Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 275-296.

Cubic Form Geometry for Surfaces in 𝕊^{3}(1)

Tsasa Lusala^{∗}

Technische Universität Berlin, Fakultät II: Mathematik und Naturwissenschaften, Institut für Mathematik Sekr. MA 8-3, Straße des 17. Juni 136, D-10623 Berlin Germany e-mail: lusala@math.TU-Berlin.DE

Abstract. We consider the traceless part C̃ of the difference tensor field C between the Levi-Civita connections of the first and the third fundamental forms for nondegenerate surface immersions in 𝕊^{3}(1). In analogy to affine differential geometry of ℝ^{n+1} where quadrics are characterized by the vanishing of a traceless cubic form, we study the condition C̃ ≡ 0, give examples and classify non-degenerate surfaces in 𝕊^{3}(1) which satisfy this condition. MSC 2000: 53B25, 53A15, 35-04, 53C24 Keywords: Non-degenerate surfaces in 3-spheres, principal curvature functions, rotational surfaces, cubic form geometry

1. Introduction

If x: M^{n} ⟶ N^{n+1} is an n-dimensional hypersurface immersion into a space N^{n+1} of constant curvature which is non-degenerate, i.e. its shape operator S has maximal rank, then we can apply methods of affine differential geometry in the following way: Besides the first fundamental form I also the third fundamental form III is a Riemannian metric; they induce Levi-Civita connections ∇^{I} and ∇^{III}, respectively. Denoting the second fundamental form of x by II we obtain a conjugate triple (∇^{I}, II, ∇^{III}) (see below). The difference tensor field C := \frac{1}{2}(∇^{I} − ∇^{III}) satisfies

2SC(u, v) = −(∇_{u}^{I}S)v.  (1)

∗This work is supported by DFG (Graduiertenkolleg “Geometrie und Nichtlineare Analysis” HU-TU Berlin).

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

276 Tsasa Lusala: Cubic Form Geometry for Surfaces in 𝕊^{3}(1)

The associated tensor field

Ĉ: (u, v, w) ⟼ II(C(u, v), w)  (2)

is the well known cubic form of the conjugate triple (∇^{I}, II, ∇^{III}) which is totally symmetric. Because of (1) and (2) the following sequence of equivalences holds:

Ĉ ≡ 0 ⟺ C ≡ 0 ⟺ ∇^{I}S ≡ 0.  (3)

Furthermore, for n = 2, the parallelity of S is equivalent to the fact that x is an isoparametric surface. In this paper we will investigate more general immersions by weakening condition (3). Namely, we study the case where only the traceless part C̃ of C vanishes (for C̃ see below). In the case that N^{n+1} is the Euclidean space ℝ^{n+1} it is known that the condition C̃ ≡ 0 characterizes those non-degenerate immersions whose images lie on a quadric; see [6], p. 117–119, and the additional remark to the proof in [3], p. 208, (2.2.b). This result belonging to affine differential geometry is a generalization of the classical theorem of Maschke. Motivated by this result we ask for the geometric meaning of the condition C̃ ≡ 0 for other spaces N^{n+1} of constant curvature. It turns out that the situation is rather complicated. Thus we restrict our considerations to non-degenerate surfaces of the Euclidean sphere 𝕊^{3}(1) of radius 1. As mentioned above the isoparametric surface immersions, which are not totally geodesic are obviously in this class. The purpose of this paper is to find further examples (see Example 3.7 and 3.6) and prove the following local classification: Theorem 1.1. Let x: M^{2} ⟶ 𝕊^{3}(1) ⊂ ℝ^{4} be a non-degenerate surface immersion of a connected and orientable C^{∞}-manifold M^{2} into 𝕊^{3}(1) satisfying the condition C̃ ≡ 0. Then there exist an open and dense subset U of M^{2} and for every point p ∈ U a neighbourhood U(p) ⊂ U, such that x_{|_{U} (p)} is one of the following types:

(1) it is totally umbilical, i.e. it describes an open part of a small sphere in 𝕊^{3}(1),

(2) it is a part of a rotational surface with an arc of an ellipse or an arc of a hyperbola as profile curve (see Example 3.6 below),

(3) it is a part of a quadratic surface of the type described in Example 3.7 below.

It should be mentioned that (2) includes the isoparametric surfaces with two distinct principal curvatures; i.e. these are isometric to Clifford tori 𝕊^{1}(r_{1}) × 𝕊^{1}(r_{2}) with 0 < r_{1}, r_{2} and r_{1}^{2} + r_{2}^{2} = 1.

From this theorem one can easily derive that the non-umbilical (C̃ ≡ 0)-surfaces lie on some quadrics of ℝ^{4} centered at 0. In fact, in this way one can get another characterization of (C̃ ≡ 0)-surfaces, which will be presented in a forthcoming paper.

Acknowledgements. I would like to thank Prof. Dr. H. Reckziegel (Cologne) for many discussions; from him I learned a lot about rotational surfaces which strongly improved the presentation of this paper. In particular he suggested the nice formulation of the main results contained in this paper. I would like also to thank Dr. Knut Pawel (his former Ph.D student), Prof. Dr. Udo Simon (TU Berlin) and Prof. Dr. L. Vrancken (U. Utrecht) for valuable comments and fruitful discussions.

2. Preliminaries

Let x: M^{2} ⟶ 𝕊^{3}(1) ⊂ ℝ^{4} be an immersion of a connected, orientable 2-dimensional C^{∞}- manifold M^{2} into 𝕊^{3}(1). Denote by y a unit normal vector field on 𝕊^{3}(1) along the immersion x, by <, > the canonical inner product of the Euclidean structure, and by ¯∇ the flat connection of ℝ^{4}. To the immersion x are associated three fundamental forms, namely: I the first fundamental form (induced metric), II the second and III the third fundamental form related by

II(u, v) = I(Su, v),  III(u, v) = I(Su, Sv),  (4)

where S is the Weingarten (shape) operator and u, v ∈ 𝔛(M^{2}) (= the C^{∞}-module of vector fields on M^{2}). The immersion x is said to be non-degenerate or regular if the shape operator S has maximal rank. In this case, the second fundamental form II is a semi-Riemannian metric while the third fundamental form III is a Riemannian metric on M^{2}. For all u, v ∈ 𝔛(M^{2}), the structure (fundamental) equations of x as immersion into ℝ^{4}, namely the Gauß equation and the Weingarten equation, are given by:

¯∇_{u}dx(v) = dx(∇_{u}^{I}v) + II(u, v)y − I(u, v)x;
dy(v) = −dx(Sv).

The structure equations above imply the following integrability conditions:

(∇_{u}^{I}S) v = (∇_{v}^{I}S) u;  (5)
R^{I}(u, v)w = I(w, v)u − I(u, w)v + II(w, v)Su − II(u, w)Sv,  (6)

where ∇^{I} and R^{I} denote the Levi-Civita connection and the Riemannian curvature tensor on M^{2} of the first fundamental form, respectively. Consequently, from the equation (4) and the Codazzi equation (5), one can check that the Levi-Civita connection ∇^{III} of the third fundamental form is given by:

∇_{u}^{III}v = S^{−1}∇_{u}^{I}(Sv).  (7)

Using (7), one can verify, for any u, v, w ∈ 𝔛(M^{2}), that

wII(u, v) = II(∇_{w}^{I}u, v) + II(u, ∇_{w}^{III}v);

in other words: the triple (∇^{I}, II, ∇^{III}) is conjugate. Therefore the Levi-Civita connection ∇^{II} of the second fundamental form satisfies

∇^{II} = \frac{1}{2}(∇^{I} + ∇^{III}).

The difference tensor C defined by

C(u, v) := \frac{1}{2}(∇_{u}^{I}v − ∇_{u}^{III}v) = −\frac{1}{2}S^{−1}(∇_{u}^{I}S)v  (8)

is a symmetric (1, 2)-tensor field and satisfies

∇^{I} = ∇^{II} + C, ∇^{III} = ∇^{II} − C.

The Tchebychev vector field T is defined by: II(u, T) := \frac{1}{2}tr[v ⟼ C(u, v)]. And finally, the (1, 2)-tensor field C̃, defined by

C̃(u, v): = C(u, v) − \frac{1}{2}[II(T, v)u + II(T, u)v + II(u, v)T]

is symmetric and traceless with respect to both variables. It is called the traceless part of C.

Proposition 2.1. Denote by K := det S the Gauß-Kronecker curvature function of the nondegenerate immersion x: M^{2} ⟶ 𝕊^{3}(1). Then the Tchebychev vector field T satisfies

T = −\frac{1}{4}S^{−1} (grad^{I} ln |K|) ,

where grad^{I} is the gradient with respect to the first fundamental form.

Proof. The Tchebychev vector field T satisfies [1]

T = −\frac{1}{4}grad^{II} ln |K|,

where grad^{II} is the gradient with respect to the second fundamental form. One has:

S(T) = −\frac{1}{4}S (grad^{II} ln |K|) = −\frac{1}{4}grad^{I} ln |K|.

3. The equation C̃ = 0

The equation C = 0 implies the equation C̃ = 0, but both equations obviously are not equivalent. As mentioned in the introduction, an immersion x: M^{2} ⟶ 𝕊^{3}(1) satisfying the condition C = 0 is isoparametric, that means that x(M^{2}) is an open part of a 2-dimensional subsphere or of a torus of 𝕊^{3}(1). In contrast, for C̃ = 0 the situation is much more difficult. Assume from now on that the regular immersion x: M^{2} ⟶ 𝕊^{3}(1) ⊂ ℝ^{4} has no umbilical points. Denote by λ_{1} and λ_{2} the two distinct principal curvature functions of the immersion x. Let (e_{1}, e_{2}) be a local I-orthonormal frame of principal vector fields on an open subset U of M^{2}: Se_{1} = λ_{1}e_{1} and Se_{2} = λ_{2}e_{2}. There exist two differentiable functions α, β ∈ C^{∞}(U) such that:

∇_{e_{1}}^{I}e_{1} = αe_{2}, ∇_{e_{2}}^{I}e_{2} = βe_{1}.

Because of ∇^{I}I = 0, I(e_{1}, e_{1}) = 1 = I(e_{2}, e_{2}) and I(e_{1}, e_{2}) = 0, one also has:

∇_{e_{1}}^{I}e_{2} = −αe_{1}, ∇_{e_{2}}^{I}e_{1} = −βe_{2}.

So the Lie bracket of e_{1} and e_{2} is:

[e_{1}, e_{2}] = −αe_{1} + βe_{2}.

With respect to the frame (e_{1}, e_{2}), the structure equations of x as immersion into ℝ^{4} are:

\begin{cases}¯∇_{e_{1}}dx(e_{1}) = αdx(e_{2}) + λ_{1}y − x, \\ ¯∇_{e_{1}}dx(e_{2}) = −αdx(e_{1}), \\ ¯∇_{e_{2}}dx(e_{1}) = −βdx(e_{2}), \\ ¯∇_{e_{2}}dx(e_{2}) = βdx(e_{1}) + λ_{2}y − x, \\ dy(e_{1}) = −λ_{1}dx(e_{1});  dy(e_{2}) = −λ_{2}dx(e_{2}). \end{cases}  (9)

Proposition 3.1. The functions α, β, λ_{1} and λ_{2} satisfy the following first order partial differential equations:

e_{1}(β) + e_{2}(α) = α^{2} + β^{2} + 1 + λ_{1}λ_{2},  (10)
e_{1}(λ_{2}) = β(λ_{2} − λ_{1}),  (11)
e_{2}(λ_{1}) = α(λ_{1} − λ_{2}).  (12)

Proof. From the integrability condition (6), one has:

R^{I}(e_{1}, e_{2})e_{2} = e_{1} + λ_{1}λ_{2}e_{1} = (1 + λ_{1}λ_{2})e_{1}.

Moreover,

R^{I}(e_{1}, e_{2})e_{2} = ∇_{e_{1}}^{I}∇_{e_{2}}^{I}e_{2} − ∇_{e_{2}}^{I}∇_{e_{1}}^{I}e_{2} − ∇_{[e_{1},e_{2}]}^{I}e_{2}
= ∇_{e_{1}}^{I}(βe_{1}) − ∇_{e_{2}}^{I}(−αe_{1}) − ∇_{−αe_{1}+βe_{2}}^{I}e_{2}
= (e_{1}(β) + e_{2}(α) − α^{2} − β^{2}) e_{1}.

So 1 + λ_{1}λ_{2} = e_{1}(β) + e_{2}(α) − α^{2} − β^{2}.

From the Codazzi equation (5), one has:

0 = (∇_{e_{1}}^{I}S)e_{2} − (∇_{e_{2}}^{I}S)e_{1}
= ∇_{e_{1}}^{I}Se_{2} − S∇_{e_{1}}^{I}e_{2} − ∇_{e_{2}}^{I}Se_{1} + S∇_{e_{2}}^{I}e_{1}
= ∇_{e_{1}}^{I}(λ_{2}e_{2}) − S(−αe_{1}) − ∇_{e_{2}}^{I}(λ_{1}e_{1}) + S(−βe_{2})
= e_{1}(λ_{2})e_{2} − αλ_{2}e_{1} + αλ_{1}e_{1} = −e_{2}(λ_{1})e_{1} + βλ_{1}e_{2} − βλ_{2}e_{2}
= (α(λ_{1} − λ_{2}) − e_{2}(λ_{1}))e_{1} + (e_{1}(λ_{2}) − β(λ_{2} − λ_{1}))e_{2}.

So e_{1}(λ_{2}) = β(λ_{2} − λ_{1}) and e_{2}(λ_{1}) = α(λ_{1} − λ_{2}).

Proposition 3.2. With respect to the frame (e_{1}, e_{2}), the components of the tensor field C and the Tchebychev vector field T are given by:

C_{11}^{1} = −\frac{1}{2}λ_{1}^{−1}e_{1}(λ_{1}),  C_{11}^{2} = −\frac{1}{2}λ_{2}^{−1}α(λ_{1} − λ_{2}),
C_{22}^{1} = −\frac{1}{2}λ_{1}^{−1}β(λ_{2} − λ_{1}), C_{22}^{2} = −\frac{1}{2}λ_{2}^{−1}e_{2}(λ_{2}),  (13)
C_{12}^{1} = −\frac{1}{2}λ_{1}^{−1}α(λ_{1} − λ_{2}), C_{12}^{2} = −\frac{1}{2}λ_{2}^{−1}β(λ_{2} − λ_{1});

T = −\frac{1}{4}[(λ_{1}^{−2}e_{1}(λ_{1}) + λ_{1}^{−1}λ_{2}^{−1}β(λ_{2} − λ_{1}))e_{1} + (λ_{1}^{−1}λ_{2}^{−1}α(λ_{1} − λ_{2}) + λ_{2}^{−2}e_{2}(λ_{2}))e_{2}]. (14)

Proof. Using the relation (8) between C, S and ∇^{I}, one has:

C(e_{1}, e_{1}) = −\frac{1}{2}S^{−1}(∇_{e_{1}}^{I}S)e_{1} = −\frac{1}{2}(λ_{1}^{−1}e_{1}(λ_{1})e_{1} + αλ_{2}^{−1}(λ_{1} − λ_{2})e_{2}).

Similarly,

C(e_{2}, e_{2}) = −\frac{1}{2}S^{−1}(∇_{e_{2}}^{I}S)e_{2} = −\frac{1}{2}(λ_{2}^{−1}e_{2}(λ_{2})e_{2} + βλ_{1}^{−1}(λ_{2} − λ_{1})e_{1}),
C(e_{1}, e_{2}) = −\frac{1}{2}S^{−1}(∇_{e_{1}}^{I}S)e_{2} = −\frac{1}{2}(βλ_{2}^{−1}(λ_{2} − λ_{1})e_{2} + αλ_{1}^{−1}(λ_{1} − λ_{2})e_{1}).

For the Tchebychev vector field we have: −grad^{I} ln |λ_{1}λ_{2}| = 4T_{1}e_{1} + 4T_{2}e_{2}, where

2T_{1} = C_{11}^{1} + C_{12}^{2} = −\frac{1}{2}λ_{1}^{−1}e_{1}(λ_{1}) − \frac{1}{2}λ_{2}^{−1}β(λ_{2} − λ_{1}),
2T_{2} = C_{12}^{1} + C_{22}^{2} = −\frac{1}{2}λ_{2}^{−1}e_{2}(λ_{2}) − \frac{1}{2}λ_{1}^{−1}α(λ_{1} − λ_{2}).

So from Proposition 2.1,

T = −\frac{1}{4}S^{−1} (grad^{I} ln |λ_{1}λ_{2}|)
= −\frac{1}{4}[(λ_{1}^{−2}e_{1}(λ_{1}) + λ_{1}^{−1}λ_{2}^{−1}β(λ_{2} − λ_{1}))e_{1} + (λ_{1}^{−1}λ_{2}^{−1}α(λ_{1} − λ_{2}) + λ_{2}^{−2}e_{2}(λ_{2}))e_{2}].

Proposition 3.3. The condition C̃ ≡ 0 is equivalent to the following equations:

e_{1}(λ_{1}) = 3βλ_{1}λ_{2}^{−1}(λ_{2} − λ_{1}),  (15)
e_{2}(λ_{2}) = 3αλ_{2}λ_{1}^{−1}(λ_{1} − λ_{2}).  (16)

Proof. The immersion x satisfies the condition C̃ ≡ 0 if and only if

C_{11}^{1} = \frac{1}{2}[T_{1} + T_{1} + T^{1}II_{11}] = \frac{3}{2}T_{1}, C_{12}^{1} = \frac{1}{2}T_{2},
C_{11}^{2} = \frac{1}{2}[T^{2}II_{11}] = \frac{1}{2}λ_{1}λ_{2}^{−1}T_{2},
C_{22}^{1} = \frac{1}{2}[T^{1}II_{22}] = \frac{1}{2}λ_{2}λ_{1}^{−1}T_{1},
C_{22}^{2} = \frac{1}{2}[T_{2} + T_{2} + T^{2}II_{22}] = \frac{3}{2}T_{2}, C_{12}^{2} = \frac{1}{2}T_{1}.

So, T_{1} = 2C_{12}^{2} = −βλ_{2}^{−1}(λ_{2} − λ_{1}), and then C_{11}^{1} = \frac{3}{2}T_{1} and C_{11}^{1} = −\frac{1}{2}λ_{1}^{−1}e_{1}(λ_{1}) implies the equation (15). For the equation (16), use T_{2} = 2C_{12}^{1} = −αλ_{1}^{−1}(λ_{1} − λ_{2}), C_{22}^{2} = \frac{3}{2}T_{2}, and C_{22}^{2} = −\frac{1}{2}λ_{2}^{−1}e_{2}(λ_{2}). The converse is obvious.

Corollary 3.4. The condition C̃ ≡ 0 implies the following equations:

e_{1}(α) = 3αβ,  (17)
e_{2}(β) = 3αβ.  (18)

Proof. From (11), (12), (15) and (16), one has:

0 = e_{1}e_{2}(λ_{1}) − e_{2}e_{1}(λ_{1}) − [e_{1}, e_{2}](λ_{1})
= (λ_{2} − λ_{1}) (−e_{1}(α) − 3λ_{1}λ_{2}^{−1}e_{2}(β) + 3αβ(3λ_{1}λ_{2}^{−1} + 1))
0 = e_{1}e_{2}(λ_{2}) − e_{2}e_{1}(λ_{2}) − [e_{1}, e_{2}](λ_{2})
= (λ_{1} − λ_{2}) (−e_{2}(β) − 3λ_{2}λ_{1}^{−1}e_{1}(α) + 3αβ(3λ_{2}λ_{1}^{−1} + 1)) .

So

0 = −e_{1}(α) − 3λ_{1}λ_{2}^{−1}e_{2}(β) + 3αβ(3λ_{1}λ_{2}^{−1} + 1)  (19)
0 = −e_{1}(β) − 3λ_{2}λ_{1}^{−1}e_{1}(α) + 3αβ(3λ_{2}λ_{1}^{−1} + 1).  (20)

From the equations (19) and (20), one gets e_{1}(α) = 3αβ = e_{2}(β).

Example 3.5. Open parts of small spheres in 𝕊^{3}(1) are trivial examples of immersions which satisfy the condition C̃ ≡ 0.

Example 3.6. (Surfaces of revolution with arcs of ellipses or arcs of hyperbolas as profile curves) Let ε ∈ {−1, 1}. Define the functions cos_{ε} and sin_{ε} by:

cos_{ε} := \{ \begin{matrix}cos if ε = 1 \\ cosh if ε = −1 \end{matrix} and sin_{ε} := \{ \begin{matrix}sin if ε = 1 \\ sinh if ε = −1. \end{matrix}  (21)

Let (C_{1}, C_{2}, C_{3}, C_{4}) be an orthonormal basis of ℝ^{4}, 0 < a < 1, 0 < b two constant reals such that b ≠ 1 if ε = 1, and I ⊂ ℝ be a non-empty open interval such that the function r defined by

r(u) = \sqrt{1 − a^{2} cos_{ε}^{2}(u) − b^{2} sin_{ε}^{2}(u)}

is real and positive on I. Then the mapping x: I × ℝ ⟶ 𝕊^{3}(1) ⊂ ℝ^{4} :

x(u, v) = r(u) · ( cos(v)C_{1} + sin(v)C_{2}) + A(u),  (22)

with

A(u) = a cos_{ε}(u)C_{3} + b sin_{ε}(u)C_{4},

is a (C̃ ≡ 0)-surface in 𝕊^{3}(1). We call x a surface of revolution with the profile curve A, which is in our cases an arc of an ellipse (ε = 1) or of a hyperbola (ε = −1). In the case ε = 1 and a = b, x describes an isoparametric surface with two distinct principal curvatures.

In order to show that the condition C̃ ≡ 0 is satisfied we compute

x_{u} = r′(u) · ( cos(v)C_{1} + sin(v)C_{2}) − a sin_{ε}(u)C_{3} + b cos_{ε}(u)C_{4},
x_{v} = r(u) · ( − sin(v)C_{1} + cos(v)C_{2}).

Defining the function σ on I by σ(u) := \sqrt{a^{2} sin_{ε}^{2}(u) + b^{2} cos_{ε}^{2}(u) − a^{2}b^{2}}, one has:

I_{11} =< x_{u}, x_{u} > = \frac{σ^{2}(u)}{r^{2}(u)},
I_{22} =< x_{v}, x_{v} > = r^{2}(u),  I_{12} =< x_{u}, x_{v} >= 0.

Since det(I) = I_{11}I_{22} ≠ 0 for every u ∈ I, the mapping x defines an immersion. The vector field

y(u, v) = \frac{abr(u)}{σ(u)} · ( cos(v)C_{1} + sin(v)C_{2}) + b(a^{2} − 1)\frac{cos_{ε}(u)}{σ(u)}C_{3} + a(b^{2} − ε)\frac{sin_{ε}(u)}{σ(u)}C_{4}, (23)

is a unit normal vector field. As < y_{u}, x_{v} >= 0 =< y_{v}, x_{u} >, the coordinate lines are lines of curvatures; therefore the principal curvature functions are

λ_{1} = −\frac{< y_{u}, x_{u} >}{< x_{u}, x_{u} >} = \frac{ab(1 − a^{2})(ε − b^{2})}{σ^{3}}, λ_{2} = −\frac{< y_{v}, x_{v} >}{< x_{v}, x_{v} >} = −\frac{ab}{σ}.

The principal curvature functions λ_{1} and λ_{2} above do not have zeros on I, i.e. the immersion x is non-degenerate. And the vector fields e_{1} = \frac{r}{σ} \frac{∂}{∂u} and e_{2} = \frac{1}{r} \frac{∂}{∂v} are orthonormal principal vector fields. They satisfy

∇_{e_{1}}^{I}e_{1} = 0 and ∇_{e_{2}}^{I}e_{2} = βe_{1},

where β(u) = −\frac{r′(u)}{σ(u)}. Moreover, one has:

e_{2}(λ_{2}) = 0 and e_{1}(λ_{1}) = \frac{3β(u)λ_{1}(λ_{2} − λ_{1})}{λ_{2}},

i.e. the equations (15) and (16) hold. Thus the immersion x satisfies C̃ ≡ 0. Note finally that the surface (22) has no umbilical points since everywhere

λ_{1} − λ_{2} = \frac{εabr^{2}}{σ^{3}} ≠ 0.

Example 3.7. Let η ∈ {−1, 1}, a_{1}, a_{2}, a_{3}, a_{4} ∈ ℝ be distinct real numbers such that

a_{1} < −ηa_{2} < 0 < a_{3} < a_{4}, a_{2} < a_{3} and a_{1}a_{2}a_{3}a_{4} = −η;

and C_{1}, C_{2}, C_{3}, C_{4} ∈ ℝ^{4} be constant orthogonal vectors in ℝ^{4} such that

< C_{1}, C_{1} > = \frac{a_{1}}{(a_{1} − a_{2})(a_{1} − a_{3})(a_{1} − a_{4})},
< C_{2}, C_{2} > = \frac{ηa_{2}}{(a_{2} − a_{3})(a_{2} − a_{4})(a_{2} − a_{1})},
< C_{3}, C_{3} > = \frac{a_{3}}{(a_{3} − a_{2})(a_{4} − a_{3})(a_{3} − a_{1})},
< C_{4}, C_{4} > = \frac{a_{4}}{(a_{4} − a_{2})(a_{4} − a_{3})(a_{4} − a_{1})}.

Define the rectangle R ⊂ ℝ^{2} by

R := \{ \begin{matrix}]a_{3}, a_{4}[×]a_{2}, a_{3}[ if η = 1 \\ ]a_{3}, a_{4}[×] − a_{2}, −a_{1}[ if η = −1. \end{matrix}

Then the mapping x: R ⟶ 𝕊^{3}(1) ⊂ ℝ^{4} defined by

x(u, v) := \sqrt{(u − a_{1})(ηv − a_{1})}C_{1} + \sqrt{(u − a_{2})(v − ηa_{2})}C_{2} + \sqrt{(u − a_{3})(a_{3} − ηv)}C_{3}
+\sqrt{(a_{4} − u)(a_{4} − ηv)}C_{4},  (24)

is a (C̃ ≡ 0)-surface in 𝕊^{3}(1). In fact,

x_{u} = \frac{1}{2}\sqrt{\frac{ηv − a_{1}}{u − a_{1}}} C_{1} + \frac{1}{2}\sqrt{\frac{v − ηa_{2}}{u − a_{2}}} C_{2} + \frac{1}{2}\sqrt{\frac{a_{3} − ηv}{u − a_{3}}} C_{3} − \frac{1}{2}\sqrt{\frac{a_{4} − ηv}{a_{4} − u}} C_{4},
x_{v} = \frac{η}{2}\sqrt{\frac{u − a_{1}}{ηv − a_{1}}} C_{1} + \frac{1}{2}\sqrt{\frac{u − a_{2}}{v − ηa_{2}}} C_{2} − \frac{η}{2}\sqrt{\frac{u − a_{3}}{a_{3} − ηv}} C_{3} − \frac{η}{2}\sqrt{\frac{a_{4} − u}{a_{4} − ηv}} C_{4}.

Since

I_{12} =< x_{u}, x_{v} > = 0,
I_{11} =< x_{u}, x_{u} > = −\frac{u(u − ηv)}{4(u − a_{1})(u − a_{2})(u − a_{3})(u − a_{4})},
I_{22} =< x_{v}, x_{v} > = \frac{ηv(u − ηv)}{4(v − ηa_{1})(v − ηa_{2})(v − ηa_{3})(v − ηa_{4})},

the mapping x defines an immersion. The vector field

y(u, v) = −η\sqrt{\frac{(u − a_{1})(ηv − a_{1})}{uv}} \frac{C_{1}}{a_{1}} − η\sqrt{\frac{(u − a_{2})(v − ηa_{2})}{uv}} \frac{C_{2}}{a_{2}} − η\sqrt{\frac{(u − a_{3})(a_{3} − ηv)}{uv}} \frac{C_{3}}{a_{3}}
−η\sqrt{\frac{(a_{4} − u)(a_{4} − ηv)}{uv}} \frac{C_{4}}{a_{4}}  (25)

is a unit normal vector field; it satisfies

y_{u} = −\frac{η}{\sqrt{u^{3}v}} x_{u}, y_{v} = −\frac{1}{\sqrt{v^{3}v}} x_{v};

therefore the principal curvature functions of the immersion x are λ_{1}(u, v) = \frac{η}{\sqrt{u^{3}v}} ≠ 0 and λ_{2}(u, v) = \frac{1}{\sqrt{v^{3}u}} ≠ 0, thus x is non-degenerate. The vector fields e_{1} = \frac{1}{\sqrt{I_{11}}} \frac{∂}{∂u} and e_{2} = \frac{1}{\sqrt{I_{22}}} \frac{∂}{∂v} are orthonormal principal vector fields such that where α and β are functions on R given by

α(u, v) = η\sqrt{\frac{(v − ηa_{1})(v − ηa_{2})(v − ηa_{3})(v − ηa_{4})}{v(ηu − v)^{3}}},

β(u, v) = −\sqrt{\frac{−(u − a_{1})(u − a_{2})(u − a_{3})(u − a_{4})}{u(u − ηv)^{3}}}.

Moreover the following equalities hold:

\frac{1}{\sqrt{I_{11}}} ∂_{u}(λ_{1}) = \frac{3βλ_{1}(λ_{2} − λ_{1})}{λ_{2}}  and  \frac{1}{\sqrt{I_{22}}} ∂_{v}(λ_{2}) = \frac{3αλ_{2}(λ_{1} − λ_{2})}{λ_{1}},

i.e. the equations (15) and (16) are satisfied, which means that the regular immersion x satisfies the condition C̃ ≡ 0.

4. Classification of (C̃ ≡ 0)-surfaces in 𝕊^{3}(1)

In this section we want to obtain a local description of the surface immersions in 𝕊^{3}(1) satisfying the condition C̃ ≡ 0. Let W be the open set of non-umbilical points on M^{2}. Assuming the situation described at the beginning of Section 3 we construct open subsets U_{1}, U_{2} and U_{3} of M^{2} by

\begin{cases}U_{1} := (M^{2} \ W)^{o}, \\ U_{3} := \{p ∈ W: α(p) ≠ 0 ≠ β(p)\}, \\ U_{2} := (W \ U_{3})^{o}, \end{cases}

where P^{o} is the topological interior of P in M^{2}. Consider the disjoint union

U := U_{1} ∪ U_{2} ∪ U_{3}.

Obviously, U is an open dense subset of M^{2}. We will show in the following that on each of the subsets U_{i} the immersion x is locally of the type (i) described in Theorem 1.1. For i = 1 this is obvious.

The I-orthonormal frame (e_{1}, e_{2}) of principal vector fields is unique up to signs and to changes of the order of the vector fields e_{1} and e_{2}. After the choice of (e_{1}, e_{2}) the functions α and β are uniquely determined. Consequently, the subsets U_{1}, U_{2} and U_{3} above are well defined.

4.1. Rotational (C̃ ≡ 0)-surface immersions in 𝕊^{3}(1)

Assume that U_{2} is non-empty. On each connected component of U_{2}, at least one of the functions α and β vanishes everywhere. Fix then α = 0 (the case β = 0 is similar because of symmetries) everywhere on a connected component.

Lemma 4.1. Let x: (M^{2}, I) ⟶ 𝕊^{3}(1) ⊂ ℝ^{4} be a non-degenerate isometric immersion without umbilical points. If there exists an I-orthonormal frame (e_{1}, e_{2}) of principal vector fields satisfying

∇_{e_{1}}^{I}e_{1} = 0 and e_{2}(λ_{2}) = 0,

where λ_{2} denotes the principal curvature function corresponding to the direction e_{2}, then for every point p ∈ M^{2} there exist local coordinates (u, v) and an orthonormal basis C_{1}, C_{2}, C_{3}, C_{4} of ℝ^{4}, such that x is described by the following parametrization of a surface of revolution

x(u, v) = r(u) · ( cos(v)C_{1} + sin(v)C_{2}) + A(u),

where A(u) is a curve in Span{C_{3}, C_{4}} and r a differentiable function satisfying

r^{2}+ < A, A >≡ 1.  (26)

Proof. From (10), (11) and (12) with α = 0 on M^{2}, one has:

\begin{matrix}e_{1}(β) = β^{2} + 1 + λ_{1}λ_{2}, \\ e_{1}(λ_{2}) = β(λ_{2} − λ_{1}), \\ e_{2}(λ_{1}) = 0 = e_{2}(λ_{2}). \end{matrix}  (27)

Furthermore,

0 = −αe_{1}(λ_{2}) + βe_{2}(λ_{2})
= [e_{1}, e_{2}](λ_{2})
= e_{1}e_{2}(λ_{2}) − e_{2}e_{1}(λ_{2})
= −(λ_{2} − λ_{1})e_{2}(β);

therefore the assumption of non-umbilicity on M^{2} implies

e_{2}(β) = 0.

Define the function r on M^{2} by:

r := \frac{1}{\sqrt{β^{2} + 1 + λ_{2}^{2}}}  (28)

Using the equations (27), one has that the function r satisfies

e_{2}(r) = 0 and e_{1}(r) = −rβ.

It follows that

[e_{1}, re_{2}] = 0.

Consequently, there is a system of local coordinates (u, v) on U such that

e_{1} = \frac{∂}{∂u} and e_{2} = r^{−1} \frac{∂}{∂v} .

This means that the manifold M^{2} locally is a warped product, with r as the warping function. The functions λ_{1}, λ_{2}, β and r depend only on u and satisfy the following system of first order ordinary differential equations, where we use the prime-notation ′ to indicate the derivatives with respect to u:

\begin{cases}β′ = β^{2} + 1 + λ_{1}λ_{2}; \\ r′ = −rβ; \\ λ′_{2} = β(λ_{2} − λ_{1}). \end{cases}  (29)

With respect to the frame (\frac{∂}{∂u}, \frac{∂}{∂v}), the structure equations are written as follows:

x_{uu} = λ_{1}y − x,  (30)
x_{uv} = −βx_{v} = r′r^{−1}x_{v} = x_{vu},  (31)
x_{vv} = r^{2} (βx_{u} + λ_{2}y − x) ,  (32)
y_{v} = −λ_{2}x_{v};  y_{u} = −λ_{1}x_{u}.  (33)

Differentiating (32) with respect to v, the immersion x(u, v) then satisfies the following equation:

x_{vvv} = −r^{2}(β^{2} + 1 + λ_{2}^{2})x_{v} = −x_{v};

for the last equality we use (28). Therefore there are vector valued functions A_{1} and A_{2} on U depending only on u such that

x_{v} = −A_{1} sin(v) + A_{2} cos(v).  (34)

Because of the linear independence of the functions 1, sin(2v) and cos(2v), and the fact that < x_{v}, x_{v} >= r^{2}, the vector valued functions A_{1} and A_{2} satisfy the following conditions:

< A_{1}(u), A_{1}(u) >=< A_{2}(u), A_{2}(u) >= r^{2}, < A_{1}(u), A_{2}(u) >= 0.

Differentiate the equation (34) with respect to u and use (31) to get

−A′_{1}(u) sin(v) + A′_{2}(u) cos(v) = \frac{r′(u)}{r(u)}[−A_{1}(u) sin(v) + A_{2}(u) cos(v)].

Using the fact that the functions cos(v) and sin(v) are linearly independent, we see that the vector valued functions A_{1}, A_{2} which depend only on u satisfy the following equations:

A′_{1}(u) = \frac{r′(u)}{r(u)}A_{1}(u); A′_{2}(u) = \frac{r′(u)}{r(u)}A_{2}(u).

Hence there are constant vectors C_{1}, C_{2} ∈ ℝ^{4} such that A_{1}(u) = r(u)C_{1}, A_{2}(u) = r(u)C_{2}. Because of < A_{1}(u), A_{1}(u) >= r^{2}(u) =< A_{2}(u), A_{2}(u) > and < A_{1}(u), A_{2}(u) >= 0, C_{1} and C_{2} are constant orthonormal vectors in ℝ^{4}.

For some vector valued function A depending only on u, the immersion x takes the form:

x(u, v) ≡ r(u) · [C_{1} cos(v) + C_{2} sin(v)] + A(u).  (35)

Since < x(u, v), x(u, v) >= 1, one has:

1 = < x(u, v), x(u, v) >
= < A(u), A(u) > +r^{2}(u) + 2r(u) [cos(v) < C_{1}, A(u) > + sin(v) < C_{2}, A(u) >] .

By the linear independence of the functions 1, cos(v) and sin(v), the vector valued function A which depends only on u, satisfies < C_{1}, A(u) >= 0 =< C_{2}, A(u) >, and the condition (26) holds.

Remark 4.2. Another way to prove Lemma 4.1 is the following: by considering α = 0, the λ_{1}-curvature lines are geodesics of M^{2}, and the λ_{2}-curvature lines are spherical bent in M^{2} (see [5]). According to a result of Hiepko (see [2]), the manifold M^{2} is then locally a warped product. The orthogonality of the vectors e_{1} and e_{2} with respect to the second fundamental form is equivalent to Nölker’s condition (D) in [4], p. 21 for an immersion of a warped product in a space of constant curvature to be a warped product of isometric immersions. So, in our situation, the case α = 0 leads to rotational surfaces.

Lemma 4.3. Let (V, <, >) be a 2-dimensional Euclidean vector space and (w_{1}, w_{2}) a basis of V. For ε ∈ {−1, 1}, let cos_{ε} and sin_{ε} be the functions as defined by (21). Then there exist an orthonormal basis (v_{1}, v_{2}) of V, s ∈ ℝ and a, b ∈ ℝ_{+} such that

∀t ∈ ℝ: cos_{ε}(t)w_{1} + sin_{ε}(t)w_{2} = a cos_{ε}(t − s)v_{1} + b sin_{ε}(t − s)v_{2}.  (36)

Proof. Let A_{ε}: ℝ ⟶ V be the curve defined by:

A_{ε}(t) := cos_{ε}(t)w_{1} + sin_{ε}(t)w_{2},

and s a point where the real function f_{ε}: t ⟼< A_{ε}(t), A_{ε}(t) > attains a minimum. One has:

0 = \frac{1}{2}f′_{ε}(s) =< A_{ε}(s), A′_{ε}(s) > .

So the vectors v_{1} := \frac{1}{∥A_{ε}(s)∥}A_{ε}(s) and v_{2} := \frac{1}{∥A′_{ε}(s)∥}A′_{ε}(s) constitute an orthonormal basis of V. By setting a := ∥A_{ε}(s)∥ and b := ∥A′_{ε}(s)∥, one verifies that the equality (36) holds.

Theorem 4.4. Let x: M^{2} ⟶ 𝕊^{3}(1) be a non-degenerate immersion without umbilical points of a connected and orientable C^{∞}-manifold M^{2} of dimension 2 into 𝕊^{3}(1). Assume that α(p) = 0, for all p ∈ M^{2}. Then the immersion x satisfies the condition C̃ ≡ 0 if and only if locally x(M^{2}) can be represented by a surface of revolution (22).

Proof. In Example (3.6) we proved that the surface (22) satisfies the condition C̃ ≡ 0. Conversely, assume that the immersion x satisfies the condition C̃ ≡ 0 with α = 0. The equation (16) implies e_{2}(λ_{2}) = 0 while ∇_{e_{1}}^{I}e_{1} = 0 from the definition of the function α in the beginning of Section 3.1. Thus, from Lemma 4.1 there are local coordinates (u, v) such that the immersion x is described by the following parametrization:

x(u, v) = r(u) · Z(v) + A(u),

where Z(v) = cos(v)C_{1} + sin(v)C_{2} with C_{1} and C_{2} constant orthonormal vectors in ℝ^{4}, and A(u) is a curve in (Span{C_{1}, C_{2}})^{⊥}. One has:

x_{u} = r′(u) · Z(v) + A′(u),  (37)
x_{v} = r(u) · Z′(v),  (38)
x_{vv} = −r(u) · Z(v),  (39)
x_{uu} = r′′(u) · Z(v) + A′′(u).  (40)

Eliminating y from the equations (30) and (32), we obtain:

λ_{2}x_{uu} − λ_{1}r^{−2}x_{vv} = −λ_{1}βx_{u} + (λ_{1} − λ_{2})x.  (41)

The substitution of (37), (39) and (40) in (41) gives:

(λ_{2}r′′ + λ_{1}βr′ + \frac{λ_{1}}{r} + (λ_{2} − λ_{1})r) · Z(v) = − (λ_{2}A′′(u) + λ_{1}βA′(u) + (λ_{2} − λ_{1})A(u)) .

Using respective derivatives of the functions β and r (see formulas (29)), one can check that

λ_{2}r′′ + λ_{1}βr′ + \frac{λ_{1}}{r} + (λ_{2} − λ_{1})r = 0.

Therefore A(u) is a solution of the following second order ordinary differential equation

A′′(u) − (−\frac{λ_{1}β}{λ_{2}} ) A′(u) + \frac{λ_{2} − λ_{1}}{λ_{2}} A(u) = 0.  (42)

Note that for α = 0 on M^{2}, by Proposition 3.3, the condition C̃ ≡ 0 is equivalent to the equation

λ′_{1} = 3λ_{1}λ_{2}^{−1}β(λ_{2} − λ_{1}).

Let ϕ be a differentiable function on M^{2} depending only on u and satisfying

ϕ′ = \sqrt{ε\frac{λ_{2} − λ_{1}}{λ_{2}}} , with ε = sign (\frac{λ_{2} − λ_{1}}{λ_{2}} ) .

One has:

\frac{ϕ′′}{ϕ′} = \frac{ε}{2ϕ′^{2}} (\frac{λ_{2}(λ′_{2} − λ′_{1}) − λ′_{2}(λ_{2} − λ_{1})}{λ_{2}^{2}})
= −\frac{βλ_{1}}{λ_{2}} ,

and then the equation (42) takes the form

A′′(u) − \frac{ϕ′′}{ϕ′} A′(u) + εϕ′^{2}A(u) = 0.

Therefore there are constant vectors C̄_{3}, C̄_{4} ∈ ℝ^{4} such that the general solution A(u) of the above equation is

A(u) = cos_{ε}(ϕ(u))C̄_{3} + sin_{ε}(ϕ(u))C̄_{4},

If the vectors C̄_{3} and C̄_{4} were not linearly independent, then there would exist a function ψ such that, up to an isometry, the immersion x would be represented by the form

x(u, v) = (r(u) cos(v), r(u) sin(v), ψ(u), 0)

which lies in a 2-dimensional great sphere, therefore it would be totally geodesic, so it would be degenerate. So the vectors C̄_{3} and C̄_{4} are linearly independent. Hence A(u) describes an arc of an ellipse or an arc of a hyperbola in the plane orthogonal to C_{1} and C_{2} if ε = 1 or ε = −1, respectively.

According to Lemma 4.3 there exist constant orthonormal vectors C_{3}, C_{4} ∈ Span{C̄_{3}, C̄_{4}}, s ∈ ℝ and a, b ∈ ℝ_{+} such that

cos_{ε}(ϕ(u))C̄_{3} + sin_{ε}(ϕ(u))C̄_{4} = a cos_{ε}(ϕ(u) − s)C_{3} + b sin_{ε}(ϕ(u) − s)C_{4}.

With the reparametrization u^{∗} = ϕ(u) − s, the immersion x takes the form of Example 3.6.

Remark 4.5. If ε = 1 and a = b in (22), then the profile curve of the surface of revolution is a circle. This case, corresponding to the situation where both functions α and β vanish everywhere, describes isoparametric surface immersions in 𝕊^{3}(1) with two distinct principal curvatures. They are non-degenerate because of 1 + λ_{1}λ_{2} = 0.

4.2. Non-rotational (C̃ ≡ 0)-surface immersions in 𝕊^{3}(1)

Assume now that the subset U_{3} is not empty (α(p) ≠ 0 ≠ β(p), for all p ∈ U_{3} ≠ ∅). By assuming that the immersion has no umbilical points, the function λ_{2} − λ_{1} has constant sign on U_{3}. Denote by η := ±1 the sign of the Gauß-Kronecker curvature function. By a suitable choice of the unit normal vector field y and of the numbering of the principal curvatures, we fix

0 < λ_{1} < λ_{2} if η = 1, or λ_{1} < 0 < λ_{2} if η = −1.

Lemma 4.6. Let x: (M^{2}, I) ⟶ 𝕊^{3}(1) ⊂ ℝ^{4} be a non-degenerate isometric immersion without umbilical points and satifying the condition C̃ ≡ 0. If there is a I-orthonormal frame (e_{1}, e_{2}) of principal vector fields such that where α and β are differentiable functions on M^{2} such that α ≠ 0 ≠ β everywhere, then for every point p ∈ M^{2} the positive functions

u := \sqrt[4]{η · \frac{λ_{2}}{λ_{1}^{3}}} and v := \sqrt[4]{η · \frac{λ_{1}}{λ_{2}^{3}}} .  (43)

define local parameters with u − ηv > 0, and there exists a polynomial

P (t) = ∏_{k=1}^{4}(t − a_{k})

with four distinct real roots a_{1} < a_{2} < a_{3} < a_{4} satisfying P (0) = −η such that the functions α and β can be expressed by

α^{2}(u, v) = \frac{η · P (ηv)}{v(u − ηv)^{3}} ,  (44)

and

β^{2}(u, v) = \frac{−P (u)}{u(u − ηv)^{3}} .  (45)

Moreover, in all cases we have 0 < a_{3} < u < a_{4} and \begin{cases}0 < a_{2} < v < a_{3} & for η = 1 \\ 0 < −a_{2} < v < −a_{1} & for η = −1 \end{cases}.

Proof. The parameters u and v as defined by (43), together with λ_{1} and λ_{2}, satisfy

\frac{λ_{1}}{λ_{2}} = η \frac{v}{u}, λ_{1} = \frac{η}{\sqrt{vu^{3}}} , λ_{2} = \frac{1}{\sqrt{uv^{3}}} and

λ_{1}λ_{2} = \frac{η}{u^{2}v^{2}} ;  (46)

and because the functions λ_{2} and λ_{2} − λ_{1} are strictly positive, one also has u − ηv > 0.

Furthermore, using (11), (12), (15), (16) to differentiate (43) we have

e_{1}(u) = −2(u − ηv)β, e_{2}(v) = 2η(u − ηv)α and e_{1}(v) = 0 = e_{2}(u);  (47)

therefore (u, v) defines a system of local curvature coordinates such that

\frac{∂}{∂u} = −\frac{1}{2(u − ηv)β} e_{1} and \frac{∂}{∂v} = \frac{η}{2(u − ηv)α}e_{2}.  (48)

Using (17) and (18), one gets the following first order partial differential equations for the function α and β:

\frac{∂α}{∂u} = \frac{−3}{2(u − ηv)} · α and \frac{∂β}{∂v} = \frac{3η}{2(u − ηv)} · β,

which actually are linear differential equations. The general solutions α ≡ α(u, v) and β ≡ β(u, v) for these equations, respectively, are

α = (\frac{1}{u − ηv} )^{\frac{3}{2}} g(v) and β = (\frac{1}{u − ηv} )^{\frac{3}{2}} f(u),  (49)

where f and g are differentiable functions, each of one variable, taking values in ℝ \ {0}.

Using (47) to compute the derivatives e_{1}(β) and e_{2}(α) in terms of u and v, one gets

e_{1}(β) = 3(u − ηv)^{−3}f^{2}(u) − (u − ηv)^{−2}(f^{2})′(u),  (50)
e_{2}(α) = 3(u − ηv)^{−3}g^{2}(v) + η(u − ηv)^{−2}(g^{2})′(v).  (51)

Inserting the equations (46), (49), (50) and (51) into the Gauß-equation (10), we get the following equation:

0 = 1 + \frac{η}{u^{2}v^{2}} − 2(u − ηv)^{−3}g̃(v) − 2(u − ηv)^{−3}f̃(u)
−η(u − ηv)^{−2}g̃′(v) + (u − ηv)^{−2}f̃′(u),

where f̃(u) := f^{2}(u) and g̃(v) := g^{2}(v). From the equation above, we have

ηg̃′(v) = \frac{(η + u^{2}v^{2})(u − ηv)^{2}}{u^{2}v^{2}} − \frac{2g̃(v)}{u − ηv} − \frac{2f̃(u)}{u − ηv} + f̃′(u).  (52)

Since g̃(v) does not depend on u, differentiating both sides of the equation (52) with respect to u, one has

0 = \frac{2(1 + u^{3}v)(u − ηv)}{vu^{3}} + \frac{2g̃(v)}{(u − ηv)^{2}} − \frac{2f̃′(u)}{u − ηv} + \frac{2f̃(u)}{(u − ηv)^{2}} + f̃′′(u).

From the equation above, one has

−2g̃(v) = \frac{2(1 + u^{3}v)(u − ηv)^{3}}{vu^{3}} + 2f̃(u) − 2(u − ηv)f̃′(u) + (u − ηv)^{2}f̃′′(u).  (53)

Once again, differentiating both sides of this last equation with respect to u, we get that the function f̃(u) satisfies the following third order ordinary differential equation:

6(η + u^{4}) + u^{4}f̃′′′(u) = 0.

Solving this differential equation, we get

f^{2}(u) = f̃(u) = \frac{−P (u)}{u}  (54)

where P is a fourth order polynomial with constant coefficients:

P (t) = ∑_{k=0}^{4} c_{k}t^{k}

such that P (0) = c_{0} = −η and c_{4} = 1. Substituting the expression (54) of P (u) into (53), we get

g^{2}(v) = g̃(v) = \frac{ηP (ηv)}{v}.

Finally we get the wanted expressions (44) and (45) for α^{2} and β^{2}, respectively. We also have

0 < −P (u) and 0 < ηP (ηv)

on the range of u and v, respectively.

Consider now the four roots a_{1}, a_{2}, a_{3}, a_{4} ∈ ℂ of the polynomial equation P (t) = 0. Following situations are possible (up to a numbering of the roots):

(i) • all the four roots are complex (not real),

• a_{1} and a_{2} are complex (not real) and a_{3} = a_{4},

• the four roots are real and equal (a_{1} = a_{2} = a_{3} = a_{4}),

• a_{1} = a_{2} ≠ a_{3} = a_{4} real or not,

(ii) • a_{1} and a_{2} are not real, and a_{3} and a_{4} are distinct real numbers,

• a_{1} = a_{2} = a_{3} ≠ a_{4} are real,

• a_{1} = a_{2} ≠ a_{3} ≠ a_{4} ≠ a_{1};

(iii) the four roots are real and distinct, let say a_{1} < a_{2} < a_{3} < a_{4}.

If the case (i) happens, the polynomial function P (t) is nowhere negative; this gives a contradiction to 0 < −P (u). And for the case (ii), the real numbers −P (t) and ηP (ηt) have distinct signs, a contradiction to the fact that f^{2}(u) > 0 and g^{2}(v) > 0. This proves that only the case (iii) can occur, i.e. the four roots are real and distinct.

Since −P (u) > 0 and ηP (ηv) > 0 everywhere, we have that

0 < a_{3} < u < a_{4} and \begin{cases}0 < a_{2} < v < a_{3} & for η = 1 \\ 0 < −a_{2} < v < −a_{1} & for η = −1 \end{cases} .

Theorem 4.7. Let x: M^{2} ⟶ 𝕊^{3}(1) be a non-degenerate immersion of a connected and orientable C^{∞}-manifold M^{2} of dimension 2 into 𝕊^{3}(1) without umbilical points. If x satisfies the condition C̃ ≡ 0 with α(p) ≠ 0 ≠ β(p) for all p ∈ M^{2}, then locally there are coordinates (u, v) such that x can be represented by a surface of the form (24).

Proof. With respect to the coordinates introduced in Lemma 4.6, because of (48), the first fundamental form of x is given by

ds^{2} = \frac{1}{4}(u − ηv) (\frac{−u}{P (u)} · du^{2} + \frac{ηv}{P (ηv)} · dv^{2}) ,  (55)

and the structure equations of x by

x_{uu} = Γ_{11}^{1}x_{u} + Γ_{11}^{2}x_{v} + II_{11}y − I_{11}x  (56)
x_{uv} = Γ_{12}^{1}x_{u} + Γ_{12}^{2}x_{v}  (57)
x_{vv} = Γ_{22}^{1}x_{u} + Γ_{22}^{2}x_{v} + II_{22}y − I_{22}x  (58)
y_{u} = −\frac{η}{\sqrt{vu^{3}}} x_{u}  (59)
y_{v} = −\frac{1}{\sqrt{uv^{3}}} x_{v},  (60)

where Γ_{ij}^{k} are Christoffel symbols associated to the first fundamental form I given by (55).

We have:

I_{11} = −\frac{u(u − ηv)}{4P (u)}, I_{22} = \frac{ηv(u − ηv)}{4P (ηv)},

II_{11} = \frac{η}{\sqrt{vu^{3}}} I_{11}, II_{22} = \frac{1}{\sqrt{uv^{3}}} I_{22},

Γ_{11}^{1} = \frac{(2u − ηv)P (u) − u(u − ηv)P′(u)}{2u(u − ηv)P (u)},

Γ_{22}^{2} = \frac{(u − 2ηv)P (ηv) − ηv(u − ηv)P′(ηv)}{2v(u − ηv)P (ηv)},

Γ_{12}^{1} = \frac{−η}{2(u − ηv)}, Γ_{12}^{2} = \frac{1}{2(u − ηv)},

Γ_{11}^{2} = \frac{−uP (ηv)}{2v(u − ηv)P (u)}, Γ_{22}^{1} = −\frac{ηvP (u)}{2u(u − ηv)P (ηv)}.

Using (57), (59) and (60) to differentiate (56) ( respectively (58)) with respect to u (respectively to v), one can see that x_{uuu} and x_{uuuu} (respectively x_{vvv} and x_{vvvv}) are combinations of x_{u}, x_{v}, x and y with coefficients depending on both variables u and v. From the expressions of x_{uu} and x_{uuu} (respectively of x_{vv} and x_{vvv}), one can express x_{v} (respectively x_{u}) and y in terms of x_{uuu}, x_{uu}, x_{u} (respectively x_{vvv}, x_{vv}, x_{v}) and x. Inserting these expressions into the expression of x_{uuuu} (respectively into x_{vvvv}), one gets the following fourth order partial differential equations:

0 = 128P (u) · x_{uuuu} + 320P′(u) · x_{uuu} + 240P′′(u) · x_{uu}  (61)
+40P′′′(u) · x_{u} − 5P^{(4)}(u) · x,
0 = 128P (ηv) · x_{vvvv} + 320P′(ηv) · x_{vvv} + 240P′′(ηv) · x_{vv}  (62)
+40P′′′(ηv) · x_{v} − 5P^{(4)}(ηv) · x.

For any fixed v, the equation (61) is a fourth order ordinary differential equation on the intervall I_{u} =]a_{3}, a_{4}[. One can check that

F_{k}(u) = \sqrt{|u − a_{k}|} (k = 1, …, 4)

are particular solutions which are linearly independent. Therefore the general solution x(u, v) is of the form

x(u, v) = ∑_{k=1}^{4} F_{k}(u)ϕ_{k}(v),  (63)

where ϕ_{k}(v) ∈ ℝ^{4}, k = 1, …, 4 are vector valued functions.

Similarly, for any fixed u, the equation (62) is a fourth order ordinary differential equation on the intervall I_{v} =]a_{2}, a_{3}[ if η = 1 or I_{v} =] − a_{2}, −a_{1}[ if η = −1, with linearly independent solutions

G_{k}(v) = \sqrt{|v − ηa_{k}|} (k = 1, …, 4).

Hence there are vector valued functions ψ_{k}(v) ∈ ℝ^{4}, k = 1, …, 4 such that

x(u, v) = ∑_{k=1}^{4} F_{k}(u)ϕ_{k}(v) = ∑_{k=1}^{4} G_{k}(v)ψ_{k}(u).

From that we derive the following linear system of 4 equations with unkowns ϕ_{k}(v), k = 1, …, 4:

\{ \begin{matrix}∑_{k=1}^{4} F_{k}(u)ϕ_{k}(v) = ∑_{i=1}^{4} G_{i}(v)ψ_{i}(u), \\ ∑_{k=1}^{4} F′_{k}(u)ϕ_{k}(v) = ∑_{i=1}^{4} G_{i}(v)ψ′_{i}(u), \\ ∑_{k=1}^{4} F′′_{k}(u)ϕ_{k}(v) = ∑_{i=1}^{4} G_{i}(v)ψ′′_{i}(u), \\ ∑_{k=1}^{4} F′′′_{k}(u)ϕ_{k}(v) = ∑_{i=1}^{4} G_{i}(v)ψ′′′_{i}(u). \end{matrix}

Because everywhere the Wronski-determinant for the functions F_{k} (k = 1, …, 4) does not vanish, there are unique functions C_{ik}(u) such that

ϕ_{k}(v) = ∑_{i=1}^{4} G_{i}(v)C_{ik}(u), k = 1, …, 4.  (64)

Differentiating the equation above with respect to u, one gets

0 = ∑_{k=1}^{4} G_{k}(v)C′_{ik}(u).

Using the linear independence of the functions G_{1}(v), G_{2}(v), G_{3}(v), G_{4}(v), we have that the functions C_{ik}(u) ≡ C_{ik}, k = 1, …, 4 are constant. Substituting (64) in (63) finally one gets

x(u, v) = ∑_{i,k} \sqrt{|u − a_{i}|}\sqrt{|v − ηa_{k}|}C_{ik}.  (65)

Inserting this expression into the equation (57), one gets

0 = ∑_{i,k} \frac{1}{\sqrt{|u − a_{i}|}\sqrt{|v − ηa_{k}|}} · (a_{k} − a_{i})C_{ik}.

By the linear independence of the functions (u, v) ⟼ \frac{1}{\sqrt{|u−a_{i}|}\sqrt{|v−ηa_{k}|}}, one deduces

(a_{k} − a_{i})C_{ik} = 0, i, k = 1, …, 4.

Therefore,

C_{ik} = 0, for i ≠ k.

Hence, (65) simplifies to

x(u, v) = ∑_{i=1}^{4} \sqrt{|u − a_{i}|}\sqrt{|v − ηa_{i}|}C_{i},

where C_{i} ∈ ℝ^{4} (i = 1, …, 4) are constant vectors in ℝ^{5}.

Since u ∈ I_{u} =]a_{3}, a_{4}[ and a_{1} < a_{2} < a_{3}, we have

F_{k}(u) = \sqrt{u − a_{k}}, k = 1, 2, 3, and F_{4}(u) = \sqrt{a_{4} − u};

and by the fact that v ∈ I_{v}, we have

G_{1}(v) = \sqrt{ηv − a_{1}}, G_{2}(v) = \sqrt{v − ηa_{2}}, G_{3}(v) = \sqrt{a_{3} − ηv}, G_{4}(v) = \sqrt{a_{4} − ηv}.

One has

1 =< x, x >= ∑_{i=1}^{4} < C_{i}, C_{i} > F_{i}^{2}(u)G_{i}^{2}(v) + 2 ∑_{i<j} < C_{i}, C_{j} > F_{i}(u)G_{i}(v)F_{j}(u)G_{j}(v).

By the linear independence of the functions 1, u, v, uv and F_{i}(u)G_{i}(v)F_{j}(u)G_{j}(v), 1 ≤ i < j ≤ 4, we get following equations:

0 = < C_{i}, C_{j} > (1 ≤ i < j ≤ 4),  (66)
0 = −a_{1} < C_{1}, C_{1} > −ηa_{2} < C_{2}, C_{2} > +a_{3} < C_{3}, C_{3} > −a_{4} < C_{4}, C_{4} >,  (67)
0 = η < C_{1}, C_{1} > + < C_{2}, C_{2} > −η < C_{3}, C_{3} > +η < C_{4}, C_{4} >,  (68)
1 = a_{1}^{2} < C_{1}, C_{1} > +ηa_{2}^{2} < C_{2}, C_{2} > −a_{3}^{2} < C_{3}, C_{3} > +a_{4}^{2} < C_{4}, C_{4} > .  (69)

From the equations (66), we have < C_{i}, C_{j} >= 0 for 1 ≤ i ≠ j ≤ 4, i.e. the vectors C_{i}, i = 1, …, 4 are orthogonal. Computing the normal vector field from the equation (56) we get

y(u, v) = \frac{1}{II_{11}} (x_{uu} − Γ_{11}^{1}x_{u} − Γ_{11}^{2}x_{v} + I_{11}x)
= −\frac{η}{\sqrt{uv}} ( ∑_{i=1}^{4} \frac{1}{a_{i}} F_{i}(u)G_{i}(v)C_{i}).

Since y is unit, we also get the following equation obtained by the same procedure as we did for x:

1 = \frac{η}{a_{1}^{2}} < C_{1}, C_{1} > +\frac{1}{a_{2}^{2}} < C_{2}, C_{2} > −\frac{η}{a_{3}^{2}} < C_{3}, C_{3} > +\frac{η}{a_{4}^{2}} < C_{4}, C_{4} > .  (70)

Solving < C_{i}, C_{i} > (i = 1, …, 4) from the equations (67), (68), (69) and (70) we get the expressions for these entities described in Example 3.7. Thus, the proof of Theorem 4.7 is complete.

Having in mind the beginning of this section we see now that only by combining Theorems 4.4 and 4.7 we get Theorem 1.1.

Remark 4.8. From the invariance of the condition C̃ ≡ 0 under polarization (see [1]), if a non-degenerate surface immersion x: M^{2} ⟶ 𝕊^{3}(1) ⊂ ℝ^{4} satisfies the condition C̃ ≡ 0, then its Gauß map y is also a (C̃ ≡ 0)-surface in 𝕊^{3}(1); for the explicit expressions of y see Example 3.6 and 3.7.

References

[1] Brito, F.; Liu, H. L.; Oliker, V.; Simon, U.; Wang, C. P.: Polar hypersurfaces in spheres. Geometry and Topology of Submanifolfds, IX, World Scientific (1999), 33–47.

[2] Hiepko, S.: Eine innere Kennzeichnung der verzerrten Produkte. Math. Ann. 241 (1979), 209–215. Zbl 0395.53021

[3] Li, A. M.; Liu, H. L.; Schwenk-Schellschmidt, A.; Simon, U.; Wang, C. P.: Cubic form

methods and relative Tchebychev hypersurfaces. Geom. Dedicata 66 (1997), 203–221.

Zbl 0878.53010

[4] Nölker, S.: Isometric immersions of warped products. Diff. Geom. and its Appl. 6 (1996), 1–30. Zbl 0881.53052

[5] Reckziegel, H.: On the eigenvalues of the shape operator of an isometric immersion into a space of constant curvature. Math. Ann. 243 (1979), 71–82. Zbl 0406.53046

[6] Simon, U.; Schwenk-Schellschmidt, A.; Viesel, H.: Introduction to the Affine Differential Geometry of Hypersurfaces. Lecture Notes, Science University of Tokyo (1991), ISBN 3 7983 1529 9. Zbl 0780.53002

Received July 24, 2001