Hyperplanar Webs and Euler Equations
We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x1, . . . , xn) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d...
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| Cite this: | Hyperplanar Webs and Euler Equations / V.V. Goldberg, V.V. Lychagin // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 276-287. — Бібліогр.: 1 назв. — англ. |
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Goldberg, V.V. Lychagin, V.V. 2010-02-23T14:29:39Z 2010-02-23T14:29:39Z 2009 Hyperplanar Webs and Euler Equations / V.V. Goldberg, V.V. Lychagin // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 276-287. — Бібліогр.: 1 назв. — англ. 1815-2910 https://nasplib.isofts.kiev.ua/handle/123456789/6308 We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x1, . . . , xn) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d ≥ n + 1) of hypersurfaces to be hyperplanar webs. These conditions are systems of generalized Euler equations, and for flat connections we give an explicit construction of their solutions. en Інститут математики НАН України Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" Hyperplanar Webs and Euler Equations Article published earlier |
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Hyperplanar Webs and Euler Equations |
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Hyperplanar Webs and Euler Equations Goldberg, V.V. Lychagin, V.V. Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" |
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Hyperplanar Webs and Euler Equations |
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Hyperplanar Webs and Euler Equations |
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Hyperplanar Webs and Euler Equations |
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Hyperplanar Webs and Euler Equations |
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hyperplanar webs and euler equations |
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Goldberg, V.V. Lychagin, V.V. |
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Goldberg, V.V. Lychagin, V.V. |
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Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" |
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Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" |
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2009 |
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English |
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Інститут математики НАН України |
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We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x1, . . . , xn) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d ≥ n + 1) of hypersurfaces to be hyperplanar webs. These conditions are systems of generalized Euler equations, and for flat connections we give an explicit construction of their solutions.
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1815-2910 |
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https://nasplib.isofts.kiev.ua/handle/123456789/6308 |
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Hyperplanar Webs and Euler Equations / V.V. Goldberg, V.V. Lychagin // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 276-287. — Бібліогр.: 1 назв. — англ. |
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AT goldbergvv hyperplanarwebsandeulerequations AT lychaginvv hyperplanarwebsandeulerequations |
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2025-11-24T21:03:24Z |
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1850494504413429760 |
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Çáiðíèê ïðàöüIí-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè2009, ò.6, �2, 276-287
V.V.Goldberg
New Jersey Institute of Technology, Newark, NJ, USA
E-mail: vladislav.goldberg@gmail.com
V.V. Lychagin
University of Tromso, Tromso, Norway
E-mail: lychagin@math.uit.no
Geodesic Webs and PDE Systems of
Euler Equations
We find necessary and sufficient conditions for the foliation defined by level
sets of a function f(x1, . . . , xn) to be totally geodesic in a torsion-free con-
nection and apply them to find the conditions for d-webs of hypersurfaces
to be geodesic, and in the case of flat connections, for d-webs (d ≥ n + 1)
of hypersurfaces to be hyperplanar webs. These conditions are systems of
generalized Euler equations, and for flat connections we give an explicit
construction of their solutions.
Keywords: web, hyperplanar web, Euler equation, foliation, connection
1. Introduction
In this paper we study necessary and sufficient conditions for
the foliation defined by level sets of a function to be totally geo-
desic in a torsion-free connection on a manifold and find necessary
and sufficient conditions for webs of hypersurfaces to be geodesic.
These conditions has the form of a second-order PDE system for
web functions. The system has an infinite pseudogroup of sym-
metries and the factorization of the system with respect to the
pseudogroup leads us to a first-order PDE system. In the pla-
nar case (cf. [1]), the system coincides with the classical Euler
equation and therefore can be solved in a constructive way. We
provide a method to solve the system in arbitrary dimension and
flat connection.
c© V.V. Goldberg, V. V. Lychagin, 2009
Hyperplanar Webs and Euler Equations 277
2. Geodesic Foliations and Flex Equations
Let Mn be a smooth manifold of dimension n. Let vector fields
∂1, . . . , ∂n form a basis in the tangent bundle, and let ω1, . . . , ωn
be the dual basis. Then
[∂i, ∂j ] =
∑
k
ckij∂k
for some functions ckij ∈ C∞ (M) , and
dωk +
∑
i<j
ckijω
i ∧ ωj = 0.
Let ∇ be a linear connection in the tangent bundle, and let Γkij be
the Christoffel symbols of second type. Then
∇i (∂j) =
∑
k
Γkij∂k,
where ∇i
def
= ∇∂i
, and
∇i
(
ωk
)
= −
∑
j
Γkijω
j.
In [1] we proved the following result.
Theorem 1. The foliation defined by the level sets of a function
f(x1, . . . , xn) is totally geodesic in a torsion-free connection ∇ if
and only if the function f satisfies the following system of PDEs:
∂i (fi)
fifi
− ∂i (fj) + ∂j (fi)
fifj
+
∂j (fj)
fjfj
=
=
∑
k
(
Γkii
fk
fifi
+ Γkjj
fk
fjfj
− (Γkij + Γkji)
fk
fifj
)
(1)
for all i < j, i, j = 1, . . . , n; here fi =
∂f
∂xi
.
278 V. V. Goldberg, V. V. Lychagin
We call such a system a flex system.
Note that conditions (1) can be used to obtain necessary and
sufficient conditions for a d-web formed by the level sets of the
functions fα(x1, . . . , xn), α = 1, . . . , d, to be a geodesic d-web,
i.e., to have the leaves of all its foliations to be totally geodesic:
one should apply conditions (1) to the all web functions fα, α =
1, . . . , d
2.1. Geodesic Webs on Manifolds of Constant Curvature.
In what follows, we shall use the following definition.
Definition 1. We call by (Flex f)ij the following function:
(Flex f)ij = f2
j fii − 2fifjfij + f2
i fjj,
where i, j = 1, . . . , n, fi =
∂f
∂xi
and fij =
∂2f
∂xi∂xj
.
It is easy to see that (Flex f)ij = (Flex f)ji, and (Flex f)ii = 0.
Proposition 1. Let (Rn, g) be a manifold of constant curvature
with the metric tensor
g =
dx2
1 + · · · + dx2
n(
1 + κ
(
x2
1 + · · · + x2
n
))2 ,
where κ is a constant. Then the level sets of a function
f(x1, . . . , xn)
are geodesics of the metric g if and only if the function f satisfies
the following PDE system:
(2) (Flex f)ij =
2κ
(
f2
i + f2
j
)
1 + κ
(
x2
1 + · · · + x2
n
)
∑
k
xkfk
for all i, j.
Proof. To prove formula (2), first note that the components of the
metric tensor g are
gii = b2, gij = 0, i 6= j,
Hyperplanar Webs and Euler Equations 279
where
b =
1
1 + κ
(
x2
1 + · · · + x2
n
) .
It follows that
gii = g−1
ii , g
ij = 0, i 6= j.
We compute Γijk using the classical formula
(3) Γkij =
1
2
gkl
(
∂gli
∂xj
+
∂glj
∂xi
− ∂gij
∂xl
)
and get
Γkii = 2κxkb, k 6= i; Γiii = −2κxib; Γkij = 0, i, j 6= k, i 6= j;
Γiij = −2κxjb, i 6= j; Γjij = −2κxib, i 6= j.
Substituting these values of Γijk into the right-hand side of for-
mula (1), we get formula (2). �
Note that if n = 2, then PDE system (2) reduces to the single
equation
Flex f =
2κ (x1f1 + x2f2)
(
f2
1 + f2
2
)
1 + κ
(
x2
1 + x2
2
) ,
where Flex f = (Flex f)12.
This formula coincides with the corresponding formula in [1].
We rewrite formula (2) as follows:
(4)
(Flex f)ij
f2
i + f2
j
= 2κb
∑
k
xkfk.
The left-hand side of equation (4) does not depend on i and j.
Thus we have
(Flex f)ij
f2
i + f2
j
=
(Flex f)kl
f2
k + f2
l
for any i, j, k, and l.
280 V. V. Goldberg, V. V. Lychagin
It follows that if
(5) (Flex f)ij = 0
for some fixed i and j, then (5) holds for any i and j.
In other words, one has the following result.
Theorem 2. Let W be a geodesic d-web on the manifold (Rn, g)
given by web-functions
{
f1, . . . , fd
}
such that (fak )2+(fal )2 6= 0 for
all a = 1, . . . , d and k, l = 1, 2, . . . , n. Assume that the intersection
of W with the plane (xi0 , xj0) , for given i0 and j0, is a linear
planar d-web. Then the intersection of W with arbitrary planes
(xi, xj) are linear webs too.
2.2. Geodesic Webs on Hypersurfaces in Rn.
Proposition 2. Let (M,g) ⊂ Rn be a hypersurface defined by
an equation xn = u (x1, . . . , xn−1) with the induced metric g and
the Levi-Civita connection ∇. Then the foliation defined by the
level sets of a function f (x1, . . . , xn−1) is totally geodesic in the
connection ∇ if and only if the function f satisfies the following
system of PDEs:
(6) (Flex f)ij =
u1f1 + · · · + un−1fn−1
1 + u2
1 + · · · + u2
n−1
(f2
j uii−2fifjuij+f
2
i ujj).
Proof. To prove formula (6), note that the metric induced by a
surface xn = u(x1, . . . , xn−1) is
g = ds2 =
n−1∑
k=1
(1 + u2
k)dx
2
k + 2
n−1∑
i,j=1(i6=j)
uiujdxidxj .
Thus the metric tensor g has the following matrix:
(gij) =
1 + u2
1 u1u2 . . . u1un−1
u2u1 1 + u2
2 . . . u2un−1
...
...
. . .
...
u1 un−1u2 . . . 1 + u2
n−1
,
Hyperplanar Webs and Euler Equations 281
and the inverse tensor g−1 has the matrix
(gij) =
1
1 +
n−1∑
k=1
(1 + u2
k)
×
×
n−1∑
k=2
(1 + u2
k) −u1u2 . . . −u1un−1
−u2u1
n−1∑
k=1(k 6=2)
(1 + u2
k) . . . −u2un−1
...
...
. . .
...
−un−1u1 −un−1u2 . . .
n−2∑
k=1
(1 + u2
k)
.
Computing Γijk by formula (3), we find that
Γkij =
ukuij
1 +
n−1∑
k=1
(1 + u2
k)
.
Applying these formulas to the right-hand side of (1), we get for-
mula (6). �
We rewrite equation (6) in the form
(7)
(Flex f)ij
f2
j uii − 2fifjuij + f2
i ujj
=
u1f1 + · · · + unfn
1 + u2
1 + · · · + u2
n
.
It follows that the left-hand side of (7) does not depend on i
and j, i.e., we have
(Flex f)ij
f2
j uii − 2fifjuij + f2
i ujj
=
(Flex f)kl
f2
l ukk − 2fkflukl + f2
kull
for any i, j, k and l. This means that if
(Flex f)ij = 0
282 V. V. Goldberg, V. V. Lychagin
for some fixed i and j, then
(Flex f)kl = 0
for any k and l.
In other words, we have a result similar to the result in Theo-
rem 2.
Theorem 3. Let W be a geodesic d-web on the hypersurface
(M,g) given by web functions
{
f1, . . . , fd
}
such that
(
faj
)2
uii − 2fai f
a
j uij + (fai )2 ujj 6= 0,
for all a = 1, . . . , d and k, l = 1, 2, . . . , n. Assume that the intersec-
tion of W with the plane (xi0 , xj0) , for given i0 and j0, is a linear
planar d-web. Then the intersection of W with arbitrary planes
(xi, xj) are linear webs too.
3. Hyperplanar Webs
In this section we consider hyperplanar geodesic webs in Rn
endowed with a flat linear connection ∇.
In what follows, we shall use coordinates x1, . . . , xn in which
the Christoffel symbols Γijk of ∇ vanish.
The following theorem gives us a criterion for a web of hyper-
surfaces to be hyperplanar.
Theorem 4. Suppose that a d-web of hypersurfaces, d ≥ n + 1,
is given locally by web functions fα(x1, . . . , xn), α = 1, ..., d. Then
the web is hyperplanar if and only if the web functions satisfy the
following PDE system:
(8) (Flex f)st = 0,
for all s < t = 1, . . . , n.
Proof. For the proof, one should apply Theorem 1 to all foliations
of the web. �
Hyperplanar Webs and Euler Equations 283
In order to integrate the above PDEs system, we introduce the
functions
As =
fs
fs+1
, s = 1, . . . , n − 1,
and the vector fields
Xs =
∂
∂xs
−As
∂
∂xs+1
, s = 1, . . . , n− 1.
Then the system can be written as
Xs (At) = 0,
where s, t = 1, . . . , n− 1.
Note that
[Xs,Xt] = 0
if the function f is a solution of (8).
Hence, the vector fields X1, . . . ,Xn−1 generate a completely in-
tegrable (n− 1)-dimensional distribution, and the functions
A1, . . . , An−1
are the first integrals of this distribution.
Moreover, the definition of the functions As shows that
Xs(f) = 0, s = 1, . . . , n − 1,
also.
As a result, we get that
As = Φs (f) , s = 1, . . . , n− 1,
for some functions Φs.
In these terms, we get the following system of equations for f :
∂f
∂xs
= Φs (f)
∂f
∂xs+1
, s = 1, . . . , n− 1,
or
(9)
∂f
∂xs
= Ψs (f)
∂f
∂xn
, s = 1, . . . , n− 1,
284 V. V. Goldberg, V. V. Lychagin
where Ψn−1 = Φn−1, and
Ψs = Φn−1 · · ·Φs
for s = 1, . . . , n− 2.
This system is a sequence of the Euler-type equations and there-
fore can be integrated. Keeping in mind that a solution of the
single Euler-type equation
∂f
∂xs
= Ψs (f)
∂f
∂xn
is given by the implicit equation
f = u0 (xn + Ψs (f)xs) ,
where u0(xn) is an initial condition, when xs = 0, and Ψs is an
arbitrary nonvanishing function, we get solutions f of system (8)
in the form:
f = u0 (xn + Ψn−1 (f)xn−1 + · · · + Ψ1 (f)x1) ,
where u0(xn) is an initial condition, when
x1 = · · · = xn−1 = 0,
and Ψs are arbitrary nonvanishing functions.
Thus, we have proved the following result.
Theorem 5. Web functions of hyperplanar webs have the form
(10) f = u0 (xn + Ψn−1 (f)xn−1 + · · · + Ψ1 (f)x1) ,
where u0(xn) are initial conditions, when x1 = · · · = xn−1 =
0, and Ψs are arbitrary nonvanishing functions.
Example 15. Assume that n = 3,
f1(x1, x2, x3) = x1, f2(x1, x2, x3) = x2, f3(x1, x2, x3) = x3,
and take u0 = x3, Ψ1(f4) = f2
4 , Ψ2(f4) = f4 in (10). Then we get
the hyperplanar 4-web with the remaining web function
f4 =
x2 − 1 ±
√
(x2 − 1)2 − 4x1x3
2x1
.
Hyperplanar Webs and Euler Equations 285
It follows that the level surfaces f4 = C of this function are defined
by the equation
x1(C
2x1 − Cx2 + x3 + C) = 0,
i.e., they form a one-parameter family of 2-planes
C2x1 − Cx2 + x3 + C = 0.
Differentiating the last equation with respect to C and excluding C,
we find that the envelope of this family is defined by the equation
(x2)
2 − 4x1x3 − 2x2 + 1 = 0.
Therefore, the envelope is the second-degree cone.
Example 16. Assume that n = 3,
f1(x1, x2, x3) = x1, f2(x1, x2, x3) = x2, f3(x1, x2, x3) = x3,
and take u0 = x3,Ψ1(f4) = 1,Ψ2(f4) = f2
4 in (10). Then we get
the linear 4-web with the remaining web function
f4 =
(
1 ±
√
1 − 4x2(x1 + x3)
2x2
)2
.
The level surfaces f4 = C2 of this function are defined by the
equation
x2(x1 + C2x2 + x3 − C) = 0,
i.e., they form a one-parameter family of 2-planes
x1 + C2x2 + x3 − C = 0.
Differentiating the last equation with respect to C and excluding C,
we find that the envelope of this family is defined by the equation
4x1x2 + 4x2x3 − 1 = 0.
Therefore, the envelope is the hyperbolic cylinder.
In the next example no one foliation of a web W3 coincides
with a foliation of coordinate lines, i.e., all four web functions are
unknown.
286 V. V. Goldberg, V. V. Lychagin
Example 17. Assume that n = 3 and take
Description 1.
(i) u01 = x3, Ψ1(f1) = f2
1 , Ψ2(f1) = f1;
(ii) u02 = x3, Ψ1(f2) = 1, Ψ2(f2) = f2
2 ;
(iii) u03 = x2
3, Ψ1(f3) = f3, Ψ2(f3) = 1;
(iv) u04 = x3, Ψ1(f4) = Ψ2(f4) = f4.
in (10). Then we get the linear 4-web with the web functions
f1 =
x2 − 1 ±
√
(x2 − 1)2 − 4x1x3
2x1
,
f2 =
(
1 ±
√
1 − 4x2(x1 + x3)
2x2
)2
(see Examples 15 and 16) and
f3 =
(
1 ±
√
1 − 4x1(x2 + x3)
2x1
)2
,
f4 =
x3
1 − x1 − x2
.
It follows that the leaves of the foliation X1 are tangent 2-planes
to the second-degree cone
(x2)
2 − 4x1x3 − 2x2 + 1 = 0
(cf. Example 15 and 16), the leaves of the foliation X2 and X3 are
tangent 2-planes to the hyperbolic cylinders
4x1x2 + 4x2x3 − 1 = 0 and 4x1x2 + 4x1x3 − 1 = 0
(cf. Example 16), and the leaves of the foliation X4 are 2-planes
of the one-parameter family of parallel 2-planes
Cx1 +Cx2 + x3 = 1,
where C is an arbitrary constant.
Hyperplanar Webs and Euler Equations 287
References
[1] Goldberg, V. V. and V. V. Lychagin Geodesic webs on a two-dimensional
manifold and Euler equations. Acta Math. Appl.–2009.–103 (to appear).
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