Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds
We study the differential-geometric aspects of generalized de Rham-Hodge complexes naturally related to integrable multidimensional differential systems of the M. Gromov type, as well as the geometric structure of the Chern characteristic classes. Special differential invariants of the Chern type ar...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509374778179584 |
|---|---|
| author | Bogolyubov, N. N. Prykarpatsky, A. K. Боголюбов, М. М. (мл.) Прикарпатський, А. К. |
| author_facet | Bogolyubov, N. N. Prykarpatsky, A. K. Боголюбов, М. М. (мл.) Прикарпатський, А. К. |
| author_sort | Bogolyubov, N. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:00Z |
| description | We study the differential-geometric aspects of generalized de Rham-Hodge complexes naturally related to integrable multidimensional differential systems of the M. Gromov type, as well as the geometric structure of the Chern characteristic classes. Special differential invariants of the Chern type are constructed, their importance for the integrability of multidimensional nonlinear differential systems on Riemannian manifolds is discussed. An example of the three-dimensional Davey-Stewartson-type nonlinear integrable differential system is considered, its Cartan type connection mapping, and related Chern-type differential invariants are analyzed. |
| first_indexed | 2026-03-24T02:40:06Z |
| format | Article |
| fulltext |
UDC 517.9
N. N. Bogolubov (jr.) (V. A. Steklov Math. Inst. Rus. Acad. Sci., Moscow, Russia),
A. K. Prykarpatsky (AGH Univ. Sci. and Technol., Krakow, Poland and Inst. Appl. Problems Mech. and
Math. Nat. Acad. Sci. Ukraine, Lviv)
GENERALIZED DE RHAM – HODGE COMPLEXES,
THE RELATED CHARACTERISTIC CHERN CLASSES
AND SOME APPLICATIONS TO INTEGRABLE
MULTIDIMENSIONAL DIFFERENTIAL SYSTEMS
ON RIEMANNIAN MANIFOLDS
UZAHAL\NENI KOMPLEKSY DE RAMA – XODÛA,
SPORIDNENI XARAKTERYSTYÇNI KLASY ÇERNA
TA DEQKI ZASTOSUVANNQ DO INTEHROVNYX
BAHATOVYMIRNYX DYFERENCIAL\NYX SYSTEM
NA RIMANOVYX MNOHOVYDAX
The differential-geometric aspects of generalized de Rham – Hodge complexes naturally related with integrable
multidimensional differential systems of M. Gromov type, as well as the geometric structure of Chern character-
istic classes are studied. Special differential invariants of the Chern type are constructed, their importance for the
integrability of multidimensional nonlinear differential systems on Riemannian manifolds is discussed. An ex-
ample of the three-dimensional Davey – Stewartson type nonlinear integrable differential system is considered,
its Cartan type connection mapping and related Chern type differential invariants are analized.
DoslidΩeno dyferencial\no-heometryçni aspekty uzahal\nenyx kompleksiv de Rama – XodΩa, wo pry-
rodnym çynom pov’qzani z intehrovnymy bahatovymirnymy dyferencial\nymy systemamy typu M. Hro-
mova, a takoΩ heometryçnu strukturu xarakterystyçnyx klasiv Çerna. Pobudovano special\ni dyfe-
rencial\ni invarianty typu Çerna ta rozhlqnuto ]x vaΩlyvist\ dlq intehrovnosti bahatovymirnyx ne-
linijnyx dyferencial\nyx system na rimanovyx mnohovydax. Rozhlqnuto pryklad tryvymirno] neli-
nijno] intehrovno] dyferencial\no] systemy typu Devi – Stgartsona i proanalizovano ]x spoluçne
vidobraΩennq ta sporidneni dyferencial\ni invarianty typu Çerna.
1. Introduction: Cartan’s connection and curvature. Consider a smooth finite-dimen-
sional Riemannian manifold M and two linear bundles over it: the tangent bundle T (M)
and a bundle E(M), endowed with some real scalar structure 〈·, ·〉E on fibers E. De-
note the related vector fields on M as T (M) and smooth sections of E(M) as E(M).
As usually [1 – 4], one can introduce on E(M) a Cartan’s connection Γ by means of a
connection mapping
dA : E(M) → T ∗(M) ⊗ E(M), (1.1)
which satisfies the following property:
dA(fα+ β) := df ⊗ α+ fdAα+ dAβ
for any smooth function f ∈ D(M) and α, β ∈ E(M). Let Λ(M) := ⊕dim M
p=0 Λp(M)
denote the usual [1, 2, 4 – 6] Grassmann algebra of differential forms on M. If to define
the associated linear bundles Λp(M,E) := Λp(M)⊗E(M) for p = 0,m, the connection
mapping (1.1) can be naturally extended on Λp(M,E) as
dA : Λp(M,E) → Λp+1(M,E), (1.2)
satisfying the related Leibnitz-rule:
dA(f (p) ∧ α(q)) := df (p) ∧ α(q) + (−1)pf (p) ∧ dAα
(q)
for any f (p) ∈ Λp(M) and α(q) ∈ Λq(M,E), q, p = 0,m.
c© N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3 327
328 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
The connection operation (1.2) possesses a very interesting and important property:
its composition d2
A := dAdA is linear over functions ring D(M):
d2
A
(
fα(p) + β(p)
)
= fd2
Aα
(p) + d2
Aβ
(p),
where f ∈ D(M) and α(p), β(p) ∈ Λp(M,E), p = 0,m, are arbitrary. The resulting
linear tensor mapping Ω(2) := d2
A : E(M) → Λ2(M,E) is called the curvature tensor and
is of great importance for geometrical analysis of integrable multidimensional differential
systems of M. Gromov type [7], generated by means of some Cartan integrable [4, 8 –
11] ideals I(α) ⊂ Λ(M,EndE) := Λ(M) ⊗ End E(M) on Riemannian manifolds
M . Moreover, one can construct the smooth integral submanifold imbedding mapping
iα : Mα → M for the ideal I(α) ⊂ Λ(M,EndE), satisfying the following determining
condition: the curvature 2-form Ω(2) ∈ Λ2(M,E) reduced upon Mα vanishes, that is
i∗αΩ(2) = 0. This implies also that the related reduced co-chain
E → Λ0(Mα, E) dα→ Λ1(Mα, E) dα→ . . .
dα→ Λmα(Mα, E) dα→ 0 (1.3)
is a de Rham complex, that is d2
α = 0, where, by definition, dα := i∗αdA and mα :=
:= dimMα. Since the submanifold Mα ⊂ M also possesses the induced from M Rie-
mannian structure gα : T (Mα) × T (Mα) → R, we can construct from (1.3) a suitably
generalized de Rham – Hodge complex of Hilbert spaces Hp
Λ(Mα), p = 0,mα, whose
properties, as it was shown before in [4, 12 – 14], make it possible to describe the so
called Delsarte – Lions transmutation operators of Volterra type, serving for constructing
integrable multidimensional differential systems on Riemannian manifolds and finding
their exact of special type solutions.
On the other side, one can consider the following generalized chain of modules overM :
E → Λ0(M,E) dA→ Λ1(M,E) dA→ . . .
dA→ Λm(M,E) → 0, (1.4)
which is not, evidently, a de Rham – Hodge complex, but determines such very impor-
tant [1, 3, 15, 16] geometric objects as the Chern characteristic classes and characters.
On these and other geometric aspects of the integrability problem of multidimensional
differential systems on Riemannian manifolds we will stay in more detail below.
2. The characteristic Chern classes and related differential invariants. The con-
nection mapping (1.2) onM one can equivalently define locally on an open neighborhood
U ⊂M by means of the following expression:
dA|U = d+A(1), (2.1)
where A(1) ∈ Λ(U,EndE) is some suitably determined EndE(U)-valued differential
1-form on U ⊂M. In local coordinates of a point u ∈ U we can write down
A(1) :=
m∑
i=1
Ai(u)dui,
where Ai(u) ∈ End E(U), i = 1,m. Making use of the representation (2.1), one can
obtain easily the following local expression for the curvature 2-form Ω(2)|U ∈ Λ2(U) ⊗
⊗ EndE(U):
Ω(2)|U = dA(1) +A(1) ∧A(1). (2.2)
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GENERALIZED DE RHAM – HODGE COMPLEXES, THE RELATED CHARACTERISTIC ... 329
The expression (2.2) is very convenient for defining on U ⊂ M the related with the
complex (1.4) local cohomology characteristic Chern classes
chj(A)|U := tr
(
Ω(2)|U
)j ∈ Λ2j(U), (2.3)
where j ∈ Z+. The expression (2.3), owing to the linear vector bundle E(M) properties,
can be invariantly extended upon the whole manifold M as correctly defined differential
forms on M, thereby determining the characteristic Chern classes
chj(A) = tr(Ω(2))j ∈ Λ2j(M) (2.4)
for j ∈ Z+, defined already on M. The following lemmas are important for further appli-
cations.
Lemma 2.2. All differential 2j-forms chj(A) ∈ Λ2j(M), j ∈ Z+, are closed,
that is
d chj(A) = 0. (2.5)
Proof. A proof is standard by means of the direct substitution of local expressions
(2.2) into (2.4) and checking (2.5).
As a consequence of this lemma we really see that inclusions
[
chj(A)
]
∈ H2j(M,R)
hold on M for all j ∈ Z+.
Lemma 2.3. The de Rham cohomology classes [chj(A)] ∈ H2j(M,R), j ∈ Z+,
do not depend on the choice of a connection mapping
dA : Λ(M,E) → Λ(M,E)
and on the choice of a Hermitian metrics on E.
Proof. The homotopy cylinder construction [3, 1, 17] if applied to two different con-
nection mappings gives right away to the independence connection choice, proving the
first half of the lemma statement. The same homotopy reasonings prove, respectively, the
independence metrics choice, the second part of the lemma statement.
As a consequence of this lemma one can define for every linear Hermitian fiber bundle
E(M) over M the set of corresponding characteristic Chern classes
chj(E,M) :=
[
chj(A)
]
for j ∈ Z+ by means of which there is determined the Chern character ch(E,M) of this
linear fiber bundle E(M):
ch(E,M) := ⊕j∈Z+(j!)−1 chj(E,M) =
[
tr exp Ω(2)
]
.
The Chern character, as is well known [1, 3, 16], finds a great deal of applications to
modern differential topology and mathematical physics problems.
Concerning applications to strongly integrable multidimensional differential systems
on Riemannian manifolds, we will consider the Cartan geometric picture, developed be-
fore in [4, 8 – 11]. Within this picture a studied nonlinear multidimensional differential
system α̂ is represented in the form of a Cartan integrable ideal I(α) ⊂ Λ(M,End E)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
330 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
with coefficients from EndE(M), where E(M) is a specially chosen Hermitian linear
fiber bundle over some suitably chosen finite-dimensional Riemannian manifold M. The
corresponding integral submanifold Mα ⊂ M of the ideal I(α), in general, is equivalent
to the set of independent variables of our multidimensional integrable nonlinear differen-
tial system.
Note also here, that we call our multidimensional differential system strongly in-
tegrable, if it allows a suitable connection Γλ parametrically dependent on λ ∈ R,
whose curvature 2-form Ω(2)
λ ∈ Λ2(M,EndE) is vanishing upon the integral subman-
ifold Mα ⊂ M of the ideal I(α) ⊂ Λ(M,EndE). The latter condition is, evidently,
equivalent to the following inclusion:
Ω(2)
λ ∈ I(α) (2.6)
for all allowed values of λ ∈ R. If the connection Γ does not depend nontrivially on
parameter λ ∈ R, a multidimensional differential system is called integrable.
On the other side, the condition (2.6) serves [4, 9] for finding the corresponding con-
nection mapping (1.2), if it a priori assumes to exist. The resulting search algorithm
details depend [4, 9, 11] strongly on the related so called holonomy group properties of
the connection Γλ, λ ∈ R, on the naturally associated with E(M) principal fiber bun-
dle P (M,G), where G is a so called structure Lie group of the connection Γλ, λ ∈ R.
Concerning the mentioned algorithm details one can consult further [4, 9, 11].
The condition (2.6) is very important regarding the result of Lemma 2.3. Really, as the
exactness relationships (2.5) hold, we can obtain under imposed conditionsH2j(M,R) =
= 0, j ∈ Z+, right away that
chj(A, λ) := dχj(A, λ)
for some suitably determined global differential (2j − 1)-form χj(A, λ) ∈ Λ2j−1(M),
j ∈ Z+, on the manifold M. Moreover, since, evidently, all degrees (Ω(2))j ∈ I(α)
too for j ∈ Z+, from the condition i∗αI(α) = 0, where iα : Mα → M is the integral
submanifold imbedding mapping, one gets easily that i∗α chj(A) = 0, or equivalently
dχj(A) = 0 (2.7)
for all j ∈ Z+, giving rise to new differential Chern type invariants on Mα. The obtained
result one can formulate as the following theorem.
Theorem 2.1. If an integrable multidimensional nonlinear differential system α̂ is
equivalent to the Cartan integrable ideal I(α) ⊂ Λ(M,EndE) on a Riemannian man-
ifold M, satisfying the cohomology conditions H2j(M,R) = 0, j ∈ Z+, then it pos-
sesses a set of differential Chern type invariants (2.7) on the suitable integral submani-
fold Mα ⊂ M of the ideal I(α). If nontrivial, these differential invariants describe, in
particular, a related moduli space of the linear fiber bundle E(M).
It is useful to mention here that most of differential invariants (2.7) reduce at higher
indices j ∈ Z+ to identical zero. Really, all of differential invariants χj(A) for j ≥
≥ [dimMα/2] + 1 are identically zero. In particular, for the case of multidimensional
integrable nonlinear differential systems, for which dimMα = 2 or 3 one gets easily, that
only one differential invariant can exist if any. Moreover, if for such differential systems
the corresponding structure Lie groups are special linear ones, for which the Lie algebras
a traceless, one easily gets that, on the whole, no invariant exists on Mα.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
GENERALIZED DE RHAM – HODGE COMPLEXES, THE RELATED CHARACTERISTIC ... 331
This result is instructive enough for the mathematical theory of geometrically inte-
grable multidimensional nonlinear differential systems, possessing suitable connection
mappings (1.2) on Λ(M,E) under additional cohomology conditions H2j(M,R) = 0,
j ∈ Z+. Namely, for these connection mappings to exist on Λ(M,E) in the higher-
dimensional case dimMα ≥ 4, the nontrivial differential invariants can prove to exist.
The latter gives rise, in particular, to some topological obstacles to be satisfied. Really,
a nontrivial differential invariant entails right away nontrivial cohomology constraints
Hs(M,R) �= 0 for some s ∈ Z+, thereby contradicting with the above zero cohomol-
ogy conditions at these values s ∈ Z+. Otherwise, if a priori H2s(M,R) �= 0 for some
s ∈ Z+, the related characteristic Chern classes chs(E,M) for these s ∈ Z+ are, in gen-
eral, strongly nontrivial, thereby defining differential (2s−1)-forms χs(A) ∈ Λ2s−1
loc (M)
only locally, owing to the classical Poincaré lemma. These local differential forms, if
reduced upon the integral submanifold Mα ⊂ M, give rise to a set of differential multi-
valued quasi-invariants χs(A) ∈ Λ2s−1
loc (Mα), where dχs(A) = 0 and whose existence
can mean, in particular, that our nonlinear differential system is in some sense ill-posed.
Remark here that related differential system structures can be studied also by means of
the corresponding exact co-chain de Rham – Hodge complexes (1.3). These aspects will
be also discussed below.
3. De Rham – Hodge theory and Delsarte – Lions transmutation operators. Con-
sider the cohomology complex (1.3) and define on spaces Λp(Mα, E), p ∈ Z+, where
Mα ⊂ M is the compact integral submanifold of the Riemannian manifold M, the stan-
dard Hodge star �-operation
� : Λp(Mα, E) → Λmα−p(Mα, E),
where mα := dimMα and for any β ∈ Λp(Mα, E) the form �β ∈ Λmα−p(Mα, E) is
such that the following [1, 4, 5, 18] conditions hold:
i) 〈γ, �β〉(mα−p) := 〈〈γ, �β〉E〉mα−p = 〈〈γ∧, β〉E , dµgα〉r for any γ ∈ Λmα−p(Mα,
E), where dµgα
is the invariant measure on Mα, induced at the dual imbedding mapping
i∗α : Λ(Mα) → Λ(Mα) from the corresponding invariant measure dµg on the Riemannian
manifoldM, endowed with the positive definite Riemannian metrics g : T (M)×T (M) →
→ R, the scalar product〈
β
(1)
1 ∧ β(1)
2 ∧ . . . ∧ β(1)
k , γ
(1)
1 ∧ γ(1)
2 ∧ . . . ∧ γ(1)
k
〉
k
:=
:= det
{〈
β
(1)
i , γ
(1)
j
〉
1
: i, j = 1, k
}
,
where
〈
β
(1)
i , γ
(1)
j
〉
1
:=
〈
ĝ−1
α β
(1)
i , ĝ−1
α γ
(1)
j
〉
gα
for any β(1)
i , γ
(1)
j ∈ Λ1(Mα), i, j = 1, k,
and ĝα : T (Mα) → T ∗(Mα) is the canonical isomorphism, generated by the correspond-
ing metrics 〈·, ·〉gαon T (Mα);
ii) (mα − p)-dimensional volume |� β | of the form �β ∈ Λmα−p(Mα, E) equals
p-dimensional volume |β| of a form β ∈ Λp(Mα, E);
iii) mα-dimensional measure 〈β∧, �β〉E ≥ 0 at a fixed orientation on Mα.
Owing to the above conditions i) – iii) one can endow the spaces Λp(Mα, E), p ∈ Z+,
with the natural scalar product
(β, γ) :=
∫
Mα
〈β∧, ∗γ〉E =
∫
Mα
〈β, γ〉(p)dµgα
(3.1)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
332 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
for any β, γ ∈ Λp(Mα, E). Subject to the scalar product (3.1) we can naturally construct
the corresponding Hilbert space
HΛ(Mα) :=
mα⊕
k=0
Hk
Λ(Mα)
well suitable for our further consideration. Notice also here that the Hodge star �-operation
satisfies the following easily checkable property: for any β, γ ∈ Hk
Λ(Mα), k = 0,mα,
(β, γ)(k) = (�β, �γ)(mα−k), (β, γ) = (�β, �γ),
that is the Hodge operation � : HΛ(Mα) → HΛ(Mα) is isometry and its standard adjoint
with respect to the scalar product (3.1) operation satisfies the condition (�)
′
= (�)−1.
Denote by d′α the formally adjoint expression to the Cartan type connection mapping
dα : HΛ(Mα) → HΛ(Mα) in the Hilbert space HΛ(Mα). Here dα := i∗αdA, where
dA : HΛ(M) → HΛ(M) is a suitable Cartan connection mapping and iα : Mα → M
is the corresponding integral submanifold imbedding mapping, associated with a given
multidimensional nonlinear integrable differential system on the Riemannian manifold
M. Making use of these operations d′α and dα in HΛ(Mα), one can naturally define
[1, 18, 19] a generalized Laplace – Hodge operator ∆α : HΛ(Mα) → HΛ(Mα) as
∆α := d′αdα + dαd
′
α. (3.2)
Take a form β ∈ HΛ(Mα) satisfying the equality
∆αβ = 0.
Such a form is called harmonic. One can also verify that a harmonic form β ∈ HΛ(Mα)
satisfies simultaneously the following two adjoint conditions:
d′αβ = 0, dαβ = 0, (3.3)
easily stemming from (3.2) and (3.3).
It is not hard to check that the following differential operation in HΛ(Mα) :
d∗α := �d′α(�)−1 (3.4)
defines also a usual [10, 15, 17, 20, 21] Cartan type connection mapping in HΛ(Mα). The
corresponding dual to (3.1) co-chain
E → Λ0(Mα, E)
d∗
α→ Λ1(Mα, E)
d∗
α→ . . .
d∗
α→ Λm(Mα, E)
d∗
α→ 0 (3.5)
is, evidently, α de Rham complex too, as the property d∗αd
∗
α = 0 holds owing to the
definition (3.4).
Denote further by Hk
Λ(α)(Mα), k = 0,mα, the cohomology groups of dα-closed and
by Hk
Λ(α∗)(Mα), k = 0,mα, the cohomology groups of d∗α-closed differential forms, re-
spectively, and by Hk
Λ(α∗α)(Mα), k = 0,mα, the abelian groups of harmonic differential
forms from the sub-spaces Hk
Λ(Mα, E), k = 0,mα. Before formulating next results, de-
fine the standard sub-spaces for harmonic forms Hk
Λ(α∗α)(Mα) and cohomology groups
Hk
Λ(α)(Mα), Hk
Λ(α∗)(Mα) for k = 0,mα. Assume also that the Laplace – Hodge opera-
tor (3.2) is elliptic in H0
Λ(Mα). Now by reasonings similar to those in [1, 14, 17 – 19] one
can formulate the following a little generalized de Rham – Hodge theorem.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
GENERALIZED DE RHAM – HODGE COMPLEXES, THE RELATED CHARACTERISTIC ... 333
Theorem 3.1. The groups of harmonic forms Hk
Λ(α∗α)(Mα), k = 0,mα, are, re-
spectively, isomorphic to the cohomology groups (Hk(Mα,R))|Σ|, k = 0,mα, where
Hk(Mα,R) is the k-th cohomology group of the manifold Mα with real coefficients,
Σ ⊂ R
p, p ∈ Z+, |Σ| := card Σ, is a set of suitable “spectral” parameters labeling
the linear space of independent d∗α-closed 0-forms from H0
Λ(α)(Mα) and, moreover, the
following direct sum decompositions
Hk
Λ(α∗α)(Mα) ⊕ ∆Hk
Λ(Mα) =
= Hk
Λ(Mα) = Hk
Λ(α∗α)(Mα) ⊕ dαHk−1
Λ (Mα) ⊕ d
′
αHk+1
Λ (Mα)
hold for any k = 0,mα.
Another variant of the statement similar to that above was formulated in [13, 14, 22,
23] and reads as the following generalized de Rham – Hodge theorem.
Theorem 3.2. The generalized cohomology groups Hk
Λ(α)(Mα), k = 0,mα, are
isomorphic, respectively, to the cohomology groups (Hk(Mα,R))|Σ|, k = 0,mα.
Proof. A proof of this theorem is based on some special sequence [14, 22] of differ-
ential Lagrange type identities.
Define the following closed subspace:
H∗
0 :=
{
ϕ(0)(λ) ∈ H0
Λ(α∗)(Mα) : d∗αϕ
(0)(λ) = 0, λ ∈ Σ
}
(3.6)
for some set Σ ⊂ R
p, where H0
Λ(α∗)(Mα) is, as above, a suitable zero-order cohomology
group space from the co-chain given by (3.5). Thereby, the dimension dim H∗
0 = card Σ
is assumed to be known.
The next lemma [14, 22 – 25] is fundamental for the proof of the above isomorphism
Theorem 3.2.
Lemma 3.1. There exists a set of differential (k + 1)-forms Z(k+1)
[
ϕ(0)(λ),
dαψ
(k)
]
∈ Λk+1(Mα,R), k = 0,mα − 1, and a set of k-forms Z(k)
[
ϕ(0)(λ), ψ(k)
]
∈
∈ Λk(Mα,R), k = 0,mα − 1, parametrized by a set Σ � λ and semilinear in (ϕ(0)(λ),
ψ(k)) ∈ H∗
0 ×Hk
Λ,(Mα), such that
Z(k+1)
[
ϕ(0)(λ), dαψ
(k)
]
= dZ(k)
[
ϕ(0)(λ), ψ(k)
]
for all k = 0,mα − 1 and λ ∈ Σ.
Proof. A proof is based on the following generalized Lagrange type identity holding
for any pair
(
ϕ(0)(λ), ψ(k)
)
∈ H∗
0 ×Hk
Λ(Mα):
0 =
〈
d∗αϕ
(0)(λ),∧(ψ(k) ∧ γ̄)
〉
E
=
〈
d∗αϕ
(0)(λ),∧ ∗ (∗)−1(ψ(k) ∧ γ̄)
〉
E
=
=
〈
� d′α(�)−1ϕ(0)(λ),∧ ∗ (∗)−1
(
ψ(k) ∧ γ̄
)〉
E
=
=
〈
d′α(�)−1ϕ(0)(λ),∧ ∗ (∗)−2
(
ψ(k) ∧ γ̄
)〉
E
=
=
〈
(�)−1ϕ(0)(λ),∧(∗)−1(∗)2dα
(
ψ(k) ∧ γ̄
)〉
E
+ Z(k+1)
[
ϕ(0)(λ), dαψ
(k)
]
∧ γ̄ =
=
〈
ϕ(0)(λ),∧
(
dαψ
(k) ∧ γ̄
)〉
E
+ dZ(k)
[
ϕ(0)(λ), ψ(k)
]
∧ γ̄, (3.7)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
334 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
where Z(k+1)
[
ϕ(0)(λ), dαψ
(k)
]
∈ Λk+1(Mα,R), k = 0,mα − 1, and Z(k)
[
ϕ(0)(λ),
ψ(k)
]
∈ Λk(Mα,R), k = 0,mα − 1, are some semilinear differential forms parametrized
by the parameter λ ∈ Σ, and γ̄ ∈ Λmα−k−1(Mα,R) is an arbitrary constant (mα−k−1)-
form. Thereby, the semilinear differential (k + 1)-forms Z(k+1)
[
ϕ(0)(λ), dαψ
(k)
]
∈
∈ Λk+1(Mα,R), k = 0,mα − 1, and k-forms Z(k)
[
ϕ(0)(λ), ψ(k)
]
∈ Λk(Mα,R), k =
= 0,mα − 1, λ ∈ Σ, constructed above, exactly constitute those searched for in the
lemma.
Based now on Lemma 3.1, one can construct the cohomology group isomorphism
claimed in Theorem 3.2 formulated above. Namely, following [14, 22, 23, 26], let us
take some singular simplicial [1, 2, 17, 19, 21] complex K(Mα) of the manifold Mα
and introduce linear mappings B(k)
λ : Hk
Λ(Mα) → Ck(Mα,R)), k = 0,mα − 1, λ ∈ Σ,
where Ck(Mα,R), k = 0,mα − 1, are as before free abelian groups over the field R
generated, respectively, by all k-chains of simplexes S(k) ∈ Ck(Mα,R), k = 0,mα − 1,
from the singular simplicial complex K(Mα), as follows:
B
(k)
λ (ψ(k)) :=
∑
S(k)∈Ck(Mα, R))
S(k)
∫
S(k)
Z(k)
[
ϕ(0)(λ), ψ(k)
]
(3.8)
with ψ(k) ∈ Hk
Λ(Mα), k = 0,mα − 1. The following theorem [14, 22, 25] based on
mappings (3.8) holds.
Theorem 3.3. The set of operations (3.8) parametrized by λ ∈ Σ realizes the coho-
mology groups isomorphism formulated in Theorem 3.2.
Proof. A proof of this theorem one can get passing over in (3.8) to the correspond-
ing cohomology Hk
Λ(α)(Mα) and homology Hk(Mα,R) groups of Mα for every k =
= 0,mα − 1. If one to take an element ψ(k) := ψ(k)(µ) ∈ Hk
Λ(α)(Mα), k = 0,mα − 1,
solving the equation dαψ
(k)(µ) = 0 with µ ∈ Σk being some set of the related “spectral”
parameters marking elements of the subspace Hk
Λ(α)(Mα), then one finds easily from
(3.8) and the identity (3.7) that
dZ(k)
[
ϕ(0)(λ), ψ(k)(µ)
]
= 0
for all pairs (λ, µ) ∈ Σ × Σk, k = 0,mα − 1. This, in particular, means owing to the
Poincaré lemma [1, 17, 20, 21] that there exist differential (k−1)-forms Ω(k−1)
[
ϕ(0)(λ),
ψ(k)(µ)
]
∈ Λk−1(Mα,R), k = 0,mα − 1, such that
Z(k)
[
ϕ(0)(λ), ψ(k)(µ)
]
= dΩ(k−1)[ϕ(0)(λ), ψ(k)(µ)]
for all pairs (ϕ(0)(λ), ψ(k)(µ)) ∈ H∗
0 × Hk
Λ(α)(Mα) parametrized by (λ, µ) ∈ Σ × Σk,
k = 0,mα − 1. As a result of passing on the right-hand side of (3.8) to the homology
groups Hk(Mα,R), k = 0,mα − 1, one gets owing to the standard Stokes theorem [1,
17, 20, 21] that the mappings
B
(k)
λ : Hk
Λ(α)(Mα) � Hk(Mα,R)
are isomorphisms for every λ ∈ Σ and λ ∈ Σ. Making further use of the Poincaré
duality [1, 5, 6, 17, 21] between the homology groups Hk(Mα,R), k = 0,mα − 1, and
the cohomology groups Hk(Mα), k = 0,mα − 1, respectively, one obtains finally the
statement claimed in Theorem 3.2, that is Hk
Λ(α)(Mα) � (Hk(Mα,R))|Σ|.
The theorem is proved.
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GENERALIZED DE RHAM – HODGE COMPLEXES, THE RELATED CHARACTERISTIC ... 335
Assume now that the Riemannian compactified manifold M = Mα × R
s, dimM =
= s + dimMα ∈ Z+, and E := R
N , where Mα �
mα×
j=1
Mα,j , Mα,j := [0, Tj) ⊂ R+,
j = 1,mα, and put
dα =
mα∑
j=1
dtj ∧ Aj(t|∂), Aj(t|∂) :=
∂
∂tj
+Aj(t),
with Aj(t), j = 1,mα, being matrices parametrically dependent on t ∈ Mmα
α . It is
assumed also that operators Aj : H0
Λ(Mα) → H0
Λ(Mα), j = 1,mα, are commuting to
each other.
Take now such a fixed pair
(
ϕ(0)(λ), ψ(0)(µ)dt
)
∈ H∗
0 ×Hmα
Λ(α)(Mα), parametrized
by elements (λ, µ) ∈ Σ×Σ, for which owing to both Theorem 3.3 and the Stokes theorem
[1, 10, 17, 20] the following equality:
B
(mα)
λ
(
ψ(0)(µ)dt
)
= S
(mα)
(t)
∫
∂S
(mα)
(t)
Ω(mα−1)
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
(3.9)
holds, where S(mα)
(t) ∈ Cmα
(Mα,R) is some fixed element parametrized by an arbitrarily
chosen point t ∈ S
(s)
(t) ⊂Mα. Consider the next integral expressions
Ω(t)(λ, µ) :=
∫
∂S
(mα)
(t)
Ω(mα−1)
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
,
Ω(t0)(λ, µ) :=
∫
∂S
(mα)
(t0)
Ω(mα−1)
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
,
with a point t0 ∈ S
(mα)
(t0)
⊂ Mα, being taken fixed, λ, µ ∈ Σ, and interpret them as the
corresponding kernels [27, 28] of the integral invertible operators of Hilbert – Schmidt
type Ω(t), Ω(t0) : L
(ρ)
2 (Σ,R) → L
(ρ)
2 (Σ,R), where ρ is some finite Borel measure on
the parameter set Σ. It assumes also above that the boundaries ∂S(mα)
(t) := σ
(mα−1)
(t) and
∂S
(mα)
(t0)
:= σ
(mα−1)
(t0)
are taken homological to each other as t → t0 ∈ Mα. Define now
the expressions
Ω± : ψ(0)(η) → ψ̃(0)(η)
for ψ(0)(η) ∈ H0
Λ(Mα), η ∈ Σ, and some ψ̃(0)(η)dt ∈ H0
Λ(Mα), where, by definition,
for any η ∈ Σ
ψ̃(0)(η) := ψ(0)(η)Ω−1
(t)Ω(t0) =
=
∫
Σ
dρ(µ)
∫
Σ
dρ(ξ)ψ(0)(µ)Ω−1
(t) (µ, ξ)Ω(t0)(ξ, η), (3.10)
being motivated by the expression (3.9). Suppose now that the elements (3.10) are ones
being related to some another Delsarte – Lions transformed cohomology group H0
Λ(α̃)(Mα),
that is the following condition:
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336 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
dα̃ψ̃
(0)(η) = 0 (3.11)
for ψ̃(0)(η) ∈ H0
Λ(α)(Mα), η ∈ Σ, and some new connection mapping in HΛ(Mα)
dα̃ :=
mα∑
j=1
dtj ∧ Ãj(t|∂),
where Ãj(t; ∂) := ∂/∂tj + Ãj(t), j = 1,mα, are parametrically dependent on t ∈Mα.
Put now
Ãj := Ω±AjΩ−1
± (3.12)
for each j = 1,mα, where Ω± : H0
Λ(Mα) → H0
Λ(Mα) are the corresponding Delsarte –
Lions transmutation operators related with some elements S±
(
σ
(mα−1)
(t) , σ
(mα−1)
(t0)
)
∈
∈ Cmα(Mα,R) related naturally with homological to each other boundaries ∂S(mα)
(t) =
= σ
(mα−1)
(t) and ∂S(mα)
(t0)
= σ
(mα−1)
(t0)
. Since all of differential expressions Aj : H0
Λ(Mα) →
→ H0
Λ(Mα), j = 1,mα, were taken commuting, the same property also holds for the
transformed operators (3.12), that is [Ãj , Ãk] = 0, k, j = 0,mα. The latter is, evidently,
equivalent owing to (3.12) to the following general expression:
dα̃ = Ω±dαΩ−1
± . (3.13)
For the conditions (3.13) and (3.11) to be satisfied, let us consider the corresponding to
(3.9) expressions
B̃
(mα)
λ (ψ̃(0)(η)dt) = S
(mα)
(t) Ω̃(t)(λ, η),
related with the corresponding external differentiation (3.13), where S(mα)
(t) ∈ Cmα(M,R)
and (λ, η) ∈ Σ × Σ. Assume further that there are also defined mappings
Ω�
± : ϕ(0)(λ) → ϕ̃(0)(λ)
with Ω�
± : H0
Λ(α)(Mα) → H0
Λ(α)(Mα), being some operators associated (but not nec-
essary adjoint!) with the corresponding Delsarte – Lions transmutation operators Ω± :
H0
Λ(Mα) → H0
Λ(Mα) and satisfying the standard relationships Ã∗
j := Ω�
±A∗
jΩ
�,−1
± ,
j = 1,mα. The proper Delsarte – Lions type operators Ω± : H0
Λ(Mα) → H0
Λ(Mα) are
related with two different realizations of the action (3.10) under the necessary conditions
dα̃ψ̃
(0)(η) = 0, d∗α̃ϕ̃
(0)(λ) = 0, (3.14)
needed to be satisfied and meaning, evidently, that the embeddings ϕ̃(0)(λ) ∈
∈ H0
Λ(α̃∗)(Mα), λ ∈ Σ, and ψ̃(0)(η) ∈ H0
Λ(α̃)(Mα), η ∈ Σ, are satisfied. Now we
need to formulate a lemma being important for the conditions (3.14) to hold.
Lemma 3.2. The following invariance property
Z(mα) = Ω(t0)Ω
−1
(t)Z
(mα)Ω−1
(t)Ω(t0) (3.15)
holds for any t and t0 ∈Mα.
As a result of (3.15) and the symmetry invariance between cohomology spaces
H0
Λ(α)(Mα) and H0
Λ(α̃)(Mα) one obtains the following pairs of related mappings:
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GENERALIZED DE RHAM – HODGE COMPLEXES, THE RELATED CHARACTERISTIC ... 337
ψ(0) = ψ̃(0)Ω̃−1
(t) Ω̃(t0), ϕ(0) = ϕ̃(0)Ω̃�,−1
(t) Ω̃�
(t0)
,
ψ̃(0) = ψ(0)Ω−1
(t)Ω(t0), ϕ̃(0) = ϕ(0)Ω�,−1
(t) Ω�
(t0)
,
(3.16)
where the integral operator kernels from L
(ρ)
2 (Σ,R) ⊗ L
(ρ)
2 (Σ,R) are defined as
Ω̃(t)(λ, µ) :=
∫
σ
(mα)
(t)
Ω̃(mα−2)
[
ϕ̃(0)(λ), ψ̃(0)(µ)dτ
]
,
Ω̃�
(t)(λ, µ) :=
∫
σ
(mα)
(t)
Ω̃
(mα−2),ᵀ[
ϕ̃(0)(λ), ψ̃(0)(µ)dτ
]
for all (λ, µ) ∈ Σ × Σ, giving rise to finding proper Delsarte – Lions transmutation oper-
ators ensuring the pure differential nature of the transformed expressions (3.12).
Note here also that owing to (3.15) and (3.16) the following operator property
Ω(t0)Ω
−1
(t)Ω(t0) + Ω̃(t0)Ω
−1
(t)Ω(t0) = 0 (3.17)
holds for any t and t0 ∈Mα meaning that Ω̃(t) = −Ω(t0).
One can now define similar to (3.6) the additional closed and dense in H0
Λ(Mα) three
subspaces
H0 :=
{
ψ(0)(µ) ∈ H0
Λ(α)(Mα) : dαψ
(0)(µ) = 0, µ ∈ Σ
}
,
H̃0 :=
{
ψ̃(0)(µ) ∈ H0
Λ(α̃)(Mα) : dα̃ψ̃
(0)(µ) = 0, µ ∈ Σ
}
,
H̃∗
0 :=
{
ϕ̃(0)(η) ∈ H0
Λ(α̃)(Mα) : d∗α̃ϕ̃
(0)(η) = 0, ϕ̃(0)(η)|Γ̃ = 0, η ∈ Σ
}
,
(3.18)
and construct the actions
Ω± : ψ(0) → ψ̃(0) := ψ(0)Ω−1
(t)Ω(t0), Ω�
±: ϕ(0) → ϕ̃(0) := ϕ(0)Ω�,−1
(t) Ω�
(t0)
(3.19)
on arbitrary but fixed pairs of elements
(
ϕ(0)(λ), ψ(0(µ)
)
∈ H∗
0 × H0, parametrized
by the set Σ, where by definition, one needs that all obtained pairs
(
ϕ̃(0)(λ), ψ̃(0)(µ)
)
,
λ, µ ∈ Σ, belong to H0
Λ(α̃)(Mα) ×H0
Λ(α̃)(Mα). Note also that related operator property
(3.17) can be compactly written down as follows:
Ω̃(t) = Ω̃(t0)Ω
−1
(t)Ω(t0) = −Ω(t0)Ω
−1
(t)Ω(t0).
Construct now from the expressions (3.19) the following operator kernels from the Hilbert
space L(ρ)
2 (Σ,R) ⊗ L
(ρ)
2 (Σ,R):
Ω(t)(λ, µ) − Ω(t0)(λ, µ) =
∫
∂S
(mα)
(t)
Ω(mα−1)
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
=
:= −
∫
∂S
(mα)
(t0)
Ω(mα−1)
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
338 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
=
∫
S
(mα)
(±) (σ
(mα−1)
(t) ,σ
(mα−1)
(t0) )
dΩ(mα−1)
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
=
=
∫
S
(mα)
(±) (σ
(mα−1)
(t) ,σ
(mα−1)
(t0) )
Z(mα)
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
,
and, similarly,
Ω©∗
(t)(λ, µ) − Ω©∗
(t0)
(λ, µ) =
∫
∂S
(mα)
(t)
Ω̄(mα−1),T
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
−
−
∫
∂S
(mα)
(t0)
Ω̄(mα−1),T
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
=
=
∫
S
(mα)
± (σ
(mα−1)
(t) ,σ
(mα−1)
(t0) )
dΩ̄(mα−1),T
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
=
=
∫
S
(mα)
± (σ
(mα−1)
(t) ,σ
(mα−1)
(t0) )
Z̄(mα−1),T
[
ϕ(0)(λ), ψ(0)(µ)dτ
]
, (3.20)
where λ, µ ∈ Σ, and by definition,mα-dimensional surfaces S(mα)
+
(
σ
(mα−1)
(t) , σ
(mα−1)
(t0)
)
and S
(mα)
−
(
σ
(mα−1)
(t) , σ
(mα−1)
(t0)
)
⊂ Cmα−1(Mα) are spanned smoothly without self-
intersection between two homological cycles σ
(mα−1)
(t) = ∂S
(mα)
(t) and σ
(mα−1)
(t0)
:=
:=∂S(mα)
(t0)
∈ Cmα−1(Mα,R) in such a way that the boundary ∂(S(mα)
+
(
σ
(mα−1)
(t) ,
σ
(mα−1)
(t0)
)
∪ S(mα)
−
(
σ
(mα−1)
(t) , σ
(mα−1)
(t0)
)
)
= ∅. Since the integral operator expressions
Ω(t0),Ω
©∗
(t0)
: L(ρ)
2 (Σ,R) → L
(ρ)
2 (Σ,R) are at a fixed point t0 ∈ Mα, evidently, constant
and assumed to be invertible, for extending the actions given (3.19) on the whole Hilbert
space H0
Λ(Mα)×H0
Λ(Mα) one can apply to them the classical constants variation ap-
proach, making use of the expressions (3.20). As a result, we obtain easily the following
Delsarte – Lions transmutation integral operator expressions
Ω± = 1−
∫
Σ×Σ
dρ(ξ)dρ(η)ψ̃(t; ξ)Ω−1
(t0)
(ξ, η)×
×
∫
S
(mα)
± (σ
(mα−1)
(t) ,σ
(mα−1)
(t) )
Z(mα)
[
ϕ(0)(η), ·
]
,
Ω©∗
± = 1−
∫
Σ×Σ
dρ(ξ)dρ(η)ϕ̃(t; η)Ω©∗ ,−1
(t0)
(ξ, η)×
×
∫
S
(mα)
± (σ
(mα−1)
(t) ,σ
(mα−1)
(t0) )
Z̄(mα),T
[
·, ψ(0)(ξ)dτ
]
(3.21)
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GENERALIZED DE RHAM – HODGE COMPLEXES, THE RELATED CHARACTERISTIC ... 339
for fixed pairs
(
ϕ(0)(ξ), ψ(0)(η)
)
∈ H∗
0 ×H0 and
(
ϕ̃(0)(λ), ψ̃(0)(µ)
)
∈ H̃∗
0 ×H̃0, λ, µ ∈
∈ Σ, being bounded invertible integral operators of Volterra type on the whole space
H×H∗.Applying the same arguments as in Section 1, one can show also that respectively
transformed sets of operators Ãj := Ω±AjΩ−1
± , j = 1,mα, and Ã∗
k := Ω©∗
± A∗
kΩ
©∗ ,−1
± ,
k = 1,mα, prove to be purely differential too. Thereby, one can formulate [4, 13, 14] the
following final theorem.
Theorem 3.4. The expressions (3.21) are bounded invertible Delsarte – Lions trans-
mutation integral operators of Volterra type onto H0
Λ(Mα)×H0
Λ(Mα), transforming, re-
spectively, given commuting sets of expressions Aj , j = 1,mα, and their formally adjoint
ones A∗
k, k = 1,mα, into the pure differential sets of expressions Ãj := Ω±AjΩ−1
± ,
j = 1,mα, and Ã∗
k := Ω©∗
± A∗
kΩ
©∗ ,−1
± , k = 1,mα. Moreover, the suitably constructed
closed subspaces H0 ⊂ H0
Λ(α)(Mα) and H̃0 ⊂ H0
Λ(α̃)(Mα) such that the operators Ω
and Ω©∗ ∈ B(H0
Λ(Mα)) depend strongly on the topological structure of the general-
ized cohomology groups H0
Λ(α)(Mα) and H0
Λ(α̃)(Mα), being parametrized by elements
S
(mα)
±
(
σ
(mα−1)
(t) , σ
(mα−1)
(t0)
)
∈ Cmα(Mα,R).
Some applications of the results obtained to multidimensional integrable nonlinear
differential systems on Riemannian manifolds we discuss in the next section.
4. An example: three-dimensional Davey – Stewartson type integrable differen-
tial system. Consider the generalized de Rham – Hodge theory for a commuting set A of
three differential expressions in a Hilbert space H0
Λ(Mα), for the special case when the
Riemannian compactified manifold M := M3
α × C̄
6 and
A :=
{
Aj :=
∂
∂tj
+Aj(t;λ|∂): tj ∈Mα,j := [0, Tj) ⊂ R+, j = 1, 3
}
,
where, by definition, the integral submanifold of a suitable multidimensional differential
system α̂ is M3
α :=
3
×
j=1
Mα,j and
A1 :=
∂
∂t1
+
−λ −u f
−ū λ −g
−f∗ −g∗ 0
, A2 =
∂
∂t2
+
−λ 0 f
0 −λ g
−f∗ g∗ 0
,
A3 =
∂
∂t3
+ i
−(λ2 + ff∗) fg∗ λf +
∂f
∂t2
−f∗g −λ2 + gg∗ λg +
∂g
∂t2
−λf∗ +
∂f∗
∂t2
λg∗ − ∂g∗
∂t2
−gg∗ + ff∗
for some smooth functions f, f∗, g, g∗, u, ū: Mα → R and arbitrary “spectral” parameter
λ ∈ R. The scalar product in H0
Λ(Mα) is given as
(ϕ,ψ) :=
∫
M3
α
dt〈ϕ,ψ〉E3
for any pair (ϕ,ψ) ∈ H0
Λ(Mα)×H0
Λ(Mα). Respectively, the Cartan connection mapping
dA : Λ(M,E) → Λ(M,E) on M is given as
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340 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
dA :=
3∑
j=1
dtj ∧ Aj + du ∧ ∂
∂u
+ dū ∧ I
∂
∂ū
+ df ∧ I
∂
∂f
+
+ df∗ ∧ I
∂
∂f∗
+ dg ∧ I
∂
∂g
+ dg∗ ∧ I
∂
∂g∗
.
It is easy to check that for all t ∈M3
α ⊂M the zero curvature condition d2
α := i∗αd
2
A = 0
holds, where dα :=
∑3
j=1
dtj ∧Aj , and commutators [Aj ,Ak] = 0 for j, k = 1, 3. The
latter is equivalent to the following three-dimensional Davey – Stewartson type integrable
differential system α̂ on the Riemannian manifold M:
du
dt3
= i
(
∂2u
∂t1∂t2
+ 2u(ff∗ + gg∗
)
,
dū
dt3
= −i
(
∂2ū
∂t1∂t2
+ 2ū(ff∗ + gg∗
)
,
augmented with such a set of compatible differential relationships:
∂g
∂t2
= − ∂g
∂t1
+ ūf,
∂g∗
∂t2
= −∂g
∗
∂t1
+ uf∗,
∂f
∂t2
=
∂f
∂t1
− ug,
∂f∗
∂t2
=
∂f∗
∂t1
− ūg∗,
∂f
∂t2
=
∂f
∂t1
− ug,
∂f∗
∂t2
=
∂f∗
∂t1
− ūg∗,
∂u
∂t2
= −2fg∗,
∂ū
∂t2
= −2f∗g,
∂(ff∗)
∂t2
− ∂(ff∗)
∂t1
=
1
2
∂(uū)
∂t2
= −
(
∂(gg∗)
∂t1
+
∂(gg∗)
∂t2
)
,
df
dt3
= i
(
∂2f
∂t21
+ (2ff∗ − uū)f − ∂u
∂t1
g
)
,
df∗
dt3
= −i
(
∂2f∗
∂t21
+ (2ff∗ − uū)f∗ − ∂ū
∂t1
g∗
)
,
dg
dt3
= i
(
∂2g
∂t21
− (2gg∗ + uū)g − ∂ū
∂t1
f
)
,
dg∗
dt3
= −i
(
∂2g∗
∂t21
− (2gg∗ + uū)g∗ − ∂u
∂t1
f∗
)
.
In particular, this means, evidently, that the corresponding generalized co-chains of mod-
ules
E → Λ0(Mα, E) dα→ Λ1(Mα, E) dα→ . . .
dα→ Λmα(Mα, E) dα→ 0,
E → Λ0(Mα, E)
d∗
α→ Λ1(Mα, E)
d∗
α→ . . .
d∗
α→ Λmα(Mα, E)
d∗
α→ 0
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
GENERALIZED DE RHAM – HODGE COMPLEXES, THE RELATED CHARACTERISTIC ... 341
are de Rham complexes. It is easy to check that all Chern type characteristic invariants
χj(α), j ∈ Z+, on the integral submanifold Mα ⊂M are trivial. Nonetheless, the Chern
character ch(E,M) of the suitable connection mapping dA : Λ(M,E) → Λ(M,E) on
the whole Riemannian manifold M is nontrivial, giving rise to some sets of relationships,
describing moduli space [3, 16, 17] of the Hermitian linear fiber bundle E(M) over the
manifold M.
Based on relationships (3.6) and (3.18), proceed to constructing closed subspaces H©∗
0
and H0, making possible to construct suitable Delsarte – Lions transmutation operators:
H0 :=
{
ψ(0)(λ, η) ∈ H0
Λ(α)(Mα) :
∂ψ(0)(λ, η)
∂tj
= Aj(t)ψ(0)(λ, η), j = 1, 3,
ψ(0)(λ, η)
∣∣∣
t=t0
= ψλ(η) ∈ E,
(λ, η) ∈ Σ × Σσ
}
,
H∗
0 :=
{
ϕ(0)(λ, η) ∈ H0
Λ(α)(Mα) :
−∂ϕ
(0)(λ, η)
∂tj
= A∗
j (t)ϕ
(0)(λ, η), j = 1, 3,
ϕ(0)(λ, η)
∣∣∣
t=t0
= ϕλ(η) ∈ E,
(λ, η) ∈ Σ × Σσ
}
(4.1)
for some “spectral” set Σσ ∈ C
p−1. By means of subspaces (4.1) one can now proceed to
construction of Delsarte – Lions transmutation operators Ω± : H0
Λ(Mα) � H0
Λ(Mα) in
the general form like (3.21) with kernels Ω(t0)(λ; ξ, η) ∈ L
(ρ)
2 (Σσ,R) ⊗ L
(ρ)
2 (Σσ,R) for
every λ ∈ Σ, being defined as
Ω(t0)(λ; ξ, η) :=
∫
σ
(mα−1)
(t0)
Ω(mα−1)
[
ϕ(0)(λ; ξ), ψ(0)(λ; η)dτ
]
,
Ω©∗
(t0)
(λ; ξ, η) :=
∫
σ
(mα−1)
(t0)
Ω̄(mα−1),T
[
ϕ(0)(λ; ξ), ψ(0)(λ; η)dτ
]
for all (λ; ξ, η) ∈ Σ×Σ3
σ.As a result one gets for the corresponding product ρ := ρσ◦ρΣ2
σ
such integral expressions:
Ω± = 1−
∫
Σ
dρσ(λ)
∫
Σσ×Σσ
dρΣσ
(ξ)dρΣσ
(η)×
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
342 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
×
∫
dτ
S
(mα)
± (σ
(mα−1)
(t) ,σ
(mα−1)
(t0) )
ψ̃(0)(λ; ξ)Ω−1
(t0;t10)
(λ; ξ, η)ϕ̄(0),T(λ; η)(·),
(4.2)
Ω©∗
± = 1−
∫
Σ
dρσ(λ)
∫
Σσ×Σσ
dρΣσ
(ξ)dρΣσ
(η)×
×
∫
dτ
S
(mα)
± (σ
(mα−1)
(t) ,σ
(mα−1)
(t0) )
ϕ̃
(0)
λ (ξ)Ω̄T,−1
(t0)
(λ; ξ, η) × ψ̄(0),T(λ; η)(·),
where S(mα)
+
(
σ
(mα−1)
(t0)
, σ
(mα−1)
(t0)
)
∈ Cmα
(Mα,R) is some smoothmα-dimensional sur-
face spanned between two homological cycles σ(mα−1)
(t) and σ
(mα−1)
(t0)
∈ K(Mα) and
S
(mα)
−
(
σ
(mα−1)
(t) , σ
(mα−1)
(t0)
)
∈ Cmα(Mα,R) is its smooth counterpart such that the bound-
ary ∂
(
S
(mα)
+
(
σ
(mα−1)
(t) , σ
(mα−1)
(t0)
)
∪S(mα)
−
(
σ
(mα−1)
(t) , σ
(mα−1)
(t0)
))
= ∅. Concerning the
related results obtained above one can construct from (4.2) the corresponding factorized
Fredholm operators Ω and Ω©∗ : H0
Λ(Mα) → H0
Λ(Mα), as follows:
Ω := Ω−1
+ Ω−, Ω©∗ := Ω©∗ −1
+ Ω©∗
− .
It is also important to notice here that kernels K̂±(Ω) and K̂±(Ω©∗ ) ∈ H0
Λ(Mα) ⊗
⊗H0
Λ(Mα) satisfy exactly the generalized [14, 27, 28] determining equations in the fol-
lowing tensor form
(Ãj,ext ⊗ 1)K̂±(Ω) = (1⊗A∗
j,ext)K̂±(Ω),
(Ã∗
j,ext ⊗ 1)K̂±(Ω©∗ ) = (1⊗Aj,ext)K̂±(Ω©∗ ).
Since, evidently, supp K̂+(Ω) ∩ supp K̂−(Ω) = ∅ and supp K̂+(Ω©∗ ) ∩
∩ supp K̂−(Ω©∗ ) = ∅, one derives from results [14, 24, 25, 29] the corresponding
Gelfand – Levitan – Marchenko equations
K̂+(Ω) + Φ̂(Ω)+K̂+(Ω) · Φ̂(Ω)=K̂−(Ω),
K̂+(Ω©∗ ) + Φ̂(Ω©∗ )+K̂+(Ω©∗ ) · Φ̂(Ω©∗ )=K̂−(Ω©∗ ),
where, by definition, Ω := 1+ Φ̂(Ω),Ω©∗ := 1 + Φ̂(Ω©∗ ), which can be solved [30, 31]
in the space B±
∞(H0
Λ(Mα)) for kernels K̂±(Ω) and K̂±(Ω©∗ ) ∈ H0
Λ(Mα) ⊗ H0
Λ(Mα)
depending on t ∈ M2
α. Thereby, Delsarte – Lions transformed differential expressions
Aj : H0
Λ(Mα) → H0
Λ(Mα), j = 1, 3, will be, evidently, commuting to each other too,
satisfying the following differential relationships:
Ãj =
∂
∂tj
+ Ω±AjΩ−1
± −
(
∂Ω±
∂tj
)
Ω−1
± :=
∂
∂tj
+ Ãj , (4.3)
where expressions for Ãj ∈ L(H0
Λ(Mα)), j = 1, 3, prove to be purely matrices. The
latter property makes it possible to construct nonlinear partial differential equations on
coefficients of differential expressions (4.3) and solve them by means of the standard pro-
cedures either of inverse spectral [32 – 36] or the Darboux – Backlund [25, 37] transforms,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
GENERALIZED DE RHAM – HODGE COMPLEXES, THE RELATED CHARACTERISTIC ... 343
producing a wide class of exact soliton like solutions. On these and related Chern type
differential invariants problems we will stay in detail elsewhere.
5. Conclusion. The study done above presents some of recent results devoted to
the development of a generalized de Rham – Hodge theory [1, 4, 14, 17, 19, 21, 22, 25]
and related differential-geometric aspects of Chern characteristic classes, concerning spe-
cial differential complexes with Cartan type connections, which give rise to effective an-
alytical tools of studying multidimensional integrable nonlinear differential systems of
M. Gromov type [7] on Riemannian manifolds. Some results on the structure of the
Delsarte – Lions transmutation operators are adapted for constructing effective transfor-
mations of Cartan type connections for multidimensional integrable Davey – Stewartson
type nonlinear differential system on a Riemannian manifold M, vanishing upon three-
dimensional integral submanifold Mα ⊂ M. The results obtained can be used for study-
ing a wide class of exact special solutions to this differential systems, having applications
[1, 3, 4, 16, 30, 32, 38] at solving some problems of modern differential topology and
mathematical physics.
6. Acknowledgements. The authors are very appreciated to the Abdus Salam ICTP
and SISSA Institutions (Trieste, Italy), for invitation and nice hospitality, where part of
the article was prepared. They also thankful to their friends and colleagues from ICTP
and SISSA for helpful and constructive discussions of the problems treated in the article.
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Received 12.09.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
|
| id | umjimathkievua-article-3310 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:40:06Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2a/5d991b0f5f5906179f95295fce3a932a.pdf |
| spelling | umjimathkievua-article-33102020-03-18T19:51:00Z Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds Узагальнені комплекси де Рама-Ходжа, споріднені характеристичні класи Черна та деякі застосування до інтегровних багатовимірних диференціальних систем на ріманових многовидах Bogolyubov, N. N. Prykarpatsky, A. K. Боголюбов, М. М. (мл.) Прикарпатський, А. К. We study the differential-geometric aspects of generalized de Rham-Hodge complexes naturally related to integrable multidimensional differential systems of the M. Gromov type, as well as the geometric structure of the Chern characteristic classes. Special differential invariants of the Chern type are constructed, their importance for the integrability of multidimensional nonlinear differential systems on Riemannian manifolds is discussed. An example of the three-dimensional Davey-Stewartson-type nonlinear integrable differential system is considered, its Cartan type connection mapping, and related Chern-type differential invariants are analyzed. Досліджено диференціально-геометричні аспекти узагальнених комплексів де Рама-Ходжа, що природним чином пов'язані з інтегровними багатовимірними диференціальними системами типу M. Громова, а також геометричну структуру характеристичних класів Черна. Побудовано спеціальні диференціальні інваріанти типу Черна та розглянуто їх важливість для інтегровності багатовимірних нелінійних диференціальних систем на ріманових многовидах. Розглянуто приклад тривимірної нелінійної інтегровної диференціальної системи типу Деві-Стюартсона і проаналізовано їх сполучне відображення та споріднені диференціальні інваріанти типу Черна. Institute of Mathematics, NAS of Ukraine 2007-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3310 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 3 (2007); 327–344 Український математичний журнал; Том 59 № 3 (2007); 327–344 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3310/3365 https://umj.imath.kiev.ua/index.php/umj/article/view/3310/3366 Copyright (c) 2007 Bogolyubov N. N.; Prykarpatsky A. K. |
| spellingShingle | Bogolyubov, N. N. Prykarpatsky, A. K. Боголюбов, М. М. (мл.) Прикарпатський, А. К. Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds |
| title | Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds |
| title_alt | Узагальнені комплекси де Рама-Ходжа, споріднені характеристичні класи Черна та деякі застосування до інтегровних багатовимірних диференціальних систем на ріманових многовидах |
| title_full | Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds |
| title_fullStr | Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds |
| title_full_unstemmed | Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds |
| title_short | Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds |
| title_sort | generalized de rham-hodge complexes, the related characteristic chern classes, and some applications to integrable multidimensional differential systems on riemannian manifolds |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3310 |
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