Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 46 (2005), No. 2, 513-521.
Biharmonic Curves in the Generalized
Heisenberg Group
Dorel Fetcu
∗
Department of Mathematics, Technical University
Iaşi, România
e-mail: dfetcu@math.tuiasi.ro
Abstract. We find the conditions under which a curve in the generalized Heisenberg group is biharmonic and non-harmonic. We give some existence and nonexistence examples of such curves.
MSC 2000: 53C43, 53C42
Keywords: Biharmonic curves, Heisenberg group
1. Introduction
First we should recall some notions and results related to the harmonic and the biharmonic
maps between Riemannian manifolds, as they are presented in [2], [9] and in [5].
Let f : M → N be a smooth map between two Riemannian manifolds (M, g) and (N, h).
Let f −1 (T N ) be the induced bundle over M of the tangent bundle, T N , defined as follows.
Denote by π : T N → N the projection. Then
[
Tf (x) N.
f −1 (T N ) = {(x, u) ∈ M × T N, π(u) = f (x), x ∈ M } =
x∈M
The set of all C ∞ -sections of f −1 (T N ), denoted by Γ(f −1 (T N )), is Γ(f −1 (T N )) = {V : M →
T N, C ∞ -map, V (x) ∈ Tf (x) N, x ∈ M }. Denote by ∇M , ∇N , the Levi-Civita connections on
(M, g) and (N, h) respectively. For a smooth map f between (M, g) and (N, h), we define
the induced connection ∇ on the induced bundle f −1 (T N ) as follows. For X ∈ χ(M ), V ∈
Γ(f −1 (T N )), define ∇X V ∈ Γ(f −1 (T N )) by ∇X V = ∇N
f∗ X V .
∗
The author wants to thank Professor Vasile Oproiu for his many valuable advices
c 2005 Heldermann Verlag
0138-4821/93 $ 2.50 514
D. Fetcu: Biharmonic Curves in the Generalized Heisenberg Group
The differential of the smooth map f can be viewed as a section of the bundle Λ1 (f −1 (T N )) =
T ∗ M ⊗ f −1 (T N ), and we denote by |df | its norm at a point x ∈ M .
Suppose that M is a compactRmanifold. Define the energy density of f by e(f ) = 21 |df |2 ,
and the energy of f by E(f ) = M e(f ) ∗1, where ∗1 is the volume form on M . The map
f is a harmonic map if it is a critical point of the energy, E(f ). In [9] it is proved that a
map f : M → N is a harmonic map if and only if it satisfies the Euler-Lagrange equation
τ (f ) = 0, where τ (f ) = trace ∇df is an element of Γ(f −1 (T N )) called the tension field
of f . The Laplacian acting on Γ(f −1 (T N )), induced by the connection ∇, is given by the
Weitzenböck formula
∆V = − trace ∇2 V,
for some V ∈ Γ(f −1 (T N )).
R
The bienergy of f is defined by E2 (f ) = 12 M |τ (f )|2 ∗ 1. We say that f is a biharmonic
map if it is a critical point of the bienergy, E2 (f ). It is proved in [5] that a map f : M → N
is a biharmonic map if and only if it satisfies the equation τ2 (f ) = 0, where
τ2 (f ) = −∆τ (f ) − trace RN (df (·), τ (f ))df (·),
(1.1)
where RN denotes the curvature tensor field on (N, h).
Note that any harmonic map is a biharmonic map and, moreover, an absolute minimum
of the bienergy functional.
2. Generalized Heisenberg group
Consider R2n+1 with the elements of the form X = (x1 , y1 , x2 , y2 , . . . , xn , yn , z). Define the
product on R2n+1 by
n
e = (x1 + x
XX
e1 , y1 + ye1 , . . . , xn + x
en , yn + yen , z + ze +
1X
(e
xi yi − yei xi )),
2 i=1
e = (e
where X = (x1 , y1 , . . . , xn , yn , z), X
x1 , ye1 , . . . , x
en , yen , ze).
2n+1
Let H2n+1 = (R
, g) be the generalized Heisenberg group endowed with the Riemannian metric g which is defined by
g=
n
X
n
h
i2
1X
(dx2i + dyi2 ) + dz +
(yi dxi − xi dyi ) .
2 i=1
i=1
(2.1)
Note that the metric g is left invariant.
We can define a global orthonormal frame field in H2n+1 by
E2i−1 =
∂
yi ∂
∂
xi ∂
∂
−
, E2i =
+
, E2n+1 =
,
∂xi
2 ∂z
∂yi
2 ∂z
∂z
for i = 1, . . . , n. The Levi-Civita connection of the metric g is given by, (see [7] for the
D. Fetcu: Biharmonic Curves in the Generalized Heisenberg Group
3-dimensional case),

∇E2i−1 E2j−1 = 0,




 ∇E2i E2j = 0,
∇E2n+1 E2i−1 = − 12 E2i ,


∇E2n+1 E2i = 12 E2i−1 ,



∇E2n+1 E2n+1 = 0,
∇E2i−1 E2j = 12 δij E2n+1 ,
∇E2i E2j−1 = − 21 δij E2n+1 ,
∇E2i−1 E2n+1 = − 12 E2i ,
∇E2i E2n+1 = 12 E2i−1 ,
515
(2.2)
for i, j = 1, . . . , n. We have too

[E2i , E2j ] = 0,
 [E2i−1 , E2j−1 ] = 0,
[E2i−1 , E2n+1 ] = 0,
[E2i , E2n+1 ] = 0,

[E2i−1 , E2j ] = δij E2n+1 .
The curvature tensor field of ∇ is
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,
and Riemann-Christoffel tensor field is
R(X, Y, Z, W ) = g(R(X, Y )W, Z),
where X, Y, Z, W ∈ χ(R2n+1 ). We will use the notations
Rabc = R(Ea , Eb )Ec , Rabcd = R(Ea , Eb , Ec , Ed ),
where a, b, c, d = 1, . . . , 2n + 1. Then the non-zero components of the curvature tensor field
and of the Riemann-Christoffel tensor field are, respectively

R(2i−1)(2j−1)(2k) = − 41 δjk E2i + 14 δik E2j ,





R(2i−1)(2j)(2k−1) = 14 δjk E2i + 12 δij E2k ,



1
1


 R(2i−1)(2j)(2k) = − 4 δik E2j−1 − 2 δij E2k−1 ,

1
 R
(2i−1)(2n+1)(2j−1) = − 4 δij E2n+1 ,
(2.3)
1

 R(2i−1)(2n+1)(2n+1) = 4 δij E2i−1 ,



R(2i)(2j)(2k−1) = − 14 δjk E2i−1 + 14 δik E2j−1 ,




R(2i)(2n+1)(2j) = − 14 δij E2n+1 ,



 R
1
(2i)(2n+1)(2n+1) = 4 δij E2i ,

R(2i−1)(2j−1)(2i)(2k) = − 12 δjk + 14 δik δij ,




R(2i−1)(2j)(2i)(2k−1) = 41 δjk + 12 δik δij ,



 R
1
1
(2i−1)(2j)(2k)(2k−1) = 2 δij + 4 δik δjk ,

R(2i)(2j−1)(2j−1)(2k) = 14 δik − 14 δjk δij ,




R(2i−1)(2n+1)(2n+1)(2j−1) = − 14 δij ,



R(2i)(2n+1)(2n+1)(2j) = − 14 δij ,
for i, j, k = 1, . . . , n.
(2.4)
516
D. Fetcu: Biharmonic Curves in the Generalized Heisenberg Group
3. Biharmonic curves in H2n+1
Let γ : I → H2n+1 be a non-inflexionar curve, parametrized by its arc length. Let {T, N1 , . . . ,
N2n } be the Frenet frame in H2n+1 defined along γ, where T = γ 0 is the unit tangent vector
field of γ, N1 is the unit normal vector field of γ, with the same direction as ∇T T and the
vectors N1 , . . . , N2n are the unit vectors obtained from the following Frenet equations for γ.

∇T T
= χ 1 N1 ,




= −χ1 T + χ2 N2 ,
 ∇ T N1
...
... ...
(3.1)


∇T N2n−1 = −χ2n−2 N2n−2 + χ2n−1 N2n ,



∇T N2n
= −χ2n−1 N2n−1 ,
where χ1 = k∇T T k = kτ (γ)k, and χ2 = χ2 (s), . . . , χ2n = χ2n (s) are real valued functions,
where s is the arc length of γ. If χk ∈ R, k = 1, . . . , 2n + 1 we say that γ is a helix.
The biharmonic equation of γ is
τ2 (γ) = ∇3T T − R(T, ∇T T )T = 0.
(3.2)
Using the Frenet equations one obtains
∇3T T = (−3χ1 χ01 )T + (χ001 − χ31 − χ1 χ22 )N1 + (2χ01 χ2 + χ1 χ02 )N2 + χ1 χ2 χ3 N3 .
(3.3)
Using (2.3) we get
R(T, ∇T T )T =
n
X
(ξ2i−1 E2i−1 + ξ2i E2i ) + ξ2n+1 E2n+1 ,
i=1
with
n
ξ2i−1
1
1 2
3 X
(−T2j−1 N12j + T2j N12j−1 ) + T2i−1 T2n+1 N12n+1 − T2n+1
N12i−1 ,
= T2i
4
4
4
j=1
n
X
3
1
1 2
ξ2i = T2i−1
(T2j−1 N12j − T2j N12j−1 ) + T2i T2n+1 N12n+1 − T2n+1
N12i ,
4
4
4
j=1
n
1X
2
2
ξ2n+1 =
(−T2j−1
N12n+1 − T2j
N12n+1 + T2j−1 T2n+1 N12j−1 + T2j T2n+1 N12j ),
4 j=1
P
P2n+1 a
where T = 2n+1
a=1 Ta Ea and N1 =
a=1 N1 Ea . After a straightforward computation, we
have
2n
X
R(T, ∇T T )T =
η k Nk ,
(3.4)
k=1
with
η1 =
n
i2 1
3h X
1
2
(T2i N12i−1 − T2i−1 N12i ) − T2n+1
− (N12n+1 )2 ,
4 i=1
4
4
(3.5)
D. Fetcu: Biharmonic Curves in the Generalized Heisenberg Group
n
n
ih X
i
3h X
1
(T2i N12i−1 − T2i−1 N12i )
(T2i Nk2i−1 − T2i−1 Nk2i ) − − N12n+1 Nk2n+1 ,
4 i=1
4
i=1
P2n+1 a
where Nk = a=1
Nk Ea .
From (3.2), (3.3) and (3.4) it follows that the biharmonic equation of γ is
ηk =
517
(3.6)
τ2 (γ) = ∇3T T − R(T, ∇T T )T = (−3χ1 χ01 )T + (χ001 − χ31 − χ1 χ22 − χ1 η1 )N1 +
(2χ01 χ2 + χ1 χ02 − χ1 η2 )N2 + (χ1 χ2 χ3 − χ1 η3 )N3 − χ1
2n
X
η k Nk ,
i=4
where ηa , a = 1, . . . , 2n are given by (3.5) and (3.6). Hence
Theorem 3.1. Let γ : I → H2n+1 be a curve, parametrized by its arc length. Then γ is a
biharmonic and non-harmonic curve if and only if

χ1 ∈ R \ {0},




 χ21 + χ22 = −η1 ,
χ02 = η2 ,
(3.7)


χ 2 χ 3 = η3 ,



ηk = 0, k = 4, . . . , 2n,
where ηk , k = 1, . . . , 2n, are given by (3.5) and (3.6).
Corollary 3.2. If χ1 ∈ R \ {0} and χ2 = 0 for a curve γ : I → H2n+1 , parametrized by
its arc length, then γ is a biharmonic and non-harmonic curve if and only if χ21 = −η1 and
ηk = 0, k = 2, . . . , 2n.
Corollary 3.3. Let γ : I → H2n+1 be a curve, parametrized by its arc length. If η1 ≥ 0 then
γ cannot be a biharmonic and non-harmonic curve.
In [7] the following two results for the usual Heisenberg group, H3 , are proved.
Theorem 3.4. Let γ be the helix given by
γ(s) = (r cos(as), r sin(as), c a s),
= r2 (1 + 14 r2 ). Then γ is a biharmonic and non-geodesic curve.
q √
Remark 3.5. In the case above if r = 1+2 5 , then γ is a biharmonic and non-harmonic
curve with χ2 = 0.
where r > 0,
1
a2
In the case of the higher dimensions, we find a similar example related to Theorem 3.1. We
consider a curve in R2n+1 , given by
γ(s) = (c1 cos(a1 s), c1 sin(a1 s), . . . , cn cos(an s), cn sin(an s), cs),
518
D. Fetcu: Biharmonic Curves in the Generalized Heisenberg Group
where ci > 0, ai 6= 0, c 6= 0, i = 1, . . . , n. Then one obtains
0
T (s) = γ (s) =
n
X
[−ci ai sin(ai s)E2i−1 + ci ai cos(ai s)E2i ] + AE2n+1 ,
i=1
P
P
where A = c − 21 ni=1 c2i ai . From kT (s)k = 1 we have A2 + ni=1 c2i a2i = 1. After a
straightforward computation, using (2.2), one obtains
∇T T =
n
X
[ci ai (A − ai ) cos(ai s)E2i−1 + ci ai (A − ai ) sin(ai s)E2i ].
i=1
From the first equation in (3.1) and from kN1 k = 1, we have
n
i1/2
hX
∈ R,
χ1 =
c2i a2i (A − ai )2
i=1
and
N1 =
n
X
i=1
ci ai |A − ai |
hP
n
2 2
j=1 cj aj (A
− aj
)2
i1/2 [cos(ai s)E2i−1 + sin(ai s)E2i ].
Note that N12n+1 = 0. Next, one obtains
n
o
1 Xn
ci ai [|A − ai |(A − 2ai ) − 2χ21 ] [sin(ai s)E2i−1 −
∇ T N1 + χ 1 T =
2χ1 i=1
n
i
X
1 h 2
2 2
cos(ai s)E2i ] +
2χ1 A −
cj aj |A − ai | E2n+1 .
2χ1
j=1
From (3.1), using kN2 k = 1 we have |χ2 |2 = kχ2 N2 k2 .
In order to find a curve which satisfies conditions of Corollary 3.2 we assume that χ2 = 0
and χ21 = −η1 . From this conditions, after a straightforward computation we get
Proposition 3.6. Let γ : I → H2n+1 be the curve defined by
γ(s) = (c1 cos(a1 s), c1 sin(a1 s), . . . , cn cos(an s), cn sin(an s), cs),
P
2 (1−A2 )
1
where ci > 0, ai 6= 0, c 6= 0. If ai = a = A − 2A
, ni=1 c2i = 4A
and c =
(2A2 −1)2
A3
,
2A2 −1
where
h 3n2 − 1 + (9n4 + 6n2 + 5)1/2 i1/2
A=±
,
6n2 + 2
then γ is a biharmonic and non-harmonic curve in H2n+1 .
Note that, for such a curve and for k 6= 1, one obtains
n
X
i=1
n
(T2i Nk2i−1
−
T2i−1 Nk2i )
χ1 X 2i−1 2i−1
χ1
=
(N1 Nk + N12i Nk2i ) = −
N12n+1 Nk2n+1 = 0.
|A − a| i=1
|A − a|
That is ηk = 0, for any k = 2, . . . , 2n.
Also, note that the Proposition 3.6 is a generalization of the result in the Remark 3.5.
Next, one obtains
519
D. Fetcu: Biharmonic Curves in the Generalized Heisenberg Group
Proposition 3.7. Let γ : I → H2n+1 be the curve defined by
γ(s) = (r cos(as), r sin(as), . . . , r cos(as), r sin(as), cs),
where r > 0, ai 6= 0, c 6= 0. If χ2 = 0, χ1 6= 0 and γ is biharmonic then n = 1.
In the following we obtain a class of biharmonic and non-harmonic curves for which the
second curvature does not necessarily vanish, (see[1] for the similar result in 3-dimensional
case).
Proposition 3.8. Let γ : I → H2n+1 , γ(s) = (x1 (s), y1 (s), . . . , xn (s), yn (s), z(s)), be the
curve with the parametric equations

√ α sin(βs + ai ) + bi ,
xi (s) = β1 sin


n


 yi (s) = − 1 sin
√ α cos(βs + ai ) + ci ,
β n
P
(3.8)
(sin α)2

z(s) = cos α + 2β
s − ni=1 2βb√i n sin α cos(βs + ai )


P


− ni=1 2βc√i n sin α cos(βs + ai ) + d,
√
cosα±
5(cos α)2 −4
√
√
with i = 1, . . . , n, where β =
, α ∈ (0, arccos 2 5 5 ] ∪ [arccos(− 2 5 5 ), π) and
2
ai , bi , ci , d ∈ R. Then γ is a biharmonic and non-harmonic curve.
Proof. The covariant derivative of the unit tangent vector field, T , of γ, is
∇T T =
n
X
0
0
E2n+1 ,
[(T2i−1
+ T2i T2n+1 )E2i−1 + (T2i0 − T2i−1 T2n+1 )E2i ] + T2n+1
i=1
and T is given by
n
sin α X
T (s) = γ 0 (s) = √
[cos(βs + ai )E2i−1 + sin(βs + ai )E2i ] + cos αE2n+1 .
n i=1
Taking into account the first Frenet equation one obtains
χ1 = | sin α(cos α − β)|
and, since we can assume, without loss of generality, that sin α(cos α − β) > 0, we have
N1 =
n X
sin(βs + ai )
cos(βs + ai ) √
√
E2i−1 −
E2i .
n
n
i=1
After a straightforward computation one obtains that η1 = (sin α)2 −
k = 2, . . . , 2n.
In order to find χ2 we obtain, using the equations (2.2),
∇ T N1 + χ 1 T =
1
4
and ηk = 0, for any
n n
i
X
1
cos(βs + ai ) h
√
β − cos α + (cos α − β)(sin α)2 E2i−1
2
n
i=1
520
D. Fetcu: Biharmonic Curves in the Generalized Heisenberg Group
i o
h1
i
sin(βs + ai ) h
1
2
√
+
β − cos α + (cos α − β)(sin α) E2i − sin α − (cos α − β) cos α E2n+1 .
2
2
n
Then, using Frenet equations, we have
χ22 = k∇T N1 + χ1 T k2 = β 2 − β cos α +
1
− (sin α)2 (cos α − β)2 .
4
Hence χ2 is a constant and, from hypothesis, one obtains
χ21 + χ22 =
1
− (sin α)2 = −η1 .
4
Since χ2 N2 = ∇T N1 + χ1 T and χ23 = k∇T N2 + χ2 N1 k2 , we obtain, after a straightforward
computation, that χ3 = 0.
Hence, all conditions from Theorem 3.1 are verified by γ and then γ is a biharmonic and
non-harmonic curve.
Remark 3.9. In the same way as above it is easy to see that all biharmonic and nonharmonic curves in H2n+1 with constant second curvature and with the unit tangent vector
field, T , of the form
n
sin α X
T (s) = √
[cos(fi (s))E2i−1 + sin(fi (s))E2i ] + cos αE2n+1 ,
n i=1
where fi are some smooth functions of the arc length, such that fi0 = fj0 , for any i, j = 1, . . . , n,
and α ∈ R, are given by Proposition 3.8.
Finally, we have
Proposition 3.10. Let γ : I → H2n+1 be the curve defined by
γ(s) = (c1 s, c2 s, . . . , c2n+1 s)
with
P2n+1
j=1
c2j = 1. Then γ is biharmonic if and only if is harmonic.
Proof. We have
0
T (s) = γ (s) =
2n+1
X
cj Ej , kT k = 1, ∇T T = c2n+1
j=1
n
X
(c2i E2i−1 − c2i−1 E2i ).
i=1
pPn
2
2
It follows that χ1 = c2n+1
i=1 (c2i−1 + c2i ), and
n X
c2i−1
c2i
qP
N1 =
E2i−1 − qP
E2i .
n
n
2
2
2
2
(c
+
c
)
(c
+
c
)
i=1
2j−1
2j
2j−1
2j
j=1
j=1
One obtains
Pn
∇ T N1 + χ 1 T =
2
k=1 (c2k−1
X
n + c22k ) − c22n+1
c
c
qP 2i−1 2n+1
·
E2i−1 +
n
2
2
2
(c
+
c
)
i=1
2j
j=1 2j−1
D. Fetcu: Biharmonic Curves in the Generalized Heisenberg Group
c2i c2n+1
qP
n
2
j=1 (c2j−1
+ c22j )
That means
E2i
n
P
χ2 =
521
v
uX
u n 2
− t (c2j−1 + c22j )E2n+1 .
j=1
(c22k−1 + c22k ) − c22n+1
k=1
2
.
Thus χ21 + χ22 = 41 . But, one obtains that η1 = 34 − c22n+1 , and from (3.7) we have that if γ
is biharmonic then χ21 + χ22 = −η1 = − 34 + c22n+1 . Thus if γ is biharmonic then χ1 = 0, and
then γ is a harmonic curve.
References
[1] Caddeo, R.; Oniciuc, C.; Piu, P.: Explicit formulas for non-geodesic biharmonic curves
of the Heisenberg group. E-version, arXiv:math.DG 0311221 v1, 2003, 16 pp.
[2] Eells, J.; Lemaire, L.; Selected topics in harmonic maps. Reg. Conf. Ser.Math. 50 (1983),
85 p.
Zbl
0515.58011
−−−−
−−−−−−−−
[3] Figueroa, C. B.; Mercuri, F.; Pedrosa, R. H. L.: Invariant surfaces of the Heisenberg
Groups. Ann. Mat. Pura Appl. IV. Ser., 177 (1999), 173–194.
Zbl
0965.53042
−−−−
−−−−−−−−
[4] Gheorghiev, Gh.; Oproiu, V.: Varietăţi diferenţiabile finit şi infinit dimensionale. (Roumanian), vol.II, Ed. Academiei Române (1979), 294 pp.
Zbl
0484.53001
−−−−
−−−−−−−−
[5] Jiang, G. Y.: 2-harmonic maps and their first and second variational formulas. Chin.
Ann. Math. Ser. A, 7 (1986), 389–402.
Zbl
0628.58008
−−−−
−−−−−−−−
[6] Martin, M.: Introducere în teoria curbelor şi suprafeţelor în spaţiul euclidian n-dimensional. (Roumanian), Universitatea Bucureşti 1979.
[7] Oniciuc, C.: Tangency and harmonicity properties. (Roumanian), Geometry Balkan
Press, E-version, 2003, 87 pp.
[8] Oproiu, V.; Papaghiuc, N.: Some results on harmonic sections of cotangent bundles.
Ann. Stiint. Univ. Al.I.Cuza Iasi, 45 (1999), 275–290.
Zbl
1011.53018
−−−−
−−−−−−−−
[9] Urakawa, H.: Calculus of Variation and Harmonic Maps. Am. Math. Soc., Providence,
Rhode Island, Translations of Mathematical Monographs, 1993, xiii, 251 p.
Zbl
0799.58001
−−−−
−−−−−−−−
Received March 23, 2004