Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 523-530.

Tensor Product Surfaces of a Euclidean Space Curve and a Euclidean Plane Curve^{∗}

Kadri Arslan^{1} Ridvan Ezentas^{1} Ion Mihai^{2} Cengizhan Murathan^{1} Cihan Özgür^{1}

1Department of Mathematics, Faculty of Art and Sciences Uludag University, Görükle 16059, Bursa, Turkey 2Faculty of Mathematics, Str. Academiei 14, 70109 Bucharest, Romania

Abstract. B.Y. Chen initiated the study of the tensor product immersion of two immersions of a given Riemannian manifold (see [3]). Inspired by Chen’s definition, F. Decruyenaere, F. Dillen, L. Verstraelen and L. Vrancken (in [4]) studied the tensor product of two immersions of, in general, different manifolds; under certain conditions, this realizes an immersion of the product manifold. In [6] tensor product surfaces of Euclidean plane curves were investigated. In the present paper, we deal with tensor product surfaces of a Euclidean space curve and a Euclidean plane curve. We classify the minimal, totally real and slant such surfaces, respectively. MSC 2000: 53C15, 53C40 Keywords: Tensor product surfaces, minimal surface, totally real surface, slant surface.

1. Tensor product immersions

Recall definitions and results of [3]. Let M and N be two differentiable manifolds and

f : M → 𝔼^{m},

∗This paper is prepared during the third named author’s visit to the Uludag University, Bursa, Turkey in July 2000. The third author is supported by the Scientific and Technical Research Council of Turkey (TÜBITAK) for NATO-PC Advanced Fellowships Programme 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

524 K. Arslan et al.: Tensor Product Surfaces …

h : N → 𝔼^{n},

two immersions. The direct sum and tensor product maps

f ⊕ h: M × N → 𝔼^{m+n},
f ⊗ h: M × N → 𝔼^{mn}

are defined by

(f ⊕ h)(p, q) = (f(p), h(q)),
(f ⊗ h)(p, q) = f(p) ⊗ h(q).

Necessary and sufficient conditions for f ⊗ h to be an immersion were obtained in [4] . It is also proved there that the pairing (⊕, ⊗) determines a structure of a semiring on the set of classes of differentiable manifolds transversally immersed in Euclidean spaces, modulo orthogonal transformations. Some subsemirings were studied in [5] by F. Decruyenaere, F. Dillen, L. Verstraelen and one of the present authors.

For many immersions f, h which are not transversal, the tensor product f ⊗ h is still worthwhile to be investigated and in many cases still produces an immersion. As such, in the following sections, we will consider the tensor product immersions, actually surfaces in 𝔼^{6}, which are obtained from a Euclidean space curve and a Euclidean plane curve.

2. Minimal tensor product surfaces

Let c_{1} : ℝ → 𝔼^{3} and c_{2} : ℝ → 𝔼^{2} be two Euclidean curves. Put c_{1}(t) = (α(t), β(t), γ(t)) and c_{2}(s) = (a(s), b(s)). Then their tensor product is given by

\begin{matrix}f = c_{1} ⊗ c_{2} : ℝ^{2} → 𝔼^{6} \\ f(t, s) = (α(t)a(s), α(t)b(s), β(t)a(s), β(t)b(s), γ(t)a(s), γ(t)b(s)). \end{matrix}

We have

\frac{∂f}{∂t} = (α^{.}(t)a(s), α^{.}(t)b(s), β^{.}(t)a(s), β^{.}(t)b(s), γ^{.}(t)a(s), γ^{.}(t)b(s)),
\frac{∂f}{∂s} = (α(t)a^{.}(s), α(t)b^{.}(s), β(t)a^{.}(s), β(t)b^{.}(s), γ(t)a^{.}(s), γ(t)b^{.}(s)),

where a^{.} means the derivative of a.

The coefficients of the Riemannian metric g induced on Imf by the Euclidean metric of 𝔼^{6} are

g_{11} = ∥ċ_{1}∥^{2}∥c_{2}∥^{2},
g_{12} = <c_{1}, ċ_{1}><c_{2}, ċ_{2}>,
g_{22} = ∥c_{1}∥^{2}∥ċ_{2}∥^{2}.

An orthonormal basis on Im c_{1} ⊗ c_{2} is given by

e_{1} = \frac{1}{∥ċ_{1}∥∥c_{2}∥} \frac{∂f}{∂t} ,

e_{2} = \frac{1}{∥ċ_{1}∥∥c_{2}∥\sqrt{∥ċ_{1}∥^{2}∥c_{1}∥^{2}∥ċ_{2}∥^{2}∥c_{2}∥^{2} − <c_{1}, ċ_{1}>^{2}<c_{2}, ċ_{2}>^{2}}} ×

[∥ċ_{1}∥^{2}∥c_{2}∥^{2} \frac{∂f}{∂s} − <c_{1}, ċ_{1}><c_{2}, ċ_{2}>\frac{∂f}{∂t}].

The normal space is spanned by

n_{1} = (−β(t)b(s), β(t)a(s), α(t)b(s), −α(t)a(s), 0, 0),
n_{2} = (0, 0 − γ(t)b(s), γ(t)a(s), β(t)b(s), −β(t)a(s)),
n_{3} = (−β^{.}(t)b^{.}(s), β^{.}(t)a^{.}(s), α^{.}(t)b^{.}(s), −α^{.}(t)a^{.}(s), 0, 0),
n_{4} = (0, 0 − γ^{.}(t)b^{.}(s), γ^{.}(t)a^{.}(s), β^{.}(t)b^{.}(s), −β^{.}(t)a^{.}(s)),

Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature vector H vanishes identically (see, for instance, Chen [1]).

In the case under consideration, Imf is minimal if and only if

h(e_{1}, e_{1}) + h(e_{2}, e_{2}) = 0,

where h denotes the second fundamental form of f, or equivalently

<h(e_{1}, e_{1}) + h(e_{2}, e_{2}), n_{i}> = 0, i ∈ {1, 2, 3, 4}.

A straightforward calculation leads to

<g_{22} \frac{∂^{2}f}{∂t^{2}} + g_{11} \frac{∂^{2}f}{∂s^{2}} − 2g_{12} \frac{∂^{2}f}{∂t∂s}, n_{i}> = 0, i ∈ \{1, 2, 3, 4\}.  (1)

We have

\frac{∂^{2}f}{∂t^{2}} = (α^{..}(t)a(s), α^{..}(t)b(s), β^{..}(t)a(s), β^{..}(t)b(s), γ^{..}(t)a(s), γ^{..}(t)b(s)),
\frac{∂^{2}f}{∂s^{2}} = (α(t)a^{..}(s), α(t)b^{..}(s), β(t)a^{..}(s), β(t)b^{..}(s), γ(t)a^{..}(s), γ(t)b^{..}(s)),
\frac{∂^{2}f}{∂t∂s} = (α^{.}(t)a^{.}(s), α^{.}(t)b^{.}(s), β^{.}(t)a^{.}(s), β^{.}(t)b^{.}(s), γ^{.}(t)a^{.}(s), γ^{.}(t)b^{.}(s)).

Since <\frac{∂^{2}f}{∂t^{2}} , n_{i}> = <\frac{∂^{2}f}{∂s^{2}} , n_{i}> = 0, i = 1, 2, (1) implies

g_{12}<\frac{∂^{2}f}{∂t∂s}, n_{1}> = g_{12}<\frac{∂^{2}f}{∂t∂s}, n_{2}> = 0.

We distinguish two cases:

a) g_{12} = 0; in this case c_{1} is spherical or c_{2} is a circle centered at the origin.

b) <\frac{∂^{2}f}{∂t∂s}, n_{1}> = <\frac{∂^{2}f}{∂t∂s}, n_{2}> = 0, which is equivalent to

(a^{.}(s)b(s) − a(s)b^{.}(s))(β^{.}(t)α(t) − α^{.}(t)β(t)) = 0,
(a^{.}(s)b(s) − a(s)b^{.}(s))(γ^{.}(t)β(t) − γ(t)β^{.}(t)) = 0.

We have two subcases: b_{1}) a^{.}(s)b(s) − a(s)b^{.}(s) = 0, i.e. c_{2} is a portion of a straight line passing through the origin; b_{2}) β^{.}(t)α(t) − α^{.}(t)β(t) = 0 and γ^{.}(t)β(t) − γ(t)β^{.}(t) = 0, i.e. c_{1} is a portion of a straight line passing through the origin. Also the case a) has two subcases: a_{1}) c_{2} is a circle centered at the origin. Then c_{2}(s) = (cos s, sin s). Using equation (1) for i = 3, 4, we get

<g_{22} \frac{∂^{2}f}{∂t^{2}} + g_{11} \frac{∂^{2}f}{∂s^{2}} , n_{3}> = <g_{22} \frac{∂^{2}f}{∂t^{2}} + g_{11} \frac{∂^{2}f}{∂s^{2}} , n_{4}> = 0,

or equivalently

∥c_{1}∥^{2} <\frac{∂^{2}f}{∂t^{2}} , n_{3}> + ∥c^{.}_{1}∥^{2} <\frac{∂^{2}f}{∂s^{2}} ,n_{3}> = 0,  (2)
∥c_{1}∥^{2} <\frac{∂^{2}f}{∂t^{2}} , n_{4}> + ∥c^{.}_{1}∥^{2} <\frac{∂^{2}f}{∂s^{2}} ,n_{4}> = 0.  (3)

We may choose t such that ∥c_{1}∥ = ∥c^{.}_{1}∥ . Then the last equations become

β^{.}(t)(α^{..}(t) − α(t)) − α^{.}(t)(β^{..}(t) − β(t)) = 0,  (4)
γ^{.}(t)(β^{..}(t) − β(t)) − β^{.}(t)(γ^{..}(t) − γ(t)) = 0.  (5)

By ∥c_{1}∥ = ∥c^{.}_{1}∥ , one has

α^{.}(t)(α^{..}(t) − α(t)) + β^{.}(t)(β^{..}(t) − β(t)) + γ^{.}(t)(γ^{..}(t) − γ(t)) = 0.  (6)

Consider the system (4)–(6). We have two subsubcases: a_{11}) β^{.}(t) = 0 ⟹ β(t) = 0 ⟹ c_{1} is an orthogonal hyperbola in the plane x^{2} = 0. a_{12}) If all the components of c_{1} are not constant, then by (4)–(6), it follows that

α^{..}(t) = α(t),

β^{..}(t) = β(t),

γ^{..}(t) = γ(t). Then

α(t) = λ_{1} cosh(t + μ_{1}),

β(t) = λ_{2} cosh(t + μ_{2}),
γ(t) = λ_{3} cosh(t + μ_{3}).

a_{2}) c_{1} is spherical, then α^{2} + β^{2} + γ^{2} = 1. Also we may assume c_{1} is parametrized by arc length, i.e. α^{.}^{2} + β^{.}^{2} + γ^{.}^{2} = 1.

Let c_{2}(s) = ρ(s)(cos s, sin s); then a(s) = ρ(s) cos s, b(s) = ρ(s) sin s. One has

a^{.}(s) = ρ^{.}(s) cos s − ρ(s) sin s, b^{.}(s) = ρ^{.}(s)sin s+ρ(s) cos s,

a^{..}(s) = ρ^{..}(s) cos s − 2ρ^{.}(s) sin s − ρ(s) cos s,
b^{..}(s) = ρ^{..}(s) sin s + 2ρ^{.}(s) cos s − ρ(s) sin s.

We have

\frac{∂^{2}f}{∂t^{2}} = (α^{..}(t)a(s), α^{..}(t)b(s), β^{..}(t)a(s), β^{..}(t)b(s), γ^{..}(t)a(s), γ^{..}(t)b(s)),
\frac{∂^{2}f}{∂s^{2}} = (α(t)a^{..}(s), α(t)b^{..}(s), β(t)a^{..}(s), β(t)b^{..}(s), γ(t)a^{..}(s), γ(t)b^{..}(s)).

Using equations (2) and (3), we get

(ρ^{2} + ρ^{.}^{2})<\frac{∂^{2}f}{∂t^{2}} , n_{3}> + ρ^{2}<\frac{∂^{2}f}{∂s^{2}} ,n_{3}> = 0,  (7)
(ρ^{2} + ρ^{.}^{2})<\frac{∂^{2}f}{∂t^{2}} , n_{4}> + ρ^{2}<\frac{∂^{2}f}{∂s^{2}} ,n_{4}> = 0.  (8)

The equation (7) becomes

(ρ^{2} + ρ^{.}^{2})(α^{..}(t)β^{.}(t)−α^{.}(t)β^{..}(t))(a^{.}(s)b(s) − a(s)b^{.}(s))

+ρ^{2}(α(t)β^{.}(t)−α^{.}(t)β(t))(a^{.}(s)b^{..}(s) − a^{..}(s)b^{.}(s)) = 0,

which leads to

−(ρ^{2} + ρ^{.}^{2})(α^{..}(t)β^{.}(t)−α^{.}(t)β^{..}(t)) + (α(t)β^{.}(t)−α^{.}(t)β(t))(2ρ^{.}^{2} − ρρ^{..} + ρ^{2}) = 0,

or equivalently

\frac{2ρ^{.}^{2} − ρρ^{..} + ρ^{2}}{ρ^{2} + ρ^{.}^{2}} = \frac{α^{..}(t)β^{.}(t)−α^{.}(t)β^{..}(t)}{α(t)β^{.}(t)−α^{.}(t)β(t)}.  (9)

The left hand term is a function of s and right hand term is a function of t, then both should be a constant, say k. Similarly from (8) we find

\frac{β^{..}(t)γ^{.}(t)−β^{.}(t)γ^{..}(t)}{β(t)γ^{.}(t)−β^{.}(t)γ(t)} = k.  (10)

If c_{1} has a constant component, which must be 0, then c_{1} is a portion of a circle. In this case c_{2} is an orthogonal hyperbola (see also [6]).

Otherwise from the equation (9) and (10) we get

\frac{α^{..}(t) − kα(t)}{α^{.}(t)} = \frac{β^{..}(t) − kβ(t)}{β^{.}(t)} = \frac{γ^{..}(t) − kγ(t)}{γ^{.}(t)} = m(t).  (11)

Since c_{1} is parametrized by arc length, we have α^{.}(t)α^{..}(t)+β^{.}(t)β^{..}(t)+γ^{.}(t)γ^{..}(t) = 0. Substituting (11) in the last relation, we obtain m(t) = 0. Then we get α^{..}(t) = kα(t), β^{..}(t) = kβ(t),

γ^{..}(t) = kγ(t).

c_{1} being spherical, we have k<0. We put k = −l^{2}, l>0. Finally we find

α(t) = ε_{1} cos lt + η_{1} sin lt,
β(t) = ε_{2} cos lt + η_{2} sin lt,
γ(t) = ε_{3} cos lt + η_{3} sin lt.

satisfying ε_{1}^{2} + ε_{2}^{2} + ε_{3}^{2} = η_{1}^{2} + η_{2}^{2} + η_{3}^{2} = 1 and ε_{1}η_{1} + ε_{2}η_{2} + ε_{3}η_{3} = 0.

Also from equation (9) one gets

ρρ^{..} − ρ^{.}^{2} = (1 + l^{2})(ρ^{2} + ρ^{.}^{2}).  (12)

Putting w = \frac{ρ^{.}}{ρ}, from (12) one obtains w = tan[(1 + l^{2})s + l_{1}], which implies

ρ(s) = \frac{l_{2}}{[cos(1 + l^{2})s + l_{1}]^{\frac{1}{1+l^{2}}}} ,

where l_{1}, l_{2} are constant. Thus c_{2} is sinusoidal spiral. In particular for l = 1, c_{2} is an orthogonal hyperbola.

Conversely, it is easily seen that in all the above discussed cases, the tensor product immersion c_{1} ⊗ c_{2} is minimal.

Summing up, the following theorem is proved.

Theorem 2.1. The tensor product immersion c_{1} ⊗ c_{2} of a Euclidean space curve and a Euclidean plane curve is a minimal surface in 𝔼^{6} if and only if either

i) c_{1} is a straight line through 0; ii) c_{2} is a straight line through 0; iii) c_{1} is a circle centered at 0 and c_{2} is an orthogonal hyperbola centered at 0; iv) c_{1} is an orthogonal hyperbola centered at 0 and c_{2} is a circle centered at 0;

v) c_{2} is a circle centered at 0 and c_{1} is given by

c_{1}(t) = (λ_{1} cosh(t + μ_{1}), λ_{2} cosh(t + μ_{2}), λ_{3} cosh(t + μ_{3}));

vi) c_{1} is given by

c_{1}(t) = (ε_{1} cos lt + η_{1} sin lt, ε_{2} cos lt + η_{2} sin lt, ε_{3} cos lt + η_{3} sin lt),

where ε_{1}^{2} + ε_{2}^{2} + ε_{3}^{2} = η_{1}^{2} + η_{2}^{2} + η_{3}^{2} = 1 and ε_{1}η_{1} + ε_{2}η_{2} + ε_{3}η_{3} = 0, and

c_{2}(s) = \frac{l_{2}}{[cos(1 + l^{2})s + l_{1}]^{\frac{1}{1+l^{2}}}} (cos s, sin s),

with l_{1}, l_{2} =constant.

3. Totally real and slant tensor product surfaces

Let c_{1} : ℝ → 𝔼^{3} and c_{2} : ℝ → 𝔼^{2} be two Euclidean curves and f = c_{1} ⊗ c_{2} their tensor product.

We identify 𝔼^{6} with ℂ^{3} and consider the standard complex structure J given by

J(y^{1}, …, y^{6}) = (−y^{2}, y^{1}, −y^{4}, y^{3}, −y^{6}, y^{5}), y^{1}, …, y^{6} ∈ ℝ.

Then Imf is a real 2-dimensional submanifold of ℂ^{3}, which is totally real, i.e. the complex structure J of 𝔼^{6} at each point transforms the tangent space to the surface into the normal space, according to the following result.

Theorem 3.1. The tensor product immersion c_{1} ⊗ c_{2} of a Euclidean space curve and a Euclidean plane curve is totally real in (ℂ^{3}, J) if and only if c_{1} is spherical or c_{2} is a portion of a straight line passing through 0.

Proof. Imf is a totally real surface if and only if J(\frac{∂f}{∂t}) is orthogonal to \frac{∂f}{∂s} and J(\frac{∂f}{∂s}) is orthogonal to \frac{∂f}{∂t}. We have

J(\frac{∂f}{∂t}) = (−α^{.}(t)b(s), α^{.}(t)a(s), −β^{.}(t)b(s), β^{.}(t)a(s), −γ^{.}(t)b(s), γ^{.}(t)a(s)),

where α^{.} means the derivative of α.

By a straightforward calculation, we obtain

<J(\frac{∂f}{∂t}), \frac{∂f}{∂s} > = −<J(\frac{∂f}{∂s}), \frac{∂f}{∂t} > = 0

if and only if

α(t)α^{.}(t) + β(t)β^{.}(t) + γ(t)γ^{.}(t) = 0

or a(s)b^{.}(s) − b(s)a^{.}(s) = 0.

Integrating these equations, we find that c_{1} is spherical or c_{2} is a portion of a straight line which contains 0, respectively.

Recall the definition of a slant surface in (ℂ^{3}, J) (see [2]). Let M be a surface in (ℂ^{3}, J). For a given orthonormal basis {e_{1}, e_{2}} of T_{x}M (x ∈ M), we put

θ(T_{x}M) = arccos <Je_{1}, e_{2}>,

which is independent of the choice of {e_{1}, e_{2}}. M is said to be slant if θ(T_{x}M) is constant along M. Totally real and complex surfaces are improper slant surfaces, with slant angles θ = \frac{π}{2} and θ = 0, respectively.

Let c_{1} : ℝ → 𝔼^{3}, c_{2} : ℝ → 𝔼^{2} be two Euclidean curves. From Theorem 3.1, we know that if c_{2} is a portion of a straight line containing 0, c_{1} ⊗ c_{2} is an improper slant surface. Otherwise, we consider polar coordinates on c_{2}. Then

c_{2}(s) = ρ_{2}(s)(cos s, sin s).

A straightforward computation leads to

<Je_{1}, e_{2}> = \frac{[a(s)b^{.}(s) − b(s)a^{.}(s)][α(t)α^{.}(t) + β(t)β^{.}(t) + γ(t)γ^{.}(t)]}{\sqrt{∥ċ_{1}∥^{2}∥c_{1}∥^{2}∥ċ_{2}∥^{2}∥c_{2}∥^{2} − <c_{1}, ċ_{1}>^{2}<c_{2}, ċ_{2}>^{2}}}.

Let A(t) = α^{2}(t) + β^{2}(t) + γ^{2}(t), B = \frac{α^{.}}{α} and R = \frac{ρ^{.}}{ρ} . Then

cos θ = \frac{1}{\sqrt{ρ^{2}(ρ^{2} + ρ^{.}^{2})A^{2} − ρ^{2}ρ^{.}^{2}(\frac{A^{.}}{2})^{2}}} ρ^{2} \frac{A^{.}}{2}.

Therefore Imf is a slant surface if and only if

B = constant and R = constant,

or equivalently

A(t) = k_{1}e^{l_{1}t},
ρ(s) = k_{2}e^{l_{2}s}.

We proved the following

Theorem 3.2. The tensor product immersion c_{1} ⊗ c_{2} of a Euclidean space curve and a Euclidean plane curve is a proper slant surface if and only if c_{2} is a logarithmic spiral curve or a circle and c_{1} satisfies α^{2}(t) + β^{2}(t) + γ^{2}(t) = k_{1}e^{l_{1}t} , for all t ∈ ℝ.

References

[1] Chen, B. Y.: Geometry of Submanifolds. M. Dekker, New York 1973.

[2] Chen, B. Y.: Geometry of Slant Submanifolds. Katholieke Universiteit Leuven 1990.

[3] Chen, B. Y.:Differential Geometry of semiring of immersions, I: General theory. Bull. Inst. Math. Acad. Sinica 21 (1993), 1–34.

[4] Decruyenaere, F.; Dillen, F.; Verstraelen, L.; Vrancken, L.: The semiring of immersions of manifolds. Beiträge Algebra Geom. 34 (1993), 209–215.

[5] Decruyenaere, F.; Dillen, F.; Mihai, I.; Verstraelen, L.: Tensor products of spherical and equivariant immersions. Bull. Belg. Math. Soc. - Simon Stevin 1 (1994), 643–648.

[6] Mihai, I.; Rosca, R.; Verstraelen, L.; Vrancken, L.: Tensor product surfaces of Euclidean planar curves. Rend. Sem. Mat. Messina 3 (1994/1995), 173–184.

[7] Rouxel, B.: A-submanifolds in Euclidean space. Kodai Math. J. 4 (1981), 181–188.

Received August 25, 2000