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Thai Journal of Mathematics

Volume 8 (2010) Number 1 : 129–140

www.math.science.cmu.ac.th/thaijournal

Online ISSN 1686-0209

Lorentzian Helicoids in Three

Dimensional Heisenberg Group

E. Turhan and G. Altay

Abstract : In this paper we study the minimal surface in three dimensional

Heisenberg group Heis3. We use Levi-Civita connections and obtain mean curva-

ture of Lorentzian Helicoid. We characterize the Lorentzian Helicoid and obtain

the condition of being minimal surface for Lorentzian Helicoid.

Keywords : Heisenberg group, Lorentzian Helicoid, Minimal surface.

2000 Mathematics Subject Classification :

53C22, 53C30, 53C50, 22Exx.

1 Introduction

The helicoid is generated by spiraling a horizontal straight line along a vertical

axis, and so, it is a ruled surface which is also foliated by helices.

In Euclidean Geometry there are two equivalent approaches from which the

notion of mean curvature of a submanifold arises. One starts with the definition

of the second fundamental form as the orthogonal component of the directional

derivative of a tangent vector field to the submanifold, and the mean curvature

appears as the trace of the second fundamental form. The other one considers

the volume functional defined on the submanifolds of the same dimension and the

mean curvature appears as the gradient of this functional. In this paper, we use

the trace of the second fundamental form when computing the mean curvature

of the surface. It is known that if M ⊂ R3is a minimal surface the Gaussian

curvature K ≤ 0 and defined on the whole plane is either a plane or has an image

under the Gauss map that omits at most two points.

In [1] it is studied that helicoids are axially symmetric minimal surfaces and

it is shown that helicoid is a minimal surface in Heisenberg space which is given

Copyright c ? 2010 by the Mathematical Association of Thailand.

reserved.

All rights

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Thai J. Math. 8(1) (2010)/ E. Turhan and G. Altay

by the metric

ds2= dx2+ dy2+

?λ

2(x2dx − x1dy) + dz

?2

, λ ?= 0.

(1.1)

In [2] it is studied that Gauss map in the Heisenberg group which is endowed

with the metric (2.1) for λ = 1.

In [6] minimal surfaces in the three dimensional Heisenberg group are studied

and the authors obtain Weierstrass representation of Heis3, which is endowed with

the metric in 2.1 for λ = 1.

In [8], it is classified that space-like ruled minimal surfaces in a three dimen-

sional Minkowski space R3

representations for space-like surfaces.

In [11] minimal surfaces and one-parameter subgroups in the three dimensional

Heisenberg groups are studied. The authors obtain a characterization of the one-

parameter subgroups. Then Frenet formulas for one-parameter subgroups of Heis3

are calculated.

In [15], it is given equation of minimal surfaces in three dimensional Minkowski

space R3

Minkowski space R3

Let M be a 2-manifold and Ω : M −→

3-manifold. We denote by g the pull-backed tensor field of ˜ g by Ω :

1. Kobayashi derives two kinds of Weierstrass- Enneper

1. Woestijne obtained the plane, the helicoid, the catenoid are minimal in

1.

?

˜

M3, ˜ g

?

an immersion into a Lorentzian

g = ˜ g (dΩ,dΩ).

Then

i. (M,g) is said to be non-degenerate if g is non-degenerate, i.e., det(g) ?= 0

on M.

ii. (M,g) is said to be a spacelike surface if g is a Riemannian metric, i.e.,

det(g) > 0.

iii. (M,g) is said to be a timelike surface if det(g) < 0.

Let M be a spacelike surface or timelike surface in˜

local unit normal vector field N such that

?

The constant ε is called the sign of M.

Definition 1.1. The second fundamental form h derived from N is defined

by

˜∇XY = ∇XY + h(X,Y )N

where X,Y ∈ χ(M) ,˜∇ and ∇ are Levi-Civita connections of

tively.

M . Then we can take a

˜ g (N,N) = ε, ε =

1

−1

M is spacelike

M is timelike

˜

M and M, respec-

Definition 1.2. A spacelike surface is said to be a maximal surface if H = 0.

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Lorentzian Helicoids in Three Dimensional Heisenberg Group

131

Definition 1.3. A timelike surface is said to be a extremal surface (or minimal

surface) if H = 0.

In [12] and [13] it is shown that three dimensional Heisenberg group has the

following left-invariant Lorentz metrics

g1 = − dx2+ dy2+ (xdy + dz)2

g2 = dx2+ dy2− (xdy + dz)2

g3 = dx2+ (xdy + dz)2− ((1 − x)dy − dz)2.

and some geometric properties of the Heisenberg group Heis3 endowed with a

Lorentz metric are studied.

Let Ω(u) be a curve parametrized by u. Then we have these possibilities;

i. g(Ω?,Ω?) > 0 and g(Ω??,Ω??) > 0. Spacelike curve with spacelike normal.

ii. g(Ω?,Ω?) > 0 and g(Ω??,Ω??) < 0. Spacelike curve with timelike normal.

iii. g(Ω?,Ω?) > 0 and g(Ω??,Ω??) = 0. Spacelike curve with null normal.

iv. g(Ω?,Ω?) < 0 and g(Ω??,Ω??) < 0. Timelike curve.

v. g(Ω?,Ω?) = 0 and g(Ω??,Ω??) > 0. Null curve

Recall that when Ω(u) is a non–null curve in Heis3with spacelike or timelike

rectifying plane, then the Frenet equations are

∇TT

∇TN

∇TB

=

=

=

κN

−?0?1κT + τB

−?1?2τN

where ?0 = g(T,T) = ±1, ?1 = g(N,N) = ±1, ?2 = g(B,B) = ±1 and

?0?1?2= −1,[5]. If Ω(u) is a nullike curve the Frenet equations are

∇TT

∇TN

∇TB

where g(T,T) = 1, g(N,N) = 0, g(B,B) = 0, g(T,N) = 0, g(T,B) = 0, g(N,B) =

1,[6].

=

=

=

κN

−κB + τT

τN

Lemma 1.1.

Heis3is isometric to one of the metrics g1, g2, g3, [12].

Each left invariant Lorentz metric on the Heisenberg group

Proposition 1.1. The left invariant Lorentz metric g3is flat, [12].

In this paper we obtain a characterization of the Lorentzian Helicoid in three

dimensional Heisenberg group which is given by the Lorentz left invariant metrics

g2= dx2+ dy2− (xdy + dz)2.

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Thai J. Math. 8(1) (2010)/ E. Turhan and G. Altay

and

g3= dx2+ (xdy + dz)2− ((1 − x)dy − dz)2

We find that the Lorentzian Helicoid has different properties in (Heis3,g2) and

.(Heis3,g3).

Heisenberg group is a matrix group which is given by:

In Heisenberg group the group multiplication is:

?

The Lie algebra of the Heis3is

The orthonormal bases of the heis3are

000

0

Heis3=

1

0

0

x

1

0

z

y

1

,x,y,z ∈ R

(x1,y1,z1) · (x2,y2,z2) =

x1+ x2,y1+ y2,z1+ z2+1

2x1y2−1

2y1x2

?

.

heis3=

0

0

0

u1

0

0

u3

u2

0

,u1,u2,u3∈ R

.

E1=

0

0

1

0

0

0

,E2=

0

0

0

0

0

0

1

0

,E3=

0

0

0

0

0

0

1

0

0

of the tangent space at the identity.

In this paper we will use the left invariant Lorentz metrics g1and g2. For the

metric

g2= dx2+ dy2− (xdy + dz)2

we have the orthonormal frames

e1=

∂

∂x, e2= x∂

∂z−

∂

∂y, e3=

∂

∂z

For the metric

g3= dx2+ (xdy + dz)2− ((1 − x)dy − dz)2

we have the orthonormal frames

e1=

∂

∂x, e2=

∂

∂y+ (1 − x)∂

∂z, e3=

∂

∂y− x∂

∂z

The element zero 0 = (0,0,0) is the unit of this group structure and the inverse

element for (z, t) is (z, t)−1= (−z, − t).

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Lorentzian Helicoids in Three Dimensional Heisenberg Group

133

Let a = (z,t) and b = (w,s). The commutator of the elements a, b ∈ H3is

equal to

[a,b]=

=

=

aba−1b−1

(z,t)(w,s)(−z,−t)(−w,−s)

(z + w − z + w,t + s − t − s + α) = (0,α)

where α ?= 0 in general. This shows that H3is not abelian. On the other hand for

any a, b, c ∈ H3, their double commutator is:

[[a,b],c] = (0,0)

This implies that H3is a nilpotent Lie group with nilpotency 2.

We know that in E3a Helicoid is a minimal regle surface. In this paper we

want to make a characterization of Helicoid in (Heis3,g).

2Lorentzian Helicoids in Three Dimensional Heisen-

berg Group

2.1 Minimal Surfaces in Lorentzian Heisenberg Group (Heis3,g2)

Lorentzian Helicoid has coordinates

metric

g2= dx2+ dy2− (xdy + dz)2.

The Lie algebra of Heis3has the orthonormal bases

∂

∂x, e2= x∂

Lie brackets of these orthonormal basis are

P (u,v) =

x = coshv cosu

y = coshv sinu

z = u

(2.1)

Let three dimensional Heisenberg group Heis3is given by the left-invariant Lorentz

e1=

∂z−

∂

∂y, e3=

∂

∂z.

(2.2)

[e1,e2] = e3,

[e1,e3] = [e2,e3] = 0.

Then we have the Levi-Civita connections

∇e1e1 = ∇e2e2 = ∇e3e3 = 0,

∇e1e2 = − ∇e2e1 =

∇e1e3 = ∇e3e1 =

∇e2e3 = ∇e3e2 = −1

1

2e3,

1

2e2,

2e1,