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38
Том 153, кн. 3
УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА
Физико-математические пауки
2011
UDK 514.763.2
TANGENT BUNDLES AND GAUGE GROUPS
M. Rahula, V. Balan
Abstract
The differentials Tka (k > 1) of a diffeomorphism a of a smooth manifold M induce in the fibers of the fiber bundles
TkM, in the corresponding tangent spaces, linear transformations, which embody the action of the gauge group Gk ■ This action extends in a natural way to the osculating subbundles Osck-1M C TkM.
Key words: diffeomorpliism of a smooth manifold, fiber bundles, action of the gauge group.
Introduction
The differential group G of a smooth manifold M induces in the tangent bundle TkM an action of the group of fc-jets of transformations. More specifically, if a is a diffeomorpliism of the manifold M, then its fc-th differential Tka is a transformation of the level TkM. Then the level TkM may be regarded as a homogeneous space Jk/Hk, where Jk is the group of fc-jets of transformations and Hk is the stabilizer of an element u(u) G TkM. The gauge group Gk is defined as a certain subgroup of the linear group GL(2kn,R), where n = dim M which is isomorphic to the stabilizer Hk. The action of the group G extends to the osculating subbundle Osck-1M c TkM.
The paper contains all the necessary definitions and founds all the previous considerations. Commented examples and groups of derived formulas are presented as exercises.
1. Tangent groups
T
smooth manifolds M1 and M2:
T(M1 x M2) = (TM1 x M2) © (M1 x TM2),
and for the smooth mapping from M1 x M2 to some smooth manifold M
A : M1 x M2 —► M : (u, v) ^ w = u ■ v,
we define the tangent mapping TA. First, by fixing the points u G M^d v G M2 we define two mappings Au and Av:
Au : M2 ^ M : v ^ u ■ v, Av : M1 ^ M : u ^ u ■ v.
Theorem 1. To the pair of vectors u1 G TuM1 and, v1 G TvM2 the mapping TA associates the vector w1 G TwM, and, we have
w = u • v ^ wi = Ml • v + и • vi
(1)
w/iere u1 ■ v = TAv(u1) rad u ■ v1 = TAu(v1). /n short, one can apply to the "product" w = u ■ v the Leibniz rule.
Proof. We specify that, by means of the tangent maps TAv and TAu, two vectors u1 G TuM1 and v1 G TvM2 are transported from the points u G M^d v G M2 to the point w G M, where their sum defines the vector w1 G Tw M. Locally, this is confirmed by the formula:
6\p ■ 3\p
w»=X"(u\va) w? = —u[ + —v?,
where ul, va, wp are the coordinates of th e points u,v,Md ui, v f,w p are the components of the vectors u 1;v 1;w 1 on the neighborhoods U1 C M1, U2 c M2, U C M, i = 1,..., dim Mi, a = 1,..., dimM2, p= 1,..., dimM. □
Using the Leibniz rule we derive a set of important formulas in coordinate free form.
Exercise 1: action of Leibniz rule. Show that the Leibniz rule can be applied to the "product" of several factors, e.g.,
(u • v • w)l = Ml • v • w + u • vi • w + u • v • wi.
Exercise 2: prolongation of Leibniz rule. Prove that for the second tangent mapping T2A, the following formulas hold true:
w = u • v, Wl = Ml • v + M • vi,
W2 = U2 • v + u • v2, (2)
Wi2 = Ml2 • v + U2 • vi + Ml • v2 + u • vi2.
Exercise 3: functional equation.
Question: how can one solve the equation (u • v)l = ul • v + u • vl with respect to ul for given vl and (u • v)i, or relative to vi for given u^d (u • v)i ? This reminds the method of integration by parts:
d(uv) = udv + vdu ^ uv = J udv + J vdu, whence either / udv = uv — vdu, ot / vdu = uv — / udv.
1.2. Coordinate-free story. The rulo (1) is easy to uso while building tangent groups and further, while studying representations of groups. If we have previously denoted the ''product" of elements by a dot, as in (1) and (2), then while denoting the product of group elements, the dot will be omitted.
To a Lie group G with composition rule 7 : (a, b) ^ c = ab, we associate the tangent group TG, having the composition law T7 :
ab =>• ci =016+ 061. (3)
The vectors a1 G TaG and b1 e TbG are transported % means of the right shift rb = 75 and of the left shift la == Ya, more exactly, % means of the tangent mappings Trb and Tla, from the points ^d b to the point c, where the sum a1b+ab1 = Trb(a1 )+Tla(b1) determines the vector c1 e TcG. This is the composition law on the tangent group TG. The unity of the group TG is the null vector from TeG. The inversion for the elements TG
a1 G TaG ^ a-1 = —a-1a1a-1 € Ta-1G. (4)
Exercise 4: unity and inverse elements. Using (3), conlirm the assertion regarding the unity of the group TG and the inversion of the elements (4). The formula (4) is obtained by solving the equation 046 + abi = 0 relative to bi for b = a-i .
Exercise 5: matrix representation. Prove that the formulas (3) can be represented in matrix form as
( a 0\ ( b 0\ = (a 0\ ( e 0\ (b 0\
Ui a) \bi b) \0 a) Va-lai + bib-i ej ' V0 b) ' ( )
The sum of vectors in TabG reduces to the sum in TGe :
aib + abi = T(1a o rb)(a-1 ai + bib-1). Explain the meaning of the equality aib + abi = (aia-1) c + c (b-1bi).
Exercise 6: second tangent group. Prove that in the second tangent group T2G the product of elements is delined by the formulas
c = ab, ci = aib + abi ,
C2 = a2b + ab2 , (6)
ci2 = ai2b + a2bi + aib2 + abi2 ,
and the inversion is performed by the rule
(a, ai, a2, ai2)-1 = (a-1, a—1, a—1, a-2), where a—1 = —a-1aia—1 ,
a—1 = —a-1a2a-1 , (7)
a-2 = —a-1ai2 a-1 + a-1a2a-1aia-1 + a-1aia-1a2a-1 .
Exercise 7: classical formulas. Reduce the formulas (6) and (7) to the well known formulas from Analysis:
(uv)' = u'v + uv', (uv)'' = u''v + 2u'v' + uv'', 1 \' fly -uu" + 2(m')2
u u2
Exercise 8: matrix relations. Using (6) and (7) prove the matrix relations:
ic 0
ci c
c2 0
\ci2 c2
a
ai
a2
0\
0
0
c!
\ai2
0 a 0
a2
aa ai a2 ai2
-i
0 a 0
a2
ai
-i
0 0
0 a
bi b2 \bi2
0 0 b bi
0
0
0
0
a-1 -1
a
0
0 0
-1
0\ 0 0
a! V«x2
Which endomorphism is involved here?
Exercise 9: logarithmic derivatives. Following the example of (5), represent the product of elements of the group T2G in the form:
a0 ai a 00 00
0 0 a ai
0 0 0 e/ -i
0 0
0 0 0
bib .In this generalization, both
0\/ e 0 0 0\ lb 0
0 0 e
0 ^ a-1a2 + b2b-1 0
V 0 0 ai aj \(a-1a2 + b2b- 1)i a-1a2 + b2b-1
where (a-1 a2 + b2b-1)i = a-1ai2 — a- 1aia- 1a2 + bi2b-1 — b2b-logarithmic derivatives (ln(uv))^d (ln(uv))'', are present.
At the unity e G G we fa the tangent vector e1 G TeG. This vector is displaced by left shifts Za over the group G to produce the left-invariant vector field aei and by right shifts ra, to produce the right-invariant vector field e1^. If at the unit e G G we
TeG
G
another, at the point a G G, is defined % some matrix A(a), which is an element of the linear group GL = GL(dim G, R). By this way, we define a homomorphism of the group G into the linear group GL:
G ^ GL : a ^ A(a).
(8)
Exercise 10: right/left shifts and inner automorphisms. Show that an 1-parametric subgroup at of the group G defines in the group G three flows corresponding to right shifts, left shifts and inner automorphisms
rat = exp tX, lat = exp tX, Aat = lat ° r—1 = exP t(X — X)
and, accordingly, the left-invariant operator X, the right-invariant operator X and the adjoint representation operator Y = XX — X. Prove this, using the formulas
Xf =(f ° rat )t = 0, XX f =(f ° lat )t = 0, Yf =(f ° Aat )t = 0,
where f is an arbitrary smooth function on G, taking into consideration that left shifts commute with right shifts.
1.3. Elements of representation theory. Wo consider a diffbroritiablo manifold, which we shall call representation space for the group G, or, in the following, simply space. A smooth mapping
A : M x G —► M : (u,a) ^ v = u ■ a
defines an action of the group G on the space M, if all the mappings
Aa : M ^ M u ^ u ■ a, Va £ G,
are transformations (diffeomorphisms) of the space M, and the mapping a ^ Aa is
GM homomorphism a ^ Aa is understood either in the sense of the equality Aab = Aa o Ab or in the sense of the equality Aab = Ab o Aa. In the first case we say that the action GM v = a ■ u v = u ■ a
v = a ■ u ^ (ab) ■ u = a ■ (b ■ u), v = u ■ a ^ u ■ (ab) = (u ■ a) ■ b.
GM
The kernel of the homomorphism a ^ Aa is called the stabilizer subgroup of the G
sists of the unity e £ G, and the mapping a ^ Aa is injective. For a fixed point u £ M, the mapping
Au : G ^ M : a ^ u ■ a
MM Au(G). When Au(G) = M, i.e., when the space M is the only orbit of the group G, we
G M M
GM
GM G
Equation 11: action by right/left shifts and inner automorphisms. Show that the actions G
actions provided by inner automorphisms are non-transitive.
The tangent map of the mapping A, i.e., TA, defines a representation of the tangent group TG on the first level TM,
TA : TM x TG ^ TM : (ui,ai) ^ vi = ui ■ a + u ■ ai
Formula (1) looks similarly:
v = u ■ a ^ vi = ui ■ a + u ■ ai.
(9)
Wo remark two particular cases:
for ai = 0 we define the action of the group G on the level TM,
ai =0 ^ ui ^ vi = ui • a;
for ui = 0 we define the action of the tan gent group TG on the space M
ui =0 ^
ei = a ai ^ vi = u ■ ai = v ■ a ai = v ■ ei.
The formula
vi = v ■ a ai
(10)
is the fundamental formula of the theory of Lie group representations.
In fact, to the vector ei = a-iai G TeG (which is an element of the Lie algebra g) at the point v G M we associate some vector vi G TvM, and since v is an arbitrary
M
Gi
ei G TeG, in the space M we have an infinite set of group operators, and all of them, as vector fields, are tangent to the corresponding orbits.
For vi = 0 the equality (10) provides the equation v • a-iai = 0, or v • ei =0, which determines in the space TeG those directions ei, along which the point v G M remains
G
which contains the point e G G, is a subgroup Hv c G called the stationary subgroup
v
In coordinates (va) on the neighbor hood U c M of the point v G M, Eq. (10) is written as a system dva = where are the forms of the left-invariant coframe
on the group G. There appears a matrix2 £ = (£a), which determines a system of forms on the group G, and in the space M, a system of basic operators X®:
The number of operators X® is equal to the dimension of G, and the number of forms is the dimension of M. Operators X® and forms are not necessarily linearly-independent. The Pfafi system £f w® = 0 for a fixed point v G M is completely integrable and defines the stabilizer Hv c G.
Exercise 12: vision from the classical theory. Show that the system = 0 is the coordinate form of the equation v ■ aTxa\ = 0.
1.4. Adjoint representation. In the groups G and TG we define the action by-left shifts:
1Since the time of S. Lie and frequently nowadays, group operators have been called infinitesimal transformations or fundamental vector fields of the group.
2The matrix J plays an essential role in the theory of Lie group representations (see, e.g., S. Lie Theorems).
la : b ^ c = ab, Tlai : bi ^ ci = (aia-i) c + abi,
ci = (oio 1) c ,
(ID
right shifts:
rn : b ^ c = b a,
Trni : b\ ^ ci = b\ a + c (a 1ai)
ci = c (a ai)
(12)
arid inner autoniorphisnis:
Aa : b ^ c = aba-1, TAni : bi ^ ci = (aia-1) c — c (aia-1) + a bi a-1,
ci = (aia 1) c — c (aia
(13)
The basic formula (10) is rewritten, for b1 = 0, in the forms (11), (12) and (13), respectively.
Inner autoniorphisnis are directly related to higher order movements.
Hence, when in the spaces A Mid B there take place the transformations a Mid b, the mapping y : A ^ B is brought into the mapping y : A ^ B. This is shown by the diagram:
A B
a 1
A
1 B
y ^ y = bya
If we set here A = B, a = b and if y is a difleomorphism, i.e., y is a transformation of the space A, then this diagram describes the transformation of the mapping y, subject
a
A
A
A 1
A
y
^ y = aya
-1
(14)
y
Exercise 13: higher order transformations. The transformation of order 2 <p ^ p is described by the 2-dimensional diagram (1.14). Show that the transformation of order 3, i.e., a transformation of transformation <p ^ Y> ¡s described by a 3-dimensional diagram and the transformation of order k is described by a corresponding k-dimensional diagram.
If the arrow a in diagram (14) is assumed to represent the 1-parametric group at of transformations of the space A, or in brief, the flow at, then we see how, to a change of the parameter t (of time), it corresponds to a change of the mapping yt = ajya-1.
We can talk then about a 1-parametric family of mappings yt in the fie Id at. y
bT of the space A (the flow bT), then we can see how this flow changes under the transformation a, i.e., bT ^ bT = abTa-1.
Exercise 14: transformation of the flow. Show that if bT is the flow of the vector field Y and bT is the flow of the field Y, then Y = TaY, and the tangent mapping Ta acts rn the field Y:
= exp tY
= abT a = exp tY,
Y ^ Y = TaY.
Exercise 15: interaction of vector fields. Let X and Y be two vector fields. The flows of these fields at = exp tX and bT = exp tY interact according to the scheme:
bT ^ atbTa—t, Y ^ TatY,
at ^ bTatb—T, X ^ TbTX.
Using derivatives of the function f
Xf =(f о at)'t=о and Yf = (f о bT)'T=0 ,
perform the differentiation (the parameters i or т from above the arrow mean differentiation relative to i for i u 0 or tо т Дог т u 0):
f о (atbTa-1) -U (Xf) о bT - X(f о bT) -U (YX - XY)f, f о (Ьтatb-1) -U (Yf) о at - Y(f о at) -U (XY - YX)f.
Check the validity of the relation (TatY)£_0 = - (ТЬтX)T_0 and establish a connection with the brackets [X, Y] = XY - YX.
If in one flow the points move along trajectories and under the influence of the other flow, this movement is transformed, and then the movement of the movement takes place, or a second-order movement. Under the influence of a third flow, the movement of second order changes its shape, then the movement of third order occurs, etc. In the infinitesimal approach this reduces to the iterations
at ^ Tat ^ T2at ^ ...
and to the corresponding vector fields on the levels
(1) (2) , (k) X ^ X ^ X ^ T kat = exp tX, k = 0,1, 2,... (15)
In this way, the flow Tkat induces a movement of order k.
1.5. Gauge groups. Let us fix on each level a point
(k) € T M, such that nk(u(k))= W(k-1), k = 0,1, 2,... In the neighborhood TkU c TkM these points are defined by their coordinates:
U : u(0) ^ (u1), TU : u(i) (m®,M1) ,
rji2j T , / i i i i \
T U : u(2) ^ (u ,u1,u2,u12) >
T3U : u(3) ^ (u\u1,u2,u12,u3,u13,u23,u123) >
UcM
ui ^ Mi o a = ai
TkU
i i i i i i i i i i i i (u , u1, u2? u12> . . . ) ^ (w > u1> w12> . . . ) = (a > a1> a2> a127 . . . ).
TU
the system
{Mi = ai,
«1 = a1 = aj «1,
with the Jacobian block-matrix
/ ai 0\ dai $2ai Ul)! whcrc = = № =
then the transformation of coordinates in the neighborhood T2U are defined by the system
ÏÏ = ai,
ÏÏ = ai = a j u i,
ÏÏ = a2 = aj u2j
^2 = a12 = ajkuiuk - 1- aj ui2
with the Jacobian block-matrix
aj (aj )i 0 0 0
aj 0 0
(«j )2 0 aj 0
\(aj) 12 (aj )2 (aj )i aj
(aj )i = ajk ui> where ( (aj)2 = ajkuf,
>j) 12 = j uÎu2 + aj k u12
etc.
When performing a lift from one level to another, U ^ TU ^ T2U Jacobian matrix is inductively built according to the scheme:
the
, a 0
a i
a1a
a 0 0 0
a1 a 0 0
a2 0 a 0
a12 a2 a1 a
(16)
with repeated, as shown above, n-dimensional blocks
(aj), ai = (aj)i, a2 = (aj)2, ai2 = (aj
) 12,
Therefore, there follows the general rule: the Jacobian matrix of the transformation of coordinates on the neighborhood, TkU is of the form
A 0 Ak A
(17)
where the block A is the Jacobian matrix on Tk-1U and Ak = dkA, k = 1,2,...
In other words, the Jacobian matrix on the neighborhood TkU consists of four blocks, where the Jacobian matrix A of the neighbor hood T k-1U is repeated on the diagonal, the upper-right block is zero, and the left-lower block is the differential of the block A taking into consideration the k-th level, i.e., Ak = dkA.
Formula (17) defines the sequence of matrices (16).
Exercise 16: inversion rule. Show that the inversion of the matrix (17) takes place according to the scheme:
(A a) = (A^A-1 e) -1
A 0 0 A
A0 ^ \Ak A
where E is the identity block. See (4) and
ai = (aia-1)a
A-1 0
0
A-1
E
-Ak A-1
= —a 1 (a1a 1).
1
The matrix (17) depends on the point u(k) G TkU. If this point is fixed, then
ai
(or the corresponding jet of the transformation).
Exercise 17: gauge group. Show that all matrices of the form (17), with the point u(k) € Tk U fixed, determine a subgroup of the linear group of order 2kn,
Gk C GL(2kn, R).
Prove the existence of the groups Gi, G2, G3 and extend to Gk ■
We call the group Gk of matrices (17) with fixed point u(k) e TkU the gauge group of order k on the manifold M. By setting k = 0,1,2, 3,..., we obtain an infinite sequence of gauge groups
G Gi ^ G2 G3 ^ ... (18)
kk of the linear group GL(n, R), which, in its turn, is embedded, in the linear group GL(2kn, R);
Gk « Tk (GL(n, R)) c GL(2kn, R), k = 0,1,2, ... (19)
In this case
dimGL(2kn, R) = (2kn)2 and dim Gk = 2kn2.
Proof. We fix the element u(k) e TkM of the k-th level. Matrices (17) generate a subgroup Gk of the linear group GL(2kn, R) (see Exercise 17). The fixing of the point u(k) does not limit the freedom of choice for the element (17) in the group Gk • Hence, the group Gk is uniquely ^^d regardless of the point u(k) e TkM. On the other side, the tangent group Tk (GL(n, R)) coincides up to an isomorphism, with the matrix Gk
assume G = GL(n,~R). □
Further, in the matrix (17), besides the point u(k) e TkU, there exists the k-jet of coordinate transformations (aj, ajii2,..., ajli2...ifc). We shall denote as Jk the group of u
Xk : Jk ^ Gk. (20)
Exercise 18: jets and gauge group. Show that for k = 2, the mapping X2 is homomorphic, .e., to a composition of 2-jets (ak , a-^) and (bk ,akl) there corresponds the product of matrices A2 ,
a, 0\ ( j 0\( akj 0
aklu1 akJ \bjlu1 bjl \(ak bj )lu1 ak bj
l r,i / ' hk „,l hk i !„i hk\,„,l
(ak bk)lu1 ak
and to the inverse 2-jet (aj ,aji) 1 = (aj , —a\aakiak), there corresponds the inverse matrix A- 1
ak 0 \-1 = / aj 0 akiui ak) \-aiakiakui aj k
Exercise 19: homogeneity of tangent space. Show that the kernel of the homomorphism Xk is the stabilizer Hu^k) of the element u(k) € Tk M in the group Jk . The tangent space Tk M is identified with the homogeneous space Jk/Hu(k).
G
is the linear group GL(n, R),
G = GL(n, R).
The second group £1 is isomorphic to the tangent group T(GL(n, R)). Its elements are block matrices of the form
a 0 a1 a
where a G GL(n, R) and a1 G gl(n,R).
The correspondence £1 T(GL(n, R)) is one-to-one. The product of elements in the group £1,
fa 0\ (b 0\ f ab 0
ya1 a) \b1 by \(ab)1 aby
reduces to the Leibniz rule in the tangent group T(GL(n,
(ab)1 = a1b + ab1,
and the inversion of elements in
1
a0 a1 a
a-1 0
-a-1a1a-1 a-1 ' '
reduces to the rule
-1 -1 -1 a- = -a 1a1a 1.
This speaks about an isomorphism between the groups T(GL(n, R)). An inner
authomorphism in £1 is generated as allows:
a0 a1 a
b0 b1 b
a0 a1 a
-1
-1
aba
(aba-1)1 aba
-1
with the block (aba-1)i = abia-1 + aia-1(aba-1) - (aba-1)aia-1, etc.
The following group £2 is isomorphic to the tangent group T2(GL(n, R)). The stair-like structure appears again:
where
a 0 0 0 b 0 00 ab 0 0 0
a1 a 0 0 b1 b 0 0 (ab)1 ab 0 0
a2 0 a 0 b2 0 b 0 (ab)2 0 ab 0
a12 a2 a1 a b12 b2 b1 b \(ab)12 (ab)2 (ab)1 ab
(ab)1 = a1 b + ab1,
(ab)2 = a2b + ab2,
(ab)12 = a12b + a2b1 + a1b2 + ab 12.
Exercise 20: logarithmic rule for gauge group. Show that while forming the blocks a-1ax ^ (a-1ai)2 = a-1ai2 — a-1a2a-1a1 ^ ...
there appears the following property of the logarithmic function
U u" (u')2 In u — ---
2
We shall further denote the Lie algebra of the group Qk by Qk ■
u
u
The general scheme is the following. An element of the group Gk is generated according to the principle:
A 0 Ak A
where A G Gk-i, Ak G Gk-i-
The product and the inversion of elements.
A0 Ak A
B 0 Bk B
AB
V(AB )k
0
AB
A0 Ak A
A
0
A- A
-i
reduce to the rules:
(AB)k = Ak B + ABk, A-1 = -A-1Ak A-1.
The Lie algebra Gk-1 is identified with the additive subgroup of the matrix group Gk ■ whose matrices have the form:
E 0 Ak E
(21)
where E is the unit block, i.e., the unity of the group Gk-1- The product and the inversion of such matrices are performed in the following way:
E0 Ak E
E0 Bk E
E0 Ak E
-i
E0
Ak + Bk E
E0 -Ak E
Gk
1
A0 Ak A
E0 Bk E
A0 Ak A
E
ABkA-1
Gk
Bk * Bk = ABk A-1.
Under such a transformation, the spectrum of the matrix Bk is preserved. The invariants will be the eigenvalues of this matrix and the corresponding symmetric polynomials, which are coefficients in the Hamilton Cayley formula.
Exercise 21: Lie algebra of the Lie group. Show that the Lie algebra of an arbitrary Lie group G may be regarded as an additive subgroup and a normal divisor of the tangent group TG. Describe the cosets of this normal divisor and the corresponding quotient group of the group TG.
G
a
tangent groups TG, T2G and T3G can be similarly put into a spacial matrices of type b), c) and d), respectively.
2. Tangent bundles and oscillators
2.1. Levels and sector-forms. The tangent functor T iterated k times associates to a smooth manifold M its k-fold tangent bundle TkM (the k-th level of M) and associates to a smooth map ^ : Mi ^ M2 the graded morphism TV : TkM1 ^ TkM2, the k-th derivative of The level TkM has a multiple vector bundle structure with k projections onto Tk-1M:
ps = Tk-sns : TkM ^ Tk-1M, s = 1, 2,..., k,
where ns is the natural projection TsM ^ Ts-1M. Local coordinates in neighborhoods
TsU C TsM, s = 1, 2,..., k, where Ts-1U = ns(TsU),
are determined automatically by those in the neighborhood U C M, the quantities (uJ) being regarded either as coordinate functions on U or as the coordinate components of the point u G U:
U: (u4), i = 1, 2,..., n = dim M, TU (u®,u1), with u4 = u4 o n1, u1 = du4, T2U: (u4, u1, u|, u12), ..... with u4 = u4 o ^1^2, u1 = du4 o n2, u2 = d(u4 o n^), u12 = d(du4), etc.
We set up the following convention: to introduce coordinates on TfcU, vie take the coordinates on Tk-1U and repeat them with an additional index k, so that a tangent vector is preceded by its point of origin. This indexing is convenient since at present the symbols with index s become fiber coordinates for the projection ps, s =1,2,..., k.
Thus, for example, under the projections ps : T3U ^ T2U, s = 1,2, 3, the coordinates with indices 1, 2 and 3 are each suppressed in turn:
(u® u1 u2 u12 u3 u13 u23 u123) P1 L P2 I \ P3
(u u2 u3 u23) (u u1 u3 u13) (u u1 u2 u12).
The level TkM
IS 3. smooth manifold of dimension 2kn and admits an important subspace of dimension (k + 1)n called the osculating bundle of M (briefly - osculator) of order k — 1 and denoted by Oscfc-1M. The bundle Oscfc-1M is determined by the equality of the projections
P1 = P2 = ... = Ps^
meaning that an element of TkM belongs to the bundle Oscfc-1M precisely when all its k projections into Tk-1M coincide. In this case all coordinates with the same
M
in T2U C T2M % the equation u1 = u2, and the second bundle Osc2M is determined in T3U C T3M by u1 = u2 = u3, u12 = u13 = u23, etc. The coordinates in Oscfc-1M will be denoted % the derivatives of the coordinate functions on U, that is (u4, du4, d2u4, .. ., dku4).
The immersion Z : OscM ^ T2M and its derivative TZ are determined in coordinates by matrix formulas:
ui u u 3 du
u1 u 2 oZ= du4 du4 , u13 u23 o TZ = d2u 4 d2u 4
u12 ^d2u4/ u123 ^d3u4/
id d d \ f d d
d(d2u®^ ydu^ du1 du|' du12 The fibres of the bundle OscM are the integral manifolds of the distribution
(dl + dldP), with ¿>/ + ¿>2= * + * a,12- 5
dul du2 4 d«i2
The functions (ul — u2) vanish on OscM.
Historically, osculating bundles were introduced under various names long before TkM
of V. Vagner [2] has been culminated in recent times in Miron Atanasin theory [3].
TkM
that the multiple fibre bundle structure demands a whole new understanding and new
approach (see [1. 4 6]). Attempts such as [7] and the so-called synthetic formulation of TkM
uGM
gent vector ^ to M, an infinitesimal displacement of the element (u, u1) G TM is
determined by the quantities (u2, u12), representing a tangent vector to TM, etc. This
TkM
motion. Clearly, the future belongs to these bundles.
White considers on the level TkM or on a k-multiple vector bundle certain sector-forms which are functions simultaneously linear on the fibres of all k projections (see [7]). In particular, the sector-forms on T2U and T3U can be written as
$ = ¥ij «1«2 + ¥>¿«12,
bijk«1«2«3 + ^¿j« 1"23 "T" Vij"2 "13 "T" Vij1
^ = ^ijk «1«2«3 + ^ij «1«23 + ^¿j «2 «13 + ^¿j «3«12 + ^i«123>
with coefficients in U. For example, in each term of ^ the index 1 (or 2 or 3 respectively) appears exactly once. This means that the function ^ is linear on the fibres of p1 (and p2 and p3).
Any scalar function can be lifted from the level Tk-1M to the level TkM by k
different projections ps : TkM ^ Tk-1M. For example, for the sector-form $ (see
T3M
$ o P1 = fiju2u3 + ^¿uJ23, $ o P2 = fiju1u3 + fiu13, $ o P3 = fiju1u2 + ^¿u12.
k
0/ White, a scalar function on TkM that is constant on the fibres of Osck-1M.
$M
M
$ = fij u1u2 + fiu12 ^
(d1 + di2)$ = fiju2 + f jiu1 = (fij + f ji)u1 — fij (u1 — u2)' dj12$ = fi ^ f(ij) =0, fi =0.
$
(u4, du^ as a 2-form $ = f [ij jdu2, A duj. Thus the sector-form $ is constant on OscM
2
If k = 3 the fibres Osc2M of dimension 3n are the integral manifolds of the distribution
<d1 + d2 + d3, d23 + d13 + d12, di123).
For the sector-form * (see above) we have
* = ^ijfc ulu2u3 + ^ij uiu23 + ^ij u 2ui3
^ij u3ui2
■ ^¿W
l23
(di+d2+di3)*
^ijfc u2u3 + j uiu3 + j uiu2 + ^ij u23 + ^ij ui3 + ^ij ui2
(d?3 + dl3 + dj2)* = juj + j+ j j
5i123^ = Vi-
The derivatives vanish on the fibres Osc2M when the following conditions hold:
¥>(ijfc) = 0, + + = o, Vi = o.
These conditions are necessary and sufficient for the sector-form * to be constant on Osc2M, but not for * to be a Cartan 3-form. However, every 3-form * = = ^¿j3 A duj A du3 can be regarded as a homogeneous sector-form that is constant on Osc2M.
The argument extends likewise to the cases when k > 3. □
White's theory of sector-forms is much more extensive than that of Cartan exterior forms. In particular, exterior differentiation is an operation on the set of sector-forms that are constant on the osculating bundles.
2.2. Gauge groups on osculating spaces. The action of the gauge group Gk
on the fc-th level Tk M extends in a natural way to the osculating bundle Osc The diagram from below shows how the block-matrix 4 x 4 reduces, for u1 to a 3 x 3 block-matrix:
3-1
Wl = W2
a 0 0 0
ai a 0 0
a2 0 a 0
Vai2 a2 ai a
0
a
da
0 0
a
The blocks of the matrix from the right side are generated in the following way:
k
l du ,
al a2
j u ajfc u
da ~ daj = a
j3d
al2
-jfci
.K.i
uî w2 + ajfc «12 ^
j2 i , 3 , I i ,2 3
da ~ aj3; du du + aj3 d u .
M.
u2
The action of the gauge group G2 on the level T2M is obviously transported to the subbundle OscM c T2M. While one passes from T2M to OscM by considering
f d d
2 l2
(ai = a2,ai2) (da, da), (d + d
Vd(du) ' d(d2u)
the transformation of the natural basis on T2 M is transported to the transformation
M
(d d1 d2 d
l2
/a 0 0 0\
ai a 0 0
a2 0 a 0
\ai2 a2 ai ay
d d
d
du d(du) d(d2u)
a 0 0> da a 0 d2 a da a
In the general case, the action of the group Gk on the le vel T kM extends in a similar way to the subbundle Osck-1M.
Резюме
M. Рахула, В. Балаи. Касательные расслоения и калибровочные группы.
Дифференциалы Tka (k > 1) диффеоморфизма a гладкого многообразия M индуцируют в слоях расслоений TkM, то есть в соответствующих касательных пространствах, линейные преобразования, заключающие в себе действие калибровочной группы Gk ■ Это действие естественным образом распространяется па соприкасающиеся подрасслое-ния Osck-1M С TkM.
Ключевые слова: диффеоморфизм гладкого многообразия, пространство расслоения, действие калибровочной группы.
References
1. Ehresmann Ch. Catégories doubles et catégories structurées // C. R. Acad. Sci. - Paris, 1958. - V. 256. - P. 1198-1201.
2. Vagner V. V. Theory of differential objects and foundations of differential geometry // Veblen O., Whitehead J.H.C. The Foundations of Differential Geometry. - Moscow: IL, 1949. - P. 135-223. (in Russian)
3. Atanasiu G., Balan V., Brînzei N., Rahula M. Second Order Differential Geometry and Applications: Miron-Atanasiu Theory. - Moscow: Librokom, 2010. - 250 p. (in Russian)
4. Pradines J. Suites exactes vectorielles doubles et connexions // C. R. Acad. Sci. - Paris, 1974. - V. 278. - P. 1587-1590.
5. Atanasiu G., Balan V., Brînzei N., Rahula M. Differential Geometric Structures: Tangent Bundles, Connections in Bundles, Exponential Law in the Jet Space. - Moscow: Librokom, 2010. - 320 p. (in Russian)
6. Rahula M. Tangent structures and analytical mechanics // Balkan J. Geom. Appl. -2011. - V. 16, No 1. - P. 122-127.
7. White E.J. The Method of Iterated Tangents with Applications in Local Riemannian Geometry. - Boston, Mass.; London: Pitman Adv. Publ. Program, 1982. - 252 p.
8. Bertram W. Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings // Memoirs of AMS. - 2008. - No 900. - 202 p.
Поступила в редакцию 17.12.10
Rahula, Maido Doctor of Physics and Mathematics, Professor Emeritus, Faculty of Mathematics and Computer Science, University of Tartu, Tartu, Estonia.
Рахула, Майдо доктор физико-математических паук, почетный профессор факультета математики и информатики Тартуского университета, г. Тарту, Эстония.
E-mail: rahula Out. ее
Balan, Vladimir Doctor of Mathematics, Professor, Faculty of Applied Sciences, University Politelinica of Bucharest, Bucharest, Romania.
Валан, Владимир доктор математических паук, профессор факультета приклад-пых паук Бухарестского политехнического университета, г. Бухарест, Румыния.
E-mail: vladimir. balanQupb.ro
