, [16], [18].

    [34] On associated Frechet vector bundles

    Ho(F), GL(F) Frechet F., () Frechet -

    F, : G Ho(F), G = lim

    Gi (i N) Frechet P = lim

    Pi(i N). :

    E = P F ( ) F. , E , P . Christoffel

    : (U) L(F,L(B,F)),

    (x)(u).y =(Te (x)

    )(y).u,

    x (U), y B, u F, () .

    , E P . 1-1 Te (. , [Pham Mau Quan: Introduction a la geometrie des varietes

    differentiables, Dunod, 1969]).

    E = P F, () : G GL(F). , E = PF . , :

    = (. - [28]), E

    .

    42

  • [37] Remarks on the cohomological classification of certain Frechet bundles

    [25], [32] [34], -

    ,

    Banach, . -

    (

    ).

    E {(Ei,X, i)}(i N), E ( ) Frechet F = lim

    Ei, Ei Banach

    Ei, . T :U U H

    o(F), {U} E. (T)

    T : U U GL(F) T = T ,

    : Ho(F) GL(F) ((gi)iN

    )= lim

    gi (. [28]).

    GL(F) Ho(F) - . , Ho(F) . , -

    VplX (F) ( ) F X, Banach, X. , Ho(F) X H0(F). ,

    VplX (F)= H1(X,Ho(F))

    10 (1st cohomologyset) X H0(F).

    ,

    ,

    . -

    (cohomologous)

    Ho(F) . , - .

    G- , Banach ( G Lie-Frechet, (Lie-Banach)

    ), ( )

    PX(G) = H1(X,G),

    :

    G X G,

    PX(G) G X.

    G =lim

    Gi ( ) Gi

    43

  • . , -

    Banach (

    -).

    [38] A generalized second order frame bundle for Frechet manifolds

    [Tangent and frame bundles of order two, Anal. Stiint. Univ. Al. I.

    Cuza (Iasi), 28 (1982), 6371], C. T. J. Dodson M. S. Radivoiovici

    2, T 2M ( : -) M . - 2

    (: ) T 2xM , x M , [] : J M , . ,

    x (0) = (0), (0) = (0), (0) = (0).

    T 2M M .

    Banach C. T. J. Dodson G.

    Galanis ([Second order tangent bundles of infinite-dimensional manifolds, J. Geom. Phys.

    52 (2004), 127136]).

    2

    Frechet .

    , ,

    Banach.

    [25]

    GL(F F) Ho(F F) (: F F). ,

    (

    Christoffel).

    , -

    . , ,

    Frechet (configurations) ,

    (evolution equations) 2

    .

    [43] Geometry in a Frechet Context: A Projective Limit Approach

    Banach,

    Frechet.

    , Frechet,

    Banach ( Frechet

    ,

    , Hausdorf, ), -

    . : ,

    , ,

    44

  • Frechet

    ,

    Lie.

    -

    . ,

    Lie-Frechet -

    (

    Frechet)

    (

    ).

    Frechet

    Banach, -

    ,

    Frechet,

    Banach, .

    , , -

    , Frechet, -

    , ,

    ,

    . -

    (Banach),

    .

    :

    1. Banach: -

    (Banach),

    Lie-Banach, ,

    . -

    ,

    .

    2. Frechet: Frechet -

    Banach, -

    ,

    Frechet.

    3. Frechet:

    Frechet, Banach,

    . Lie-Frechet,

    , ( Maurer-Cartan)

    , .

    4. :

    .

    , .

    ,

    (. 5), .

    .

    45

  • .

    5. : -

    Banach.

    GL(F) ( F Frechet ) -

    -. ,

    Lie-Banach H0(F), .

    6. () : -

    , J(E) - Banach E. Frechet,

    Banach Frechet. -

    . F. Takens [A global version of the

    inverse problem of the calculus of variations, J. Differential Geom. 14(1979), 543562],

    , J(E), . -

    ()

    Frechet, -,

    H(F).

    7. : -

    .

    , , -

    Frechet.

    Banach,

    Frechet .

    8. :

    T 2M . (. [39]

    Banach M , .

    T 2M , .

    .

    5.3.

    [22] From principal connections to connections on principal sheaves

    Geometry of Vector Sheaves. An Abstract Treatment of Differential

    Geometry (Kluwer, 1998), . ,

    - ,

    (sheaves), (

    46

  • [26] ).

    A- ( - ).

    ,

    ,

    , (

    ) [26]

    .

    , = (P,X,G, ) () ,

    1(P,G) ( G Lie G) {i

    1(Ui,G) | i I}

    j = Ad(g1ij ) i + g

    1ij dgij , (10)

    (gij) Z1(U , G) P ()

    U = {Ui | i I} X. , (i), -

    /. , U ,

    1(Ui,G) = 1X(Ui)A(Ui) L(Ui), i I. (11)

    1X ( ) 1- X R [ 1X(U)

    = 1(U,R), U X ], A X [ A(U) = C(U,R)] L ( Lie) X Lie G [L(U) = C(U,G)].

    , ( )

    C(X,G) f 7 f1 df 1(X,G),

    ,

    U : C(U,G) 1(U,G); U X ,

    [ (11)]

    : G 1X A L (12)

    G () X G (. G(U) = C(U,G)).

    , C- P P, - G, (ij) Z

    1(U ,G)., (11) i i (

    1XAL)(Ui),

    (10)

    j = Ad(1ij ) i + (ij), (13)

    Ad : G Aut(L) G Lie G.

    47

  • , -

    (

    ):

    P D : P 1 A L,

    D(s g) = Ad(g1) D(s) + (g); s P(U), g G(U),

    U X. D (i) C

    0(U ,1 A L) (13).

    , P ( ) D P. ( ). D - , .

    , [26],

    .

    [23] , [26] -

    .

    [26] Connections on principal sheaves

    -

    . , -

    .

    , (. ..

    R. Sikorski, M. A. Mostow ..). ,

    .

    1989, . , ,

    ,

    (, , ).

    (X,A), .

    . -

    ( [25]) Geometry of Vector Sheaves,

    ,

    , -

    .

    , -

    . , (X,A), (A, d,1), 1 A- d : A 1 , Lie G. :

    i) A- Lie L, . : G Aut(L).

    48

  • ii) Maurer-Cartan, . ()

    : G 1 A L,

    (g h) = (h1) (g) + (h); g, h G(U),

    U X. -

    X Lie G (. [22]), GL(n,A) , Lie .

    (P,X, ) Lie - G. G P ( ) P G( . [30]).

    P D : P 1 A L,

    D(s g) = (g1) D(s) + (g); (s, g) P(U) G(U),

    U X . :

    D 0- () C0(U ,1 A L), -

    = (g1 ) + (g). (14)

    U = {U X | I} P, .

    : P|U G|U

    (g) Z1(U ,G).

    (.

    [22]) :

    Atiyah

    [a(P)] = [(g)] H1(X,1 A L) .

    () -

    () P|U , Maurer-Cartan G|U .

    , D C(P). - P ( )

    g := (g1) + (g),

    1 A L. A. Aragnol ([Ann. Sci. Ecole Norm. Sup. 75 (1958), 257407]),

    , ,

    .

    -

    .

    49

  • [23] Transformations of sheaf connections

    (. [26])

    .

    , ,

    ,

    .

    [7]

    . , ,

    ,

    ,

    () .

    ,

    ,

    . ,

    Lie G (G,L, , ) G (G,L, , ) (, ), : G G : L L Lie, ,

    G X L

    - G X L G

    - G

    L

    ?

    - L

    ?

    1 A L

    ?

    1 - 1 A L

    ?

    1

    . L L .

    , P (P,G,X, ) P (P ,G,X, ) (f, , , idX), (, ) Lie f : P P

    f(p g) = f(p) (g); (p, g) P X G.

    .

    :

    D, D P P , , (f, , , idX ) - ,

    :

    1. , .

    (1 ) D = D f.

    50

  • 2. 0- (h) C0(U ,G),

    g = h (g) h1 ,

    (1 ) = (h1 )

    +

    (h),

    , I.

    -

    (pull-back) . -

    .

    (

    ) .

    A-( ) -

    . ,

    A- - (. [27]),

    :

    A- ( F )

    ( f , F ).

    ,

    moduli , Chern-Weil .

    [27] A. Mallios A-connections as connections on principal sheaves

    (X,A) (A, ,1), . (E ,X, ) n, A- .

    ( A) -

    E X E E AX E E

    , E An, , U = {U}, I, X A|U-

    : E|U An|U

    = (A|U)n, I. (15)

    (U), , .

    ( ) A-,. C- : E E A

    1, Leibniz-

    Koszul

    ( s) = (s) + s ,

    s E(U), A(U) U X . A- E D

    . (

    51

  • ) P(E) E , -

    U 7 IsoA|U (An|U , E|U ) ,

    U B, E :

    V B I : V U.

    P(E) GL(n,A).

    GL(n,A) Lie (. [26])

    := Ad : GL(n,A) Aut(Mn(A))

    Maurer-Cartan

    : GL(n,A)Mn(A)A 1 = Mn(

    1),

    (g) := g1 g, g GL(n,A)(U) = GL(n,A(U)).

    : Mn(A) 1(Mn(A))

    . GL(n,A) U 7 GL(n,A(U)), Mn(A) U 7 Mn(A(U)) A . (: , d, [26]).

    , E P(E), .

    E = P(E) GL(n,A) An,

    U 7 P(E)(U) An(U)/

    (f, a) (f , a) ! g GL(n,A)(U) : f = f g, a = g1(a),

    GL(n,A)(U) = IsoA|U (An|U ,A

    n|U ) P(E)(U) = IsoA|U (An|U , E

    n|U ).

    :

    A- E 11 - D P(E).

    1- = (ij) (14), I 1 i, j n.

    [29] Topological algebras and abstract differential geometry

    52

  • GL(n,A). P(E) - E , [27].

    , -

    D : P 1(Mn(A)) = 1 A Mn(A),

    D. , (A, d,1, d1,2 := 1 1), d1 : 1 2 K- (K = R, C) :

    d1( ) = (d) + d1 ; A(U), 1(Mn(A))(U),

    d1 d = 0,

    D : 1(Mn(A)) 2(Mn(A))

    D = d1 +1

    2[, ] = d1 + .

    , d1 d1 1-. ( ) 1(Mn(A))(U) Mn(

    1(U)), U X.

    , D - () R RD : P 2(Mn(A)) R := D D. R (. [31] ) , -

    , - , :

    -

    w

    2.

    -

    Frobenius Im() = ker(D), () = z. , :

    Frobenius, .

    , 2 .

    .

    ,

    :

    53

  • 1) Frobenius, . , -

    ( )

    .

    2) () de Rham

    0 C Ad 1

    d1 2

    .

    (

    ) Frobenius.

    (

    G) [42], [47]. .

    [30] Vector sheaves associated with principal sheaves

    A. Grothendieck [A general theory of fibre spaces with structural sheaf, Kansas

    Univ., 1958], (P,X, ) G (G,X, G).

    , (. -

    , , [27])

    P : G, x X U X x G|U - U : P|U

    G|U .

    , U = {U} X G|U-

    : P|U G|U (: ). ,

    , ,

    G Lie (. [26]). , , (P,G,X, ) G -

    P. [36].

    : G GL(n,A), G P GL(n,A) ( [27])

    U 7 GL(n,A(U)). , A X.

    (P) , E n. , E : U X, P(U)An(U), :

    (s, a) (t, b) ! g G(U) : t = s g b = (g1) a.

    Q(U) :=(P(U) An(U)

    )/G(U) -

    , E U 7 Q(U). (-

    n) (E ,X, ). E P(E) E , id : GL(n,A) GL(n,A).

    54

  • , Lie [. [23] 1 (p. 50)] P ( [26]) A- ( . , . [27]) .

    , -

    : G H, .

    [31] On the geometry of associated sheaves

    (P,G,X, p) : G H Lie - [. 1 (p. 50)]. [30],

    Q(U) :=(P(U) H(U)

    )/G(U)

    Q U 7 Q(U). Q = (P X H)/G, () P X H P H X.

    :

    i) Q H, Q (Q,H,X, q).

    ii) P Q

    (, , idX ) : (P,G,X, p) (Q,H,X, q).

    iii) DP P DQ Q, (, , idX )-, .

    P

    - Q

    1 A LG

    DP

    ?

    1 - 1 A LH

    DQ

    ?

    3

    (, ), : LG LH Lie Lie G H (. [23]).

    Lie

    (, ) : (G,L, p, ) (Gl(n,A),Mn(A),Ad,

    ).

    , [30], E , n. , , Qn (Qn,GL(n,A),X, n

    ).

    :

    55

  • i) Qn - P(E) E ,

    (, idGl(n,A), idX

    ).

    ii) P P(E) (F, , idX ), F = , : P Qn .

    iii) DP P DP() Dn P() Qn -, DP() = Dn.

    [36] On the cohomology and geometry of principal sheaves

    (P,G,X, ), [30]. U = {U} X, :

    P|U G|U , (s) P

    (g) Z1(U ,G). s P(U) := (U,G) g G(U U).

    , Grothendieck, -

    (g) Z1(U ,G) ,

    . ,

    X G, .

    , PX(G) ( -) ,

    PX(G) = H1(X,G),

    X - G. .

    -

    .

    (. [23], [30] [31]),

    ,

    () nA(X)= H1

    (X,GL(n,A)

    ).

    n X.

    (), , Geometry of Vector Sheaves

    . .

    ,

    G , -

    G 1(L) (. [26]). ,

    PX(G)D = H1

    (X,G

    (L)

    ).

    56

  • Pf

    - P

    1(L)

    DD-

    4

    , ,

    f : P P D,D P P , ,

    4 .

    Maxwell, . .

    Maxwell E 1 (: line sheaf ) A- .

    [40] Grassmann sheaves and the classification of vector sheaves

    -

    (vector sheaves). -

    . , -

    .

    X , A (A, A,X) , n E (E , ,X) A-. [27].

    (,

    , n) , . (classifying spaces),

    Grassmann. ,

    , , -

    (pull-backs)

    . ,

    ,

    .

    Grassmann. , k n N k-Grassmann GA(k, n), ( )

    GA(k, n)(U) :={S An|U : S = A

    k|U},

    U X.

    k-Grassmann GA(k, n) - k A.

    (fiber product)

    A :=

    iN

    Ai,

    57

  • U 7

    iN

    Ai(U), U X ,

    Ai = A, i N,

    X , E - X A.

    , (universal) Grassmann n GA(n)

    GA(U) :={S A|U : S = A

    n|U}.

    , :

    . n X - Grassmann GA(n).

    [42] Geometry of principal sheaves

    -

    .

    -

    , Chern-Weil, Riemann

    ( ),

    moduli , -

    , , / ,

    , ..

    -

    (

    ).

    :

    prwteuousec desmec

    dianusmatikec desmec

    -

    prwteuonta dragmata?

    dianusmatika dragmata?

    ?

    -

    58

  • , -

    ( ) ,

    , .

    , ,

    . 1950 A. Einstein [The mea-

    ning of relativity]

    .

    J. Nestruev ( Bourbaki), -

    Smooth Manifolds and Observables, Springer, 2002 (

    ) -

    .

    (., .., )

    ()

    ,

    (gauge theory), ,

    .

    [26], .

    .

    , -

    , [26], [47],

    .

    1. :

    ,

    .

    2. :

    . .

    , , -

    . -

    ,

    , (

    , ,

    ..). .

    , [The category of differential triads, Bull. Greek Math. Soc. 44 (2000),

    129141, .

    3. Lie :

    Lie. ,

    Maurer-Cartan -

    ( ) .

    .

    4. : A.

    Grothendieck, .

    ,

    59

  • Lie .

    5. :

    . (. 27).

    ,

    . ( )

    , -

    ,

    .

    6. :

    . -

    -

    ,

    . -

    Atiyah, .

    ( ),

    ,

    .

    7. :

    5.

    , . , -

    , (. [27]).

    8. : -

    . -

    Maurer-Cartan. ()

    Cartan Bianchi. -

    , .

    , ,

    , Frobenius.

    9. Chern-Weil:

    Chern-Weil .

    10. :

    . ,

    Riemann, ..

    [46] On the geometry of the sheaf of frames of a vector sheaf

    [27], -

    .

    [47] Flat principal sheaves

    [26] [29]

    ,

    Lie G, GL(n,A), [29].

    [26], D : G 1 A L := 1(L)

    (A, d,1, d1,2) (datum ,

    60

  • . ), . ,

    K- (K = R, C)

    D : 1(L) 2(L)

    D() := d1 + = d1 +1

    2[, ]

    d1 1(L). , D -

    D = 0 (16)

    D((g)

    )= (g) D(), (g, ) G X

    1(L). (17)

    (16), (17) , , ,

    X Lie GL(n,A).

    [29], D - () R RD : P 2 A L =:

    2(L) R := D D. R G- ,

    P- P 2(L), .

    R HomG(P,2(L)) = HomG(P,

    2(L))(X).

    := R(s) = d1 + ,

    = (g1 ) ,

    (s) P, (g) Z1(U ,G) P ()

    .

    (A, d,1, d1,2, d2,3) (Bianchidatum) () Bianchi

    d2R = [R,D],

    d2 [ , ] Hom(P,p(L)). , ,

    , .

    ,

    ,

    .

    :

    I. (. ).

    61

  • II. () ( D , U = {U X| I}, (s)I , . D(s) = 0)

    III. (. -

    D|1(U) = D = , ).

    IV. - (: - , (g ) (g) = 0).

    VI. (. , .

    (g) Z1(U , GX), GX X G -

    G ).

    2 (p. 53) [29],

    , :

    1 GX G

    1(L) (18)

    , ()

    (V) [() () (IV)] = () (19)

    Frobenius, -

    (19) .

    1 ker()i G

    1(L)

    D 2(L).

    0 ker()i G

    ker(D) 0,

    ( )

    0 Ho(X, ker())i Ho(X,G)

    Ho(X, ker(D))

    (20)

    H1(X, ker())i H1(X,G).

    : Frobenius, ker(D)(X), U X () hU G(U), U U , (hU ) = |U . () ()UV := hU hV , U , V U . , ,

    (20) . , () = () , . g Ho(X,G) G(X), = (g1) + (g).

    62

  • , G GX (18) , (21) :

    0 Ho(X,GX )i Ho(X,G)

    Ho(X, ker(D))

    H1(X,GX )i H1(X,G).

    . [42], ,

    i(L) () D.

    5.4.

    [33] Connections on A-frame bundles

    [25], [27], -

    (-

    ) .

    A-, -

    m- A. , ,

    . A- A-

    , (.

    .: [ A-. ,

    . , 1986] [Differentiation in modules over topological -algebras, J. Math.Anal. Appl. 170 (1992)]).

    , -

    A- P , (F (E), GLA(P ), B, ), F (E) =xB LisA(Ex, Ex) GLA(P ) A- P .

    F (E), GLA(P ) Lie. , , A Q-, . Ao A. :

    A Q-, GLA(P ) Lie F (E) . , A-

    : (X,TX) (X,E) (X,E)

    E 1-1 1

    (F (E), LA(P )big) F (E).

    : - () Christoffel : U L

    2A(M,P ;P ),

    U U X.

    ,x(v) h := (x)((v), h

    ), (21)

    63

  • x U, v TxX, h P : TxX M ,

    1(U, LA(P )),

    = Ad(g1 ) + g

    1 dg , (22)

    Christoffel.

    (22)

    () Co(U ,1

    (U, LA(P

    )))

    ,

    |1(U) := Ad(g1 )

    + g1 dg, (23)

    g : 1(U) GL(P ) , p = s((p))

    g p 1(U).

    , (), Christoffel (21), .

    , A-, A , -

    C(X) C(X). , A Q-, M(A) = X . , (), A Q-. , GLA(P ) Lie, (23) . ,

    , . , -

    [26]. Frechet (.[25]),

    F (E) , .

    LA(P ), , A-.

    F(E). , L(X,LA(P )

    ),

    () A- LA(TX,X LA(P )

    ),

    L(X,LA(P )

    )(U) =

    (U,LA

    (TX,X LA(P )

    ))

    ={s : U x 7 sx LA

    (TxX,LA(P )

    )},

    :

    ()

    D : F(E) L(X,LA(P )

    )

    D(s) = , s F(E),

    D(s g) = Ad(g1) D(s) + (g),

    s F(E)(U) g GL(P ). , GL(P ) X GLA(P ), - F (E) . , Ad (sheafification) .

    64

  • , :

    A ( Q), E 1-1 D () (F(E),GL(P ),X,

    ). A Q-, D

    1-1

    () F (E).

    L(X,LA(P )

    ) 1- X

    Lie LA(P ) 1AL, L

    Lie

    X LA(P ),

    1(U,LA(P )) 6= 1(U)A(U) L(U),

    U X [ (11) [22], ].

    [35] Universal connections on groupoids

    , , -

    ,

    . , ,

    , -

    , ..

    Lie, .

    ( ), , : B (, ) : B B , U G U , Lie G.

    ,

    Lie

    0 L(G)i L

    q TB 0. (24)

    , , (E,B, p), E Lie [ , ] - q : E TB

    q [s, t] = [q s, q t],

    [s, f t] = f [s, t] + (q s) t ,

    s, t E f C(B). (24) Lie L Lie

    L :=

    xB

    Txx; x := 1(x),

    x x, G ,

    G =

    xB

    xx ; xx :=

    1(x) 1(x).

    65

  • -

    ,

    . , Ngo Van Que ([Ann. Inst. Fourier (Grenoble) 17 (1967),157223])

    Q1 ={j1xs | s : s(x) = x, x B

    },

    1 : Q1 B : j1xs 7 x, (

    Ngo Van Que)

    Q1 1-1 .

    , J1 := 1

    J1 Lie, , :

    S = p2 TS,

    S ( S Q1). p2 L(J1) = T (Q1)B L 2 .

    ( ) P. Garca

    ([Rend. Sem. Mat. Univ. Padova 47 (1972), 227242]) L.

    Cordero C. T. Dodson M. de Leon ([Differential Geometry of Frame Bundles, Kluwer,

    1989]).

    . -

    , Garca,

    (P,G,B, ) PPG

    P . , , P (E) E, E.

    [39] Isomorphism classes for Banach vector bundle structures of second order

    [38], 2 (:

    ) T 2M , C.T.J.Dodson and M.S. Radivoiovici [Tangent and Frame bundles of order two, Anal. Stiint. Univ.

    "Al. I. Cuza" 28 (1982), 6371], , Banach, C.T.J.

    Dodson and G.N. Galanis [Bundles of acceleration on Banach manifolds (Invited paper).

    World Congress of Nonlinear Analysts, Orlando, June 30-July 7, 2004].

    M T 2M . - -

    .

    Christoffel . -

    T 2g : T 2M T 2N g : M N . - Tg : TM TN , T 2g T 2xg : T

    2xM T

    2g(x)N , (x M ) ,

    (T 2g, g) .

    66

  • M N , - T 2M T 2N g-, T 2xg (), T 2g .

    , -

    (. ), M N , g- Tg M =N T (Tg).

    Christoffel ( ),

    .

    .

    M , , g M . g-, , T 2M , .

    , : (M,) (M,) , g : M M , g-. , :

    [(M,)]g - T 2M . , (M,), [(M,)]g .

    (. L. Man-

    giarotti and M. Modugno [Fibred spaces, jet spaces and connections for field theories, Proc.

    International Meeting on Geometry and Physics, Florence, 12-15 October 1982, Ed. M.

    Modugno, Pitagora Editrice, Bologna, 1983, 135165], M. Modugno [Systems of vector

    valued forms on a fibred manifold and applications to gauge theories. In Proc. Conference

    Differential Geomeometric Methods in Mathematical Physics, Salamanca 1985, Lecture

    Notes in Mathematics 1251 (1987), Springer, pp. 238264],

    2,

    (Banach) .

    5.5.

    [48] I.

    .

    ( 3) -

    (4). 1 2

    , ( ) -

    , , Lie .

    67

  • N. Bourbaki, -

    [Varietes differentielles et analytiques, Fascicule de resultats, 17],

    .

    [49]

    , , S. G. Krein

    N. I. Jackin (S. G. KrenN. I. kin). ,

    , . ,

    , ,

    .

    Dx = , 1(X,G) , (25)

    X , G Lie D (), . f : X U G (25), Df := f1 df (. [16]).

    ( Lie) -

    . , : R GL(n,R) (25).

    [50] , :

    Floquet-Liapunov

    ()

    ,

    [16], [17] [18]. , -

    [49].

    , :

    1. : , -

    , .

    2. : -

    [49].

    3. :

    , , ,

    (. [12].

    4.

    , FloquetLiapunov,

    .

    [51]

    . .

    . :

    68

  • 1. ( 0),

    (affine) ,

    , (-

    1).

    2. , , ,

    / ( 2).

    3. P2(R) (- 3).

    4. Desargues ,

    ( 4 5).

    5. .

    Pascal Brianchon ( 7).

    6. :

    . R Desargues .

    . P2(R), - P2(D) D. . -

    , P P2(R), R P.

    , , , -

    .

    [52]

    .

    1986 :

    . : (affine) ,

    , , . , -

    , ,

    .

    . : , .

    1987,

    ( . )

    , -

    -. , ,

    , ,

    .

    [53] :

    Lie

    ,

    , ,

    .

    69

  • Lie.

    :

    1. : , -

    .

    2. : , ,

    , .

    3. : , Lie, ,

    , .

    4. Lie: Lie, Lie Lie,

    Lie, , .

    ,

    (

    ).

    , .

    [54]

    -

    , , :

    . : , , ,

    .

    . : , -

    , Whitney, (pull-back) -

    , , VB(X), .

    . : , ,

    , , ,

    , .

    IV. : ,

    , , , -

    .

    V. : ,

    ,

    Serre-Swan.

    VI. Riemann .

    (

    SerreSwan).

    [55]

    . -

    :

    1. : ,

    , Frenet-Serret,

    .

    70

  • 2. : , -

    ,

    , .

    3. Gauss: , Gauss

    , Gauss, -

    - Meusnier+.

    , ,

    . , ,

    (

    ).

    [56]

    . -

    /

    . -

    , 2012.

    / -

    / . -

    , -

    ,

    . / / ( -

    ) .

    ( )

    , -

    .

    :

    1 .

    , ,

    ,

    (,

    ). ,

    .

    2 .

    .

    , Hilbert.

    -

    .

    , , ,

    . ,

    , ,

    . , -

    / /

    ()

    .

    71

  • 3 .

    ,

    . /

    () -

    () . -

    (

    , ). -

    ,

    Frenet-Serret,

    .

    4 , -

    (

    Egregium Gauss) . -

    ,

    ( ) Einstein,

    .

    , /

    , , -

    .

    /,

    , , .

    , ,

    () , , -

    ( tablet).

    72

    GENIKA STOIQEIA KAI SPOUDESAtomik'a Stoiqe'iaEpisthmoniko'i T'itloiAkadhmak'h Ex'elixhEreunhtik'a Endiaf'erontaSpoud'ec kai Ereuna sthn Ell'adaSpoud'ec kai Ereuna sto Exwterik'oJerin'a Sqole'ia kai Semin'ariaUpotrof'iecM'eloc Episthmonik'wn Etairei'wnX'enec Gl'wssec

    AKADHMAIKH DRASTHRIOTHTADidaktik'h Drasthri'othtaDiorg'anwsh Seminar'iwnSummetoq'h se Sun'edriaDial'exeicEp'ibleyh Didaktorik'wn Diatrib'wnEp'ibleyh Diplwmatik'wn Ergasi'wnAllec Ereunhtik'ec Drasthri'othtecEreunhtik'a Progr'ammataM'eloc Akadhmak'wn Epitrop'wnAllec Panepisthmiak'ec Drasthri'othtec

    EPISTHMONIKES ERGASIESDiatrib'ecErgas'iec se periodik'a kai praktik'a sunedr'iwn me krit'hEreunhtik'a bibl'iaAllec ergas'iecMonograf'iec kai Didaktik'a Bibl'iaShmei'wseic

    ANAFORESANALUSH EPISTHMONIKOU ERGOUAn'alush ergasi'wn thc om'adac AAn'alush ergasi'wn om'adac BAn'alush ergasi'wn om'adac GAn'alush ergasi'wn om'adac DPerieq'omeno thc suggrafik'hc drasthri'othtac