Compatible metrics of constant Riemannian curvature: local geometry, nonlinear equations and integrability

12 Pages·0.15 MB·English

by  O. I. Mokhov

Compatible metrics of constant
Riemannian curvature: local geometry,
nonlinear equations and integrability1
2
0
0
2
O. I. Mokhov

n
a
J

9
2 1 Introduction

]
G Inthepresentpaper,thenonlinearequationsdescribingallthenonsingularpencilsofmetrics
D ofconstantRiemanniancurvaturearederivedandthe integrabilityofthese nonlinearequa-
tions by the method of inverse scattering problem is proved. These results were announced
.
h
in our previous paper [1]. For the flat pencils of metrics the corresponding statements and
t
a proofs were presented in the presentauthor’s work [2], [3], where the method of integrating
m
the nonlinear equations for the nonsingular flat pencils of metrics was proposed. In [4] the
[ Lax pair for the nonsingular flat pencils of metrics was demonstrated. This Lax pair is

generalized to the case of arbitrary nonsingular pencils of metrics of constant Riemannian
1
curvature (see examples and interesting applications in [4]).
v
0 Inthispaper,itisprovedthatallthenonsingularpairsofcompatiblemetricsofconstant
8 Riemanniancurvature are describedby special integrablereductions of nonlinear equations
2 definingorthogonalcurvilinearcoordinatesystemsinthespacesofconstantcurvature. Note
1
thatthe problemofdescriptionforthe pencils ofmetrics ofconstantRiemanniancurvature
0
is equivalent to the problem of description for compatible nonlocal Poisson brackets of
2
0 hydrodynamic type generated by metrics of constant Riemannian curvature (compatible
/ Mokhov–Ferapontov brackets [5]) playing an important role in the theory of systems of
h
t hydrodynamic type.
a Recall that two pseudo-Riemannian contravariant metrics gij(u) and gij(u) are called
m 1 2
compatibleifforanylinearcombinationofthesemetricsgij(u)=λ1g1ij(u)+λ2g2ij(u),where
v: λ1 and λ2 are arbitrary constants for which det(gij(u)) 0, the coefficients of the corre-
6≡
i sponding Levi-Civita connections and the components of the corresponding tensors of Rie-
X
r mannian curvature are related by the same linear formula: Γikj(u)=λ1Γi1j,k(u)+λ2Γi2j,k(u)
a andRkijl(u)=λ1R1ij,kl(u)+λ2R2ij,kl(u)(in this case,we shallsayalso that the metrics g1ij(u)
and gij(u) form a pencil of metrics) [1]. Flat pencils of metrics, that is nothing but com-
2
patible nondegenerate local Poissonbrackets of hydrodynamic type (compatible Dubrovin–
Novikovbrackets[6]),wereintroducedin[7]. Twopseudo-Riemanniancontravariantmetrics
g1ij(u)andg2ij(u)ofconstantRiemanniancurvatureK1 andK2 respectivelyarecalledcom-
patibleifanylinearcombinationofthesemetricsgij(u)=λ1g1ij(u)+λ2g2ij(u),whereλ1 and
λ2 are arbitrary constants for which det(gij(u)) 0, is a metric of constant Riemannian
6≡
1This work was supported by the Alexander von Humboldt Foundation (Germany), the RussianFoun-
dationforBasicResearch(grantNo. 99–01–00010) andtheINTAS(grantNo. 99–1782).
1
curvature λ1K1 +λ2K2 and the coefficients of the corresponding Levi-Civita connections
are related by the same linear formula: Γikj(u) = λ1Γi1j,k(u)+λ2Γi2j,k(u) [1]. In this case,
we shall also say that the metrics gij(u) and gij(u) form a pencil of metrics of constant
1 2
Riemannian curvature[1]. Itisobviousthatallthesedefinitions aremutuallyconsistent,so
that if compatible metrics are metrics of constant Riemannian curvature, then they form a
pencilofmetrics ofconstantRiemanniancurvature,andifcompatiblemetrics areflat,then
theyforma flatpencilofmetrics. Apairofpseudo-Riemannianmetricsgij(u)andgij(u)is
1 2
callednonsingulariftheeigenvaluesofthispairofmetrics,thatis,therootsoftheequation
det(gij(u) λgij(u))=0, aredistinct (a pencil ofmetrics whichis formedby a nonsingular
1 − 2
pair of metrics is also called nonsingular). In [1], it is proved that an arbitrary nonsingular
pair of metrics is compatible if and only if there exist local coordinates u = (u1,...,uN)
such that both the metrics are diagonal in these coordinates and have the following special
form (one can consider that one of the metrics, here gij(u), is an arbitrary diagonal met-
2
ric): gij(u) = gi(u)δij and gij(u)= fi(ui)gi(u)δij, where fi(ui), i = 1,...,N, are functions
2 1
of single variable (generally speaking, complex). It is obvious that the eigenvalues of the
considered pair of metrics are given by the functions fi(ui), i=1,...,N.
2 Equations for nonsingular pairs of compatible
metrics of constant Riemannian curvature
Considerthe problemofdescriptionfor nonsingularpairsof compatible metrics of constant
Riemannian curvature. As is proved in [1] (see also Introduction above), it is sufficient
to classify pairs of diagonal metrics of constant Riemannian curvature of the special type
gij(u) = gi(u)δij and gij(u) = fi(ui)gi(u)δij, where fi(ui), i = 1,...,N, are arbitrary
2 1
(generally speaking, complex) functions of single variable, gi(u), i = 1,...,N, are arbitrary
functions.
TheproblemofdescriptionforN N diagonalmetricsofconstantRiemanniancurvature
K, that is, metrics gij(u) = gi(u)δi×j such that their tensor of Riemannian curvature has
2
the form Rij = K(δiδj δiδj), where K = const, is the classical problem of differential
kl l k − k l
geometry. Thisproblemisequivalenttotheproblemofdescriptionoforthogonalcurvilinear
coordinate systems in N-dimensional spaces of constant curvature K. Recently Zakharov
showed that the Lam´e equations describing orthogonal curvilinear coordinate systems in
Euclideanorpseudo-EuclideanN-dimensionalspacesareintegratedbythemethodofinverse
scattering problem [8] (see also an algebraic-geometric approach in [9]). Similarly, the
equations describing orthogonal curvilinear coordinate systems in N-dimensional spaces of
constant curvature are integrated by the method of inverse scattering problem.
Introduce the standard classical notation
εi N
gi(u)= , ds2 = εi(H (u))2(dui)2, (2.1)
(H (u))2 i
i i=1
X
1 ∂H
k
β (u)= , i=k, (2.2)
ik H (u) ∂ui 6
i
2
where H (u) are the Lam´e coefficients and β (u) are the rotation coefficients, εi = 1,
i ik
±
i = 1,...,N. Although, in our case, all the functions are, generally speaking, complex, we
shall use formulae, which are convenient for using also in the purely real case.
Theorem 2.1 Nonsingular pairs of compatible metrics of constant Riemannian curvature
K1 and K2 are described by the following consistent integrable nonlinear systems:
∂β
ij
=β β , i=j, i=k, j =k, (2.3)
∂uk ik kj 6 6 6
∂β ∂β
εi ∂uiij +εj ∂ujji + εsβsiβsj =−K2HiHj, i6=j, (2.4)
s6=i, s6=j
X
∂β 1 ∂β 1
εifi(ui) ij + εi(fi)′β +εjfj(uj) ji + εj(fj)′β +
∂ui 2 ij ∂uj 2 ji
εsfs(us)βsiβsj = K1HiHj, i=j, (2.5)
− 6
s6=i, s6=j
X
where fi(ui), i=1,...,N, are arbitrary given functions of one variable.
Remark 2.1 Equations (2.3) are the well-known n-wave equations (the Darboux equations
[10]), and equations (2.3), (2.4) describe orthogonal curvilinear coordinate systems in N-
dimensional spaces of constant curvature K2, in particular, for K2 = 0 we get the famous
Lam´eequations. Equations (2.5) defineanontrivial nonlineardifferential reductionof equa-
tions (2.3), (2.4).
Remark 2.2 If there are coinciding nonzero constants among the functions fi(ui), then
equations (2.3)–(2.5) also describe a pencil of metrics of constant Riemannian curvature,
but, obviously, this pencil is not nonsingular.
The system of equations (2.2)–(2.5) is consistent for any functions fi(ui) what can be
easilycheckedbydirectcalculations. Moreover,thegeneralsolutionofthenonlinearsystem
(2.2)–(2.5)dependonN2 arbitraryfunctionsofonevariable(thefunctionsβ (u)andH (u)
ij j
can be arbitrarily defined on the jth coordinate line ).
Let us consider the conditions that the diagonal metrics gij(u)=gi(u)δij and gij(u)=
2 1
fi(ui)gi(u)δij, where fi(ui), i = 1,...,N, are arbitrary functions of one indicated variable
(these functions must not be only equal to zero identically), are metrics of constant Rie-
mannian curvature K2 and K1 respectively.
Recall that for any diagonal metric Γi (u)=0 if all the indices i,j,k are distinct. It is
jk
also obvious that Rij(u)=0 if all the indices i,j,k,l are distinct. In addition, by virtue of
kl
the well-known symmetries of the tensor of Riemannian curvature, we have:
Rii(u)=Rij (u)=0, Rij(u)= Rij(u)=Rji(u)= Rji(u).
kl kk il − li li − il
3
Thus, it is sufficient to consider the condition Rkijl(u) = K2(δliδkj −δkiδlj) (the necessary
and sufficient condition that the corresponding metric is a metric of constant Riemannian
curvature K2) only for the following components of the tensor of Riemannian curvature:
Rij(u), where i=j, i=l:
il 6 6
Riilj =−K2δlj, i6=j, i6=l.
For an arbitrary diagonal metric gij(u)=gi(u)δij, we get
2
1 ∂gi
Γi (u)=Γi (u)= , for all i,k;
2,ik 2,ki −2gi(u)∂uk
1 gi(u) ∂gj
Γi (u)= , i=j,
2,jj 2(gj(u))2∂ui 6
Rij (u)=gi(u)Rj (u)=
2,il 2,iil
∂Γj ∂Γj N N
gi(u) 2,il 2,ii + Γj (u)Γs (u) Γj (u)Γs (u) . (2.6)
∂ui − ∂ul 2,si 2,il − 2,sl 2,ii !
s=1 s=1
X X
It is necessary to consider two cases separately.
1) j =l.
6
Rij =0, i=j, i=l, j =l.
il 6 6 6
∂Γj
Rij (u)=gi(u) 2,ii +Γj (u)Γi (u) Γj (u)Γj (u) Γj (u)Γl (u) =
2,il − ∂ul 2,ii 2,il − 2,jl 2,ii − 2,ll 2,ii !
1 ∂ gj(u) ∂gi 1 gj(u) ∂gi ∂gi
gi(u)
−2 ∂ul (gi(u))2∂uj − 4(gi(u))2∂uj ∂ul
(cid:18) (cid:19)
1 ∂gi ∂gj 1 gj(u) ∂gl ∂gi
+ =0. (2.7)
4gi(u)∂uj ∂ul − 4gi(u)gl(u)∂uj ∂ul
Equations (2.7) are equivalent to the equations
∂2H 1 ∂H ∂H 1 ∂H ∂H
i i j k i
= + , i=j, i=k, j =k, (2.8)
∂uj∂uk H (u) ∂uj ∂uk H (u) ∂uj ∂uk 6 6 6
j k
which are equivalent, in turn, to equations (2.3).
2) j =l.
Riijj =−K2, i6=j.
4
∂Γj ∂Γj
Rij (u)=gi(u) 2,ij 2,ii +Γj (u)Γi (u)+
2,ij ∂ui − ∂uj 2,ii 2,ij
N
Γj (u)Γj (u) Γj (u)Γs (u) =
2,ji 2,ij − 2,sj 2,ii !
s=1
X
1 ∂ 1 ∂gj 1 ∂ gj(u) ∂gi 1 gj(u) ∂gi ∂gi
gi(u) gi(u)
−2 ∂ui gj(u)∂ui − 2 ∂uj (gi(u))2∂uj − 4(gi(u))2∂uj ∂uj
(cid:18) (cid:19) (cid:18) (cid:19)
1 gi(u) ∂gj ∂gj 1 ∂gj ∂gi 1 gs(u) ∂gj ∂gi
+4(gj(u))2 ∂ui ∂ui − 4gj(u)∂ui∂ui + 4gi(u)gj(u)∂us∂us =−K2. (2.9)
s6=i
X
Equations (2.9) are equivalent to the equations
∂ 1 ∂H ∂ 1 ∂H
εi j +εj i +
∂ui H (u) ∂ui ∂uj H (u) ∂uj
(cid:18) i (cid:19) (cid:18) j (cid:19)
εs ∂H ∂H
i j
(H (u))2 ∂us ∂us =−K2HiHj, i6=j, (2.10)
s
s6=i, s6=j
X
which are equivalent, in turn, to equations (2.4).
The condition that the metric gij(u) = fi(ui)gi(u)δij is also a metric of constant Rie-
1
mannian curvature gives N(N 1)/2 additional equations (2.5). Note that in this case
−
components (2.7)of the tensorof Riemanniancurvature vanishautomatically,and the con-
dition (2.9) gives the corresponding N(N 1)/2 equations.
Actually, for the metric gij(u) = fi(ui−)gi(u)δij, the Lam´e coefficients and the rotation
1
coefficients have the form
H (u) ǫiεi
H (u)= i , fi(ui)gi(u)= , ǫi = 1, (2.11)
i
ǫifi(ui) (H (u))2 ±
i
e p1 ∂Hk
βik(u)= H (u) ∂ui = e
i
e
e ǫifi(ui) 1 ∂H ǫifi(ui)
k
pǫkfk(uke)(cid:18)Hi(u) ∂ui (cid:19)= pǫkfk(uk)βik(u), i6=k. (2.12)
p p
Accordingly, equations (2.3) are automatically satisfied also for the rotation coefficients
β (u), and equations (2.4) for β (u) give equations (2.5).
ik ik
Note that, for our further purposes, it is more convenient to write the system (2.3)–
(e2.5) namely in this form in oerder to emphasize the reduction (2.11), (2.12) playing an
important role in our method of integrating this system (see the next section), although it
is easy to show that equations (2.4), (2.5) for nonsingular pairs of metrics (that is, all the
functionsfi(ui)mustbedistinctalsointhecaseifthey areconstants)areequivalenttothe
following equations (in particular, it is more convenient to use these equations for checking
5
the consistency of system (2.2)–(2.5))
∂β 1 (fi(ui))′ εiεj (fj(uj))′
ij
= β + β
∂ui 2(fj(uj) fi(ui)) ij 2 (fj(uj) fi(ui)) ji−
− −
εiεs(fj(uj)−fs(us))β β +εi K1−K2fj(uj) H H , i=j. (2.13)
− (fj(uj) fi(ui)) si sj (fj(uj) fi(ui)) i j 6
s6=i,s6=j − −
X
Example 2.1 If all the functions fi(ui) are arbitrary distinct nonzero constants: fi(ui)=
ci, ci =const=0,i=1,...,N,ci =cj for i=j, then the integrable system (2.2)–(2.5) takes
6 6 6
the following form:
∂β
ij
=β β , i=j, i=k, j =k, (2.14)
∂uk ik kj 6 6 6
∂βij = εiεs(cj −cs)β β +εiK1−K2cjH H , i=j, (2.15)
∂ui − (cj ci) si sj (cj ci) i j 6
s6=i,s6=j − −
X
∂H
j
=β H , i=j. (2.16)
∂ui ij i 6
In the flat case, for K1 = K2 = 0, system (2.14), (2.15) was considered in [4] (it is
relatedtoacurioustripleofpairwisecommutingMonge–Amp`ereequations). Sothesystem
of equations (2.14)–(2.16) is also of special interest for applications.
3 Compatible flat metrics and the Zakharov method
of differential reductions
In this section, we demonstrate the method of integrating the system of equations (2.3)–
(2.5) in the flat case (for K1 = K2 = 0) (see [2], [3]), that is, the following system (in this
section εi =1, i=1,...,N in all formulae):
∂β
ij
=β β , i=j, i=k, j =k, (3.1)
∂uk ik kj 6 6 6
∂β ∂β
ij ji
+ + β β =0, i=j, (3.2)
∂ui ∂uj si sj 6
s6=i, s6=j
X
∂β 1 ∂β 1
fi(ui) ij + (fi)′β +fj(uj) ji + (fj)′β +
∂ui 2 ij ∂uj 2 ji
fs(us)β β =0, i=j, (3.3)
si sj
6
s6=i, s6=j
X
where fi(ui), i=1,...,N, are arbitrary given functions of one variable.
Recall the Zakharov method for integrating the Lam´e equations (3.1) and (3.2) [8].
6
We must choose a matrix function F (s,s′,u) with some special properties and solve
ij
the linear integral equation
∞
K (s,s′,u)=F (s,s′,u)+ K (s,q,u)F (q,s′,u)dq. (3.4)
ij ij il lj
Zs l
X
Then we obtain a one-parameter family of solutions of the Lam´e equations by the formula
β (s,u)=K (s,s,u). (3.5)
ij ji
In particular, if F (s,s′,u)=f (s ui,s′ uj), where f (x,y) is an arbitrary matrix
ij ij ij
− −
function of two variables, then formula (3.5) produces solutions of equations (3.1). To
constructsolutionsofequations(3.2),Zakharovproposedtoimposeonthe“dressingmatrix
function” F (s ui,s′ uj) a certain additional linear differential relation. If the function
ij
− −
F (s ui,s′ uj) satisfies the Zakharov differential relation (we shall present it below),
ij
− −
then the rotation coefficients β (u) additionally satisfy equations (3.2).
ij
Let us describe our scheme of integrating all the system of equations (3.1)–(3.3).
Lemma 3.1 If both the function F (s ui,s′ uj) and the function
ij
− −
fj(uj s′)
F (s ui,s′ uj)= − F (s ui,s′ uj) (3.6)
ij ij
− − fi(ui s) − −
p
−
satisfy the Zakharovedifferential relation,pthen the corresponding rotation coefficients βij(u)
(3.5) satisfy all the equations (3.1)–(3.3).
Actually, if K (s,s′,u) is the corresponding to the function F (s ui,s′ uj) solution of
ij ij
− −
the linear integral equation (3.4), then
fj(uj s′)
K (s,s′,u)= − K (s,s′,u) (3.7)
ij ij
fi(ui s)
p
−
is the corresponding to funection (3.6) solpution of equation (3.4). It is easy to prove multi-
plying the integral equation (3.4) by fj(uj s′)/ fi(ui s):
− −
p p
∞
K (s,s′,u)=F (s ui,s′ uj)+ K (s,q,u)F (q ul,s′ uj)dq. (3.8)
ij ij il lj
− − − −
Zs l
X
e e e e
Then both β (s,u) = K (s,s,u) and β (s,u) = K (s,s,u) satisfy the Lam´e equations
ij ji ij ji
(3.1) and (3.2). Besides, we have
e e
fi(ui s) fi(ui s)
β (s,u)=K (s,s,u)= − K (s,s,u)= − β (s,u). (3.9)
ij ji ji ij
fj(uj s) fj(uj s)
p p
− −
Thues,inthiscaese,therotatiopncoefficientsβ (u)satisfypexactlyallequations(3.1)–(3.3),
ij
that is, they generate the corresponding compatible flat metrics.
7
The Zakharov reduction is given by the following differential relations [8]:
∂F (s,s′,u) ∂F (s′,s,u)
ij ji
+ =0. (3.10)
∂s′ ∂s
To resolve these relations for the matrix function F (s ui,s′ uj), we can introduce
ij
− −
N(N 1)/2 arbitrary functions of two variables Φ (x,y), i<j, and put for i<j
ij
−
∂Φ (s ui,s′ uj)
F (s ui,s′ uj)= ij − − ,
ij
− − ∂s
∂Φ (s′ ui,s uj)
F (s ui,s′ uj)= ij − − , (3.11)
ji
− − − ∂s
and, besides, put for any i
∂Φ (s ui,s′ ui)
F (s ui,s′ ui)= ii − − , (3.12)
ii
− − ∂s
where Φ (x,y), i=1,...,N, are arbitrary skew-symmetric functions:
ii
Φ (x,y)= Φ (y,x), (3.13)
ii ii
−
see [8].
For the function
fj(uj s′)
F (s ui,s′ uj)= − F (s ui,s′ uj) (3.14)
ij ij
− − fi(ui s) − −
p
−
e p
the Zakharovdifferential relations (3.10) give exactly N(N 1)/2 linear partial differential
−
equationsofthesecondorderforN(N 1)/2functionsoftwovariablesΦ (s ui,s′ uj), i<
ij
− − −
j:
∂ fj(uj s′)∂Φ (s ui,s′ uj)
ij
− − −
∂s′ pfi(ui s) ∂s !−
−
∂ pfi(ui s) ∂Φ (s ui,s′ uj)
ij
− − − =0, i<j, (3.15)
∂s pfj(uj s′) ∂s′ !
−
p
which are equivalent to the equations
∂2Φ (s ui,s′ uj) ∂Φ (s ui,s′ uj)dfi(ui s)
2 ij − − fi(ui s) fj(uj s′) + ij − − −
∂ui∂uj − − − ∂uj dui −
∂Φ (s ui,s′ uj)dfj(cid:0)(uj s′) (cid:1)
ij
− − − =0, i<j. (3.16)
∂ui duj
It is interesting that all these equations (3.16) for the functions Φ (s ui,s′ uj) are
ij
− −
ofthe same type andcoincide with the single equationin the two-componentcase (N =2),
8
which was derived directly from the conditions of vanishing the corresponding tensors of
Riemannian curvature (see [1]).
Besides, for N functions Φ (s ui,s′ ui), we have also N linear partial differential
ii
− −
equations of the second order, which are derived from the Zakharov differential relations
(3.10):
∂ fi(ui s′)∂Φ (s ui,s′ ui) ∂ fi(ui s) ∂Φ (s′ ui,s ui)
ii ii
− − − + − − − =0
∂s′ pfi(ui s) ∂s ! ∂s pfi(ui s′) ∂s′ !
− −
(3.17)
p p
or equivalently
∂2Φ (s ui,s′ ui)
2 ii − − fi(ui s) fi(ui s′)
∂s∂s′ − − − −
∂Φ (s ui,s′ ui)dfi(cid:0)(ui s′) ∂Φ (s ui,(cid:1)s′ ui)dfi(ui s)
ii ii
− − − + − − − =0. (3.18)
∂s ds′ ∂s′ ds
Any solution of the linear partial differential equations of the second order (3.16) and
(3.18)generatesaone-parameterfamilyofsolutionsofsystem(3.1)–(3.3)bylinearrelations
and explicit formulae (3.11), (3.12), (3.4), (3.5). Thus system (3.1)–(3.3) is linearized.
The method ofdifferential reductionsis applicablealso to integratingarbitrarynonsingular
pencils of metrics of constant Riemannian curvature (2.2)–(2.5). In the next section, we
present the Lax pair with a spectral parameter for system (2.2)–(2.5), what also gives the
possibility to integrate the system.
Note that equations (3.16),(3.18) canbe easily integratedexplicitly in the special func-
tions. If the functions fi(ui) and fj(uj) are not constants, then, as is provedin [1], for the
description of all the corresponding pencils of metrics, it is sufficient to consider the case
fi(ui) = ui, fj(uj) = uj, since in this case the functions fi(ui) and fj(uj) can be chosen
as the corresponding new local coordinates. Accordingly equations (3.16) belong to one of
the following three types:
∂2F 1 ∂F 1 ∂F
= , (3.19)
∂u1∂u2 2(u1 u2)∂u1 − 2(u1 u2)∂u2
− −
∂2F 1 ∂F
= , (3.20)
∂u1∂u2 −2u2∂u1
∂2F
=0. (3.21)
∂u1∂u2
The general solutions of equations (3.20) and (3.21) are the functions
F(u1,u2)=g(u1)/√u2+h(u2) and F(u1,u2)=g(u1)+h(u2)
respectively, where g(u1) and h(u2) are arbitrary functions.
Afterthechangeofvariablesu1 =t+r,u2 =t r,equation(3.19)becomestheequation
−
∂2F ∂2F 1∂F
= + , (3.22)
∂t2 ∂r2 r ∂r
9
which is well known in the classical geometry (one more well-known Darboux equation)
and in the classical mathematical physics (the equation of axially symmetric oscillations of
gas (r is the radial coordinate) or the equation for solutions of the wave equation which
correspond to processes of radiation on the plane (x,y), that is, solutions depending only
on the radial coordinate r = x2+y2 and the time t).
In particular, the functions
p
+1 ψ(t+xr)
v(t,r)=const dx, (3.23)
Z−1 √1−x2
where ψ(t) is an arbitrary function, satisfy to the Darboux equation (3.22) with the initial
conditions
∂F
F(t,0)=ψ(t), (t,0)=0. (3.24)
∂r
Forthe function ψ(t,z)=ψ(t), givenonthe plane (t,z),the functionv(t,r) givesthe mean
value over the circle of radius r with the centre at the point (t,z).
Thegeneralsolutionsofequations(3.22)canbeconstructedbythemethodofseparation
of variables: F(t,r)=S(t)R(r),
S′′ R′′ 1R′
= + , (3.25)
S R r R
S′′ =aS, rR′′+R′ arR=0, a=const. (3.26)
−
It is easy to write the general solutions in terms of the special Bessel and modified Bessel
functions or degenerate hypergeometric functions.
4 Lax pair for nonsingular pencils of metrics
of constant Riemannian curvature
The Lax pair for nonsingular flat pencils of metrics (3.1)–(3.3) was demonstrated by Fer-
apontov in [4]:
∂ϕ (λ+fi)
i
= β ϕ , i=j, (4.1)
∂uj s(λ+fj) ij j 6
∂ϕ (λ+fk)
i
= β ϕ , (4.2)
∂ui − s(λ+fi) ki k
k6=i
X
where λ is the spectral parameter.
NotethatthelinearproblemfortheLam´eequations(3.1),(3.2)waswellknownDarboux
yet [10], see also, for example, [8], [9]:
∂ϕ
i
=β ϕ , i=j, (4.3)
∂uj ij j 6
∂ϕ
i
= β ϕ . (4.4)
∂ui − ki k
k6=i
X
10