ΥΠΟΜΝΗΜΑ ΕΠΙΣΤΗΜΟΝΙΚΗΣ...
Transcript of ΥΠΟΜΝΗΜΑ ΕΠΙΣΤΗΜΟΝΙΚΗΣ...
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2017
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2005,
( 2006).
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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
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.
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13
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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
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