A class of real-analytic surfaces in the 3-Euclidean space

Naoya Ando

Abstract. A smooth surface S in R^{3} is called parallel curved if there exists a plane in R^{3} parallel to a principal direction at each point of S. In this paper, the classification of the connected, complete, real-analytic, embedded, parallel curved surfaces is obtained and it is seen that the index of an isolated umbilical point on a real-analytic, parallel curved surface is equal to one or zero.

1. Introduction

Let S be a smooth surface in the 3-dimensional Euclidean space R^{3} and Π_{S} the set of the planes in R^{3} parallel to a principal direction at each point of S. If Π_{S} ≠ ∅, then S is called parallel curved. For example, a plane, a cylinder and a round sphere are parallel curved. More generally, a surface of revolution is also parallel curved. The main purposes of this paper are (1) to give the classification of the connected, complete, real-analytic, em-

bedded, parallel curved surfaces;

(2) to study the index of an isolated umbilical point(see [3, pp. 137]) on a real-analytic, parallel curved surface.

A pair {C_{1}, C_{2}} of two simple, real-analytic curves in R^{3} is called generating if the following hold:

(1) There exists a point p(C_{1}, C_{2}) ∈R^{3} satisfying C_{1} ∩ C_{2} = {p(C_{1}, C_{2})};

(2) There exist planes P_{1} and P_{2} such that (a) P_{1} and P_{2} are perpendicular to each other,

(b) for i = 1, 2, C_{i} ⊂ P_{i} holds. We shall prove the following in Section 4 of this paper.

Proposition 1.1. For a generating pair {C_{1}, C_{2}}, there exists a parallel curved surface S_{0} containing a neighborhood of p(C_{1}, C_{2}) in C_{1} ∪ C_{2} and satisfying P_{1} ∈ Π_{S_{0}}. In addition, if S′_{0} and S′′_{0} are such surfaces, then S′_{0} ∩ S′′_{0} is also such a surface.

The maximal surface of such surfaces as in Proposition 1.1 is denoted by S_{C_{1},C_{2}}. We shall prove the following in Section 6 of this paper.

Theorem 1.2. Let S be a connected, complete, real-analytic, embedded, parallel curved surface in R^{3}. Then S is homeomorphic to a sphere, a plane, a cylinder, or a torus. In addition,

(1) if S is homeomorphic to a sphere, then S is a surface of revolution which crosses its axis of rotation at just two points; (2) if S is homeomorphic to a plane, then one of the following holds:

(a) S is a surface of revolution which crosses its axis of rotation at just one point,

(b) S = S_{C_{1},C_{2}} holds, where {C_{1}, C_{2}} is a generating pair each element

of which is isometric to R;

(3) if S is homeomorphic to a cylinder, then S = S_{C_{1},C_{2}} holds, where {C_{1}, C_{2}} is a generating pair such that one of C_{1} and C_{2} is isometric to R and the other isometric to a simple closed curve;

(4) if S is homeomorphic to a torus, then S = S_{C_{1},C_{2}} holds, where {C_{1}, C_{2}} is a generating pair each element of which is isometric to a simple closed curve.

Let S be a smooth surface in R^{3} with an isolated umbilical point p_{0} and ind_{p_{0}}(S) the index of p_{0} on S. It is known that if S is a surface with constant mean curvature, then ind_{p_{0}}(S) < 0 holds([3. pp. 139]). More generally, if S is a Weingarten surface, then the same result is obtained([2]). In [1], we have obtained ind_{o}(G_{φ}) ∈ {1+i−k/2}_{i=0}^{[k/2]}, where φ is a homogeneous polynomial in two real variables of degree k ≧ 3 such that on the graph G_{φ} of φ, the origin o := (0, 0, 0) of R^{3} is an isolated umbilical point. In addition, we have proved in [1] that each number of {1 + i − k/2}_{i=0}^{[k/2]} may be the index of o on the graph of a homogeneous polynomial of degree k. In Section 7 of this paper, we shall prove the following.

Theorem 1.3. Let S be a real-analytic, parallel curved surface in R^{3} with an isolated umbilical point p_{0}. Then ind_{p_{0}}(S) is equal to one or zero. In addition,

(1) if ind_{p_{0}}(S) = 1, then the tangent plane T_{p_{0}}(S) at p_{0} is an element of Π_{S} and S is part of a surface of revolution such that p_{0} lies on the axis of rotation,

(2) if ind_{p_{0}}(S) = 0, then T_{p_{0}}(S) ∉ Π_{S} holds.

Remark 1.4. Theorem 1.3 does not give any negative answer to Loewner’s conjecture([4]).

2. Preliminaries

Let f be a smooth function of two real variables x, y and G_{f} the graph of f. We may consider the natural coordinates (x, y) on the xy-plane as

coordinates on G_{f}. We set p_{f} := \frac{∂f}{∂x}, q_{f} := \frac{∂f}{∂y} , and

E_{f} := 1 + p_{f}^{2},  F_{f} := p_{f}q_{f},  G_{f} := 1 + q_{f}^{2}.

The first fundamental form of G_{f} is the symmetric tensor field I_{f} on G_{f} of type (0, 2) represented in terms of the coordinates (x, y) as

I_{f} := E_{f}dx^{2} + 2F_{f}dxdy + G_{f}dy^{2},

where

dx^{2} := dx ⊗ dx, dxdy := \frac{1}{2}(dx ⊗ dy + dy ⊗ dx), dy^{2} := dy ⊗ dy.

We set r_{f} := \frac{∂^{2}f}{∂x^{2}} , s_{f} := \frac{∂^{2}f}{∂x∂y} , t_{f} := \frac{∂^{2}f}{∂y^{2}} and

L_{f} := \frac{r_{f}}{\sqrt{det(I_{f})}}, M_{f} := \frac{s_{f}}{\sqrt{det(I_{f})}}, N_{f} := \frac{t_{f}}{\sqrt{det(I_{f})}},

where det(I_{f}) := E_{f}G_{f} − F_{f}^{2}. The second fundamental form of G_{f} is the symmetric tensor field II_{f} on G_{f} of type (0, 2) represented in terms of the coordinates (x, y) as

II_{f} := L_{f}dx^{2} + 2M_{f}dxdy + N_{f}dy^{2}.

Let T_{(x_{0},y_{0})}(G_{f}) be the tangent plane to G_{f} at (x_{0}, y_{0}) and U_{(x_{0},y_{0})}(G_{f}) the subset of T_{(x_{0},y_{0})}(G_{f}) defined by

U_{(x_{0},y_{0})}(G_{f}) := {u ∈ T_{(x_{0},y_{0})}(G_{f}) ; I_{f,(x_{0},y_{0})}(u, u) = 1},

and L_{(x_{0},y_{0})}(G_{f}) the set of the one-dimensional subspaces of T_{(x_{0},y_{0})}(G_{f}). Let ν_{f,(x_{0},y_{0})} be the function on U_{(x_{0},y_{0})}(G_{f}) defined by

ν_{f,(x_{0},y_{0})}(u) := II_{f,(x_{0},y_{0})}(u, u)

for any u∈ U_{(x_{0},y_{0})}(G_{f}). Suppose that ν_{f,(x_{0},y_{0})} attains an extremum at u_{0} ∈ U_{(x_{0},y_{0})}(G_{f}). Then the extremum ν_{f,(x_{0},y_{0})}(u_{0}) is called a principal curvature of G_{f} at (x_{0}, y_{0}) and the element of L_{(x_{0},y_{0})}(G_{f}) determined by u_{0} is called a principal direction of G_{f} at (x_{0}, y_{0}). The set of the principal directions at (x_{0}, y_{0}) is represented by P D_{(x_{0},y_{0})}(G_{f}). The Weingarten map of G_{f} is the tensor field W_{f} on G_{f} of type (1, 1) satisfying

[W_{f}(\frac{∂}{∂x}), W_{f}(\frac{∂}{∂y} )] = [\frac{∂}{∂x}, \frac{∂}{∂y} ]W_{f},

where

W_{f} := (\begin{matrix}E_{f} & F_{f} \\ F_{f} & G_{f} \end{matrix})^{−1} (\begin{matrix}L_{f} & M_{f} \\ M_{f} & N_{f} \end{matrix}) .

By Lagrange’s method of indeterminate coefficients, we obtain

Proposition 2.1. The principal curvatures and the principal directions of G_{f} at (x_{0}, y_{0}) are given by the eigenvalues and the eigenspaces of W_{f,(x_{0},y_{0})}, respectively.

Gaussian curvature K_{f}(x_{0}, y_{0}) of G_{f} at (x_{0}, y_{0}) is given by K_{f}(x_{0}, y_{0}) := det (W_{f,(x_{0},y_{0})}). We obtain

K_{f} = \frac{det(II_{f})}{det(I_{f})},

where det(II_{f}) := L_{f}N_{f} − M_{f}^{2}.

Let PD_{f} be the symmetric tensor field on G_{f} of type (0, 2) represented in terms of the coordinates (x, y) as

PD_{f} := \frac{1}{\sqrt{det(I_{f})}}\{A_{f}dx^{2} + B_{f}dxdy + C_{f}dy^{2}\},

where

A_{f} := F_{f}L_{f} − E_{f}M_{f}, B_{f} := G_{f}L_{f} − E_{f}N_{f}, C_{f} := G_{f}M_{f} − F_{f}N_{f}.

For v_{1} , v_{2} ∈ T_{(x_{0},y_{0})}(G_{f}), the following holds:

\frac{1}{2}\{W_{f,(x_{0},y_{0})}(v_{1}) ∧ v_{2} + W_{f,(x_{0},y_{0})}(v_{2}) ∧ v_{1}\}

= PD_{f,(x_{0},y_{0})}(v_{1}, v_{2})\frac{1}{\sqrt{det (I_{f,(x_{0},y_{0})})}}(\frac{∂}{∂x} ∧ \frac{∂}{∂y} )_{(x_{0},y_{0})}.

Therefore from Proposition2.1, we obtain

Proposition 2.2. A tangent vector v_{0} ∈ T_{(x_{0},y_{0})}(G_{f}) is in an element of P D_{(x_{0},y_{0})}(G_{f}) if and only if PD_{f,(x_{0},y_{0})}(v_{0}, v_{0}) = 0 holds.

A point (x_{0}, y_{0}) is called an umbilical point of G_{f} if ν_{f,(x_{0},y_{0})} is a constant function. By Proposition 2.2, we obtain

Proposition 2.3. The following are mutually equivalent:

(1) (x_{0}, y_{0}) is an umbilical point of G_{f},

(2) P D_{(x_{0},y_{0})}(G_{f}) = L_{(x_{0},y_{0})}(G_{f}),

(3) A_{f}(x_{0}, y_{0}) = B_{f}(x_{0}, y_{0}) = C_{f}(x_{0}, y_{0}) = 0.

3. A parallel curved surface, I

Let Pl(R^{3}) be the set of the planes in R^{3} and L(R^{3}) the set of the lines in R^{3}. For X_{1}, X_{2} ∈Pl(R^{3})⊔L(R^{3}), X_{1} ∥ X_{2} means that X_{1} and X_{2} are parallel in R^{3} to each other, and X_{1} ⊥ X_{2} means that X_{1} and X_{2} are perpendicular in R^{3} to each other. Let S be a connected, real-analytic, embedded, parallel curved surface in R^{3} and Π_{S} the set of the elements of P l(R^{3}) parallel to a principal direction at each point of S. For P ∈ Π_{S}, we set

Ξ_{S,P} := {p ∈ S ; T_{p}(S) ∦ P }.

Suppose that there exists an element P_{0} of Π_{S} satisfying Ξ_{S,P_{0}} = ∅. Then we see that S is part of an element of Pl(R^{3}). In the following, suppose that for any P ∈ Π_{S}, Ξ_{S,P} ≠ ∅ holds. Then Ξ_{S,P} is an open set of S and therefore Ξ_{S,P} is also a surface in R^{3}.

For P ∈ Π_{S} and for p ∈ Ξ_{S,P}, let P_{p}^{⊥}(S, P) be the element of P l(R^{3}) satisfying p ∈ P_{p}^{⊥}(S, P), P_{p}^{⊥}(S, P) ⊥ P and P_{p}^{⊥}(S, P) ⊥ T_{p}(Ξ_{S,P}), and l_{p}(S, P) the connected component of Ξ_{S,P} ∩ P_{p}^{⊥}(S, P) containing p. We shall prove

Proposition 3.1. The following hold:

(1) The set l_{p}(S, P) is a real-analytic curve;

(2) For any q ∈ l_{p}(S, P), the tangent line T_{q}(l_{p}(S, P)) to l_{p}(S, P) at q satisfies T_{q}(l_{p}(S, P)) ∈ P D_{q}(Ξ_{S,P}) and T_{q}(l_{p}(S, P)) ∦ P;

(3) For q ∈ l_{p}(S, P), let T_{q}^{⊥}(l_{p}(S, P)) be the element of P D_{q}(Ξ_{S,P}) satisfying T_{q}^{⊥}(l_{p}(S, P)) ∥ P. Then for q_{1}, q_{2} ∈ l_{p}(S, P), T_{q_{1}}^{⊥}(l_{p}(S, P)) ∥ T_{q_{2}}^{⊥}(l_{p}(S, P)) holds.

Proof. For orthogonal coordinates (x, y, z) on R^{3}, the x-axis(resp. the yaxis, the z-axis) and the xy-plane(resp. the yz-plane, the xz-plane) are denoted by A_{x}(resp. A_{y}, A_{z}) and by P_{xy}(resp. P_{yz}, P_{xz}), respectively. Suppose that (x, y, z) satisfies p = o, A_{x} ⊂ T_{o}(S) and P_{xz} ∥ P. Then we obtain A_{z} ∩T_{o}(S) = {o}. Therefore there exist two positive numbers x_{0}, y_{0} > 0 and a real-analytic function f on a neighborhood U_{x_{0},y_{0}} := (−x_{0}, x_{0}) × (−y_{0}, y_{0}) of o in P_{xy} such that G_{f} is a neighborhood of o in Ξ_{S,P}. The function f satisfies f(0, 0) = p_{f}(0, 0) = 0. We see that at each point of G_{f}, the tangent

vector \frac{∂}{∂x} is in a principal direction. Then from Proposition 2.2 we obtain

s_{f}(1 + p_{f}^{2}) = p_{f}q_{f}r_{f}  (1)

on U_{x_{0},y_{0}}. We may represent f as

f(x, y) := ∑_{i,j=0}^{∞} a_{ij}x^{i}y^{j},  (2)

where a_{ij} ∈R and where a_{00} = a_{10} = 0. Then at (0, y) ∈ U_{x_{0},y_{0}}, we may rewrite (1) into

\begin{matrix}( ∑_{j=0}^{∞}(j + 1)a_{1j+1}y^{j}) × (1 + ( ∑_{j=0}^{∞} a_{1j}y^{j})^{2}) \\ = 2( ∑_{j=0}^{∞} a_{1j}y^{j}) × ( ∑_{j=0}^{∞}(j + 1)a_{0j+1}y^{j}) × ( ∑_{j=0}^{∞} a_{2j}y^{j}). \end{matrix}  (3)

Since a_{10} = 0, we obtain a_{11} = 0. Generally, by (3) we see that if each element of {a_{1k}}_{k=0}^{j−1} for a natural number j ∈N is equal to zero, then a_{1j} is also equal to zero. Therefore we obtain a_{1j} = 0 for any j ∈N∪{0}. Then for any y ∈ (−y_{0}, y_{0}), p_{f}(0, y) = 0 holds. This implies T_{(0,y)}(G_{f}) ∥ A_{x}. Hence we obtain (3) of Proposition 3.1. Noticing P_{o}^{⊥}(S, P) = P_{yz}, we see that for any y ∈ (−y_{0}, y_{0}), T_{(0,y)}(G_{f}) ⊥ P_{o}^{⊥}(S, P) holds. Therefore we obtain (1) of Proposition 3.1. Since p_{f}(0, y) = s_{f}(0, y) = 0 hold for any y ∈ (−y_{0}, y_{0}),

we see by Proposition 2.2 that (\frac{∂}{∂y} )_{(0,y)} is in an element of P D_{(0,y)}(G_{f}).

Then we obtain (2) of Proposition 3.1. □

By (2) of Proposition 3.1, we obtain Corollary 3.2. Let f be a real-analytic function on an open set U of R^{2} such that G_{f} is a parallel curved surface satisfying P_{xy} ∈ Π_{G_{f}}. Then A_{f} = −C_{f} holds at each point of U.

4. A generating pair

Let P be a plane in R^{3} and Γ_{P} the set of the simple, real-analytic curves in P, and for C ∈ Γ_{P} and for a point p ∈ C, let L_{p}^{⊥}(C) be the line in P normal to C at p. Then for C, C̃ ∈ Γ_{P}, we write C ∼ C̃ if and only if there exists a continuous bijective map α_{C,C̃} from C onto C̃ satisfying L_{α_{C,C̃}(p)}^{⊥}(C̃) = L_{p}^{⊥}(C) for any p ∈ C. It is seen that ∼ is an equivalence relation in Γ_{P}. We denote by Γ_{C} the equivalence class of C ∈ Γ_{P}, and by Reg(C) the connected component of the set ⋃_{C′∈Γ_{C}} C′ containing C. We immediately obtain

Lemma 4.1. For C ∈ Γ_{P} and for each p ∈ C, there exists a neighborhood U_{p} of p in C such that Reg(U_{p}) is a domain of P .

We denote by Proj_{P} the map from R^{3} onto P such that if l ∈L(R^{3}) satisfies l ⊥ P, then Proj_{P} (l) consists of the only one point of l ∩ P.

Proof of Proposition 1.1 . Let {C_{1}, C_{2}} be a generating pair and P_{1}, P_{2} planes satisfying P_{1} ⊥ P_{2} and C_{i} ⊂ P_{i} for i = 1, 2, and P_{3} the plane satisfying P_{1} ⊥ P_{3}, P_{2} ⊥ P_{3} and p(C_{1}, C_{2}) ∈ P_{3}. Then by Lemma 4.1, we see that for i = 1, 2, there exists a neighborhood U_{i} of p(C_{1}, C_{2}) in C_{i} such that

(1) Proj_{P_{1}}(U_{2}) ⊂ Reg(U_{1}) holds,

(2) Proj_{P_{3}} embeds each connected component of U_{2} \ {p(C_{1}, C_{2})} into P_{3}.

Then there exists the only one surface S_{0} such that if P ∈Pl(R^{3}) satisfies P ∥ P_{1} and P ∩U_{2} ≠ ∅, then each connected component of Proj_{P_{1}}(S_{0} ∩P) is an element of Γ_{U_{1}}. For each p ∈ S_{0}\U_{1}, let (x, y, z) be orthogonal coordinates on R^{3} satisfying p = o, A_{x} ⊂ T_{o}(S_{0}) and P_{xz} ∥ P_{1}. Then referring to the proof of Proposition 3.1, we obtain A_{x} ⊂ P D_{o}(S_{0}). Therefore we see that S_{0} is a parallel curved surface containing U_{1} ∪ U_{2} and satisfying P_{1} ∈ Π_{S_{0}}. It is clear that if S′_{0} and S′′_{0} are parallel curved surfaces containing a neighborhood of p(C_{1}, C_{2}) in C_{1} ∪ C_{2} and satisfying P_{1} ∈ Π_{S_{0}}, then S′_{0} ∩ S′′_{0} is also such a surface. Hence we have proved Proposition 1.1. □

5. A parallel curved surface, II

Suppose that for P ∈ Π_{S}, there exists a point p_{0} of S \Ξ_{S,P}. The purpose of this section is to prove the following two propositions.

Proposition 5.1. Suppose that T_{p_{0}}(S) ∩ V_{p_{0}} ≠ {p_{0}} holds for any neighborhood V_{p_{0}} of p_{0} in S. Then the following hold:

(1) There exist a neighborhood U_{p_{0}} of p_{0} in S and a real-analytic curve C_{p_{0}} in U_{p_{0}} satisfying T_{p_{0}}(S) ∩ U_{p_{0}} = C_{p_{0}};

(2) For p ∈ C_{p_{0}}, let P_{p}^{⊥}(C_{p_{0}}) be the plane normal to C_{p_{0}} at p. Then there exists a neighborhood U_{p} of p in S such that

(a) P_{p}^{⊥}(C_{p_{0}}) ∩ U_{p} is a real-analytic curve,

(b) for each point q of P_{p}^{⊥}(C_{p_{0}}) ∩ U_{p}, T_{q}(S) ⊥ P_{p}^{⊥}(C_{p_{0}}) holds.

Proposition 5.2. If for a neighborhood V_{p_{0}} of p_{0} in S, T_{p_{0}}(S)∩V_{p_{0}} = {p_{0}} holds, then S is part of a surface of revolution such that p_{0} lies on its axis of rotation.

Let (x, y, z) be orthogonal coordinates in R^{3} satisfying p_{0} = o and T_{o}(S) = P_{xy}. Then there exist two positive numbers x_{0}, y_{0} > 0 and a real-analytic function f on a neighborhood U_{x_{0},y_{0}} of o in P_{xy} such that G_{f} is a neighborhood of o in S. At each point of U_{x_{0},y_{0}}, the following holds:

PD_{f}(Grad_{f}^{⊥}, Grad_{f}^{⊥}) = 0,  (4)

where Grad_{f}^{⊥} is the vector field on G_{f} defined by Grad_{f}^{⊥} := −q_{f} \frac{∂}{∂x} +p_{f} \frac{∂}{∂y}.

We set

Hess_{f} := (\begin{matrix}r_{f} & s_{f} \\ s_{f} & t_{f} \end{matrix}) , grad_{f} := ( \begin{matrix}p_{f} \\ q_{f} \end{matrix} ) , grad_{f}^{⊥} := ( \begin{matrix}−q_{f} \\ p_{f} \end{matrix} ) .

Let ⟨ , ⟩ be the scalar product in R^{2}. Then we see that (4) holds at each point if and only if f is a solution of the following differential equation:

⟨ Hess_{f} grad_{f}, grad_{f}^{⊥} ⟩ = 0.  (5)

For any ϕ ∈R, we set u_{ϕ}(x, y) := (cos ϕ)x+(sin ϕ)y. A function f is said to be of one-variable if there exist a number ϕ_{0} ∈R and a real-analytic function F_{f}^{(1)} on a neighborhood of 0 in R such that f(x, y) = F_{f}^{(1)}(u_{ϕ_{0}}(x, y)) holds on U_{x_{0},y_{0}}. A function f is said to be radial if there exists a real-analytic function F_{f}^{(2)} on a neighborhood of 0 in R such that f(x, y) = F_{f}^{(2)}(r^{2}) holds on U_{x_{0},y_{0}}, where r := \sqrt{x^{2} + y^{2}}. We shall prove

Lemma 5.3. Let φ be a homogeneous polynomial. Then φ is a solution of equation (5) if and only if φ is of one-variable or radial.

Proof. For a homogeneous polynomial φ, the following holds:

⟨ Hess_{φ} grad_{φ}, grad_{φ}^{⊥} ⟩ = \frac{det(Hess_{φ})}{deg(φ) − 1}(xq_{φ} − yp_{φ}).

Since det(Hess_{φ}) ≡ 0(resp. xq_{φ} ≡ yp_{φ}) holds if and only if φ is of onevariable(resp. radial), we obtain Lemma 5.3. □

For homogeneous polynomials φ_{1}, φ_{2}, φ_{3}, we set

F_{φ_{1},φ_{2},φ_{3}} := ⟨ Hess_{φ_{1}} grad_{φ_{2}}, grad_{φ_{3}}^{⊥} ⟩,
Φ_{φ_{1},φ_{2},φ_{3}} := ∑_{{j_{1},j_{2},j_{3}}={1,2,3}} F_{φ_{j_{1}},φ_{j_{2}},φ_{j_{3}}} .

We shall prove

Lemma 5.4. Suppose that φ_{1} and φ_{2} are radial. Then φ_{3} is also radial if and only if Φ_{φ_{1},φ_{2},φ_{3}} ≡ 0 holds.

Proof. If we set φ_{j} = r^{2k_{j}} for j = 1, 2, then we obtain

F_{φ_{j_{1}},φ_{j_{2}},φ_{3}} = −4k_{1}k_{2}(2k_{j_{1}} − 1)r^{2(k_{1}+k_{2}−2)}(xq_{φ_{3}} − yp_{φ_{3}}),
F_{φ_{j_{1}},φ_{3},φ_{j_{2}}} = 4k_{1}k_{2}r^{2(k_{1}+k_{2}−2)}(xq_{φ_{3}} − yp_{φ_{3}}),
F_{φ_{3},φ_{j_{1}},φ_{j_{2}}} = 4k_{1}k_{2}(deg(φ_{3}) − 1)r^{2(k_{1}+k_{2}−2)}(xq_{φ_{3}} − yp_{φ_{3}}),

where {j_{1}, j_{2}} = {1, 2}. Therefore we see that Φ_{φ_{1},φ_{2},φ_{3}} ≡ 0 holds if and only if φ_{3} is radial. □

We may represent f as f(x, y) := ∑_{i≧2} f^{(i)}(x, y), where

f^{(i)}(x, y) := \frac{1}{i!} ∑_{j=0}^{i}_{i}C_{j} \frac{∂^{i}f}{∂x^{j}∂y^{i−j}} (0, 0)x^{j}y^{i−j}.

We call f^{(i)} the i-th component of f. We set

I_{f} := {i_{0} ∈ N ; f^{(i_{0})} ≠ 0}, deg(f) := min I_{f}.

Then we may represent ⟨ Hess_{f} grad_{f}, grad_{f}^{⊥} ⟩ as

⟨ Hess_{f} grad_{f}, grad_{f}^{⊥} ⟩ = ∑_{j_{1},j_{2},j_{3}∈I_{f}} F_{f^{(j_{1})},f^{(j_{2})},f^{(j_{3})}}.

We see that the (3 deg(f) − 4)-th component of ⟨ Hess_{f} grad_{f}, grad_{f}^{⊥} ⟩ is given by F_{f^{(deg(f))},f^{(deg(f))},f^{(deg(f))}}. Therefore by Lemma 5.3, we see that if f satisfies (5), then f^{(deg(f))} is of one-variable or radial. If I_{f} = {deg(f)}, then we obtain Proposition 5.1 and Proposition 5.2.

Suppose that I_{f} ≠ {deg(f)} and that I_{f} is a finite set. Then set n := ♯I_{f} and let i_{1}, …, i_{n} be the integers satisfying i_{1} < ⋯ < i_{n} and I_{f} = {i_{j}}_{j=1}^{n}. If f^{(i_{1})}, …, f^{(i_{j})} are radial for j ∈ {1, …, n − 1}, then we see by Lemma 5.4 that f^{(i_{j+1})} is also radial. Therefore we see that if f^{(deg(f))} is radial, then f is also radial. If I_{f} is an infinite set, then we obtain the same result. Hence if f^{(deg(f))} is radial, then T_{o}(S) ∩ G_{f} = {o} holds and S is part of a surfece of revolution such that o lies on its axis of rotation.

Suppose that I_{f} ≠ {deg(f)} and that f^{(deg(f))} is of one-variable. Then we may suppose that f^{(deg(f))}(x, y) = x^{deg(f)}. We shall prove Lemma 5.5. Suppose that f is not of one-variable. Then for any (u_{0}, v_{0}) ∈ G_{f}, there exists the only one element P_{(u_{0},v_{0})} ∈ P l(R^{3}) such that

(1) (u_{0}, v_{0}) ∈ P_{(u_{0},v_{0})},

(2) P_{(u_{0},v_{0})} ⊥ P_{xy},

(3) for each point (u, v) ∈ of P_{(u_{0},v_{0})} ∩ G_{f}, P_{(u_{0},v_{0})} ⊥ T_{(u,v)}(G_{f}) holds.

Proof. We represent f as in (2). Then by (5), we obtain a_{n1} = 0 for any n ∈N∪{0}. Therefore we see that for (0, 0) ∈ U_{x_{0},y_{0}}, P_{xz} satisfies (1) ∼ (3) in Lemma 5.5. Then noticing Proposition 3.1, we see that for any (u_{0}, v_{0}) ∈ U_{x_{0},y_{0}}, there exists an element P ∈ P l(R^{3}) satisfying (1) ∼ (3) in Lemma 5.5. In addition, we see that at any point (u_{0}, v_{0}) ∈ G_{f} ∩ Ξ_{S,P}, P_{(u_{0},v_{0})}^{⊥}(G_{f}, P_{xy}) is the only one element of P l(R^{3}) satisfying (1) ∼ (3) in Lemma 5.5. Therefore we have only to show that P_{xz} is the only one element of P l(R^{3}) satisfying (1) ∼ (3) in Lemma 5.5 at (0, 0). We easily see that for any ϕ ∈ (−π/2, π/2)\{0}, the element of P l(R^{3}) determined by u_{ϕ}(x, y) = 0 does not satisfy (1) ∼ (3) in Lemma 5.5 at (0, 0). Suppose that P_{yz} satisfies (1) ∼ (3) in Lemma 5.5 at (0, 0). Then for any (u_{0}, v_{0}) ∈ G_{f} ∩Ξ_{S,P}, P_{(u_{0},v_{0})}∩ G_{f} intersects G_{f} ∩ P_{xz} or G_{f} ∩ P_{yz} orthogonally. Therefore P_{(u_{0},v_{0})} ∥ P_{xz} or

P_{(u_{0},v_{0})} ∥ P_{yz} holds and each of \frac{∂}{∂x} and \frac{∂}{∂y} is in a principal direction at each

point of G_{f}. Then we see that f is of one-variable. Hence we have proved Lemma 5.5. □

We may suppose that for any u_{0} ∈ (−x_{0}, x_{0}) \ {0}, (u_{0}, 0) ∈ Ξ_{S,P} holds. Then by Lemma 5.5, we see that for any u_{0} ∈ (−x_{0}, x_{0}) \ {0}, there exists a real-analytic curve C_{u_{0}} on G_{f} such that

(1) (u_{0}, 0) ∈ C_{u_{0}},

(2) C_{u_{0}} is contained in the plane determined by z = f(u_{0}, 0),

(3) Reg ( Proj_{P_{xy}}(C_{u_{0}})) is a neighborhood of o in P_{xy}.

Therefore we see that V_{o} ∩ P_{xy} ≠ {o} holds for any neighborhood V_{o} of o in S and that at o, (1) and (2) of Proposition 5.1 hold. Hence we have proved Proposition 5.1 and Proposition 5.2.

By Proposition 3.1, Proposition 5.1 and by Proposition 5.2, we obtain

Proposition 5.6. For a connected, real-analytic, embedded, parallel curved surface S in R^{3}, there exists a generating pair {C_{1}, C_{2}} satisfying S ⊂ S_{C_{1},C_{2}}.

6. Classification

In this section, let S be complete.

Suppose that there exists an element P_{0} of Π_{S} satisfying Ξ_{S,P_{0}} = S. Then from Proposition 3.1, we see that for any p ∈ S, l_{p}(S, P_{0}) is isometric to R. For any p ∈ S, there exists the element P ∈ Π_{S} satisfying p ∈ P and P ∥ P_{0}. Then S∩P is isometric to R or to a simple closed curve. Therefore we obtain Proposition 6.1. Let S be a connected, complete, real-analytic, embedded, parallel curved surface in R^{3} satisfying Ξ_{S,P_{0}} = S for some P_{0} ∈ Π_{S}. Then there exists a generating pair {C_{1}, C_{2}} such that

(1) P_{1} ∥ P_{0} holds,

(2) C_{1} is isometric to R or to a simple closed curve,

(3) C_{2} is isometric to R,

(4) S = S_{C_{1},C_{2}}. Then S is homeomorphic to a plane or to a cylinder.

Suppose that Ξ_{S,P} ≠ S and Ξ_{S,P} ≠ ∅ hold for any P ∈ Π_{S}. Then for any P ∈ Π_{S} and for any p ∈ Ξ_{S,P}, we see from Proposition 3.1, Proposition 5.1 and from Proposition 5.2 that the connected component of P_{p}^{⊥}(S, P)∩S containing p is a real-analytic curve isometric to R or to a simple closed curve. There exists the element P′ of Pl(R^{3}) satisfying p ∈ P′ and P′ ∥ P. Then by Proposition 5.1, we see that the connected component of S ∩ P′ containing p is a real-analytic curve isometric to R or to a simple closed curve.

Suppose that P_{p}^{⊥}(S, P) ∩ S and S ∩ P′ cross at two points q_{1}, q_{2}. Then we may suppose that on the domain I_{P_{p}^{⊥}(S,P)∩S}(resp. I_{S∩P′}) of P_{p}^{⊥}(S, P) ∩ S(resp. S ∩ P′) determined by q_{1} and q_{2}, there exists no point of P_{p}^{⊥}(S, P) ∩ S ∩ P′. Then by Proposition 3.1, we see that there exists the only one point p_{0} of I_{P_{p}^{⊥}(S,P)∩S} such that for any q ∈ I_{S∩P′}, P_{q}^{⊥}(S, P) ∩ I_{P_{p}^{⊥}(S,P)∩S} = {p_{0}} holds. Then p_{0} ∉ Ξ_{S,P} holds. By Proposition 5.1 and by Proposition 5.2, we see that S is a surface of revolution and that p_{0} lies on its axis of rotation. Therefore we obtain

Proposition 6.2. Let S be a connected, complete, real-analytic, embedded, parallel curved surface satisfying Ξ_{S,P} ≠ S and Ξ_{S,P} ≠ ∅ for any P ∈ Π_{S}. Then one of the following holds:

(1) S is a surface of revolution with an axis of rotation A_{S} satisfying ♯S ∩ A_{S} ∈ {1, 2}. Then S is homeomorphic to a plane or to a sphere,

(2) There exists a generating pair {C_{1}, C_{2}} such that (a) for i = 1, 2, C_{i} is isometric to R or to a simple closed curve,

(b) S = S_{C_{1},C_{2}}.

Then S is homeomorphic to a plane, a cylinder or to a torus.

By Proposition 6.1 and by Proposition 6.2, we may construct Theorem 1.2.

7. An isolated umbilical point of a parallel curved surface

Firstly, we shall give examples of parallel curved surfaces with an isolated umbilical point.

Example 7.1. Let f be a real-analytic, radial function on a neighborhood of o in R^{2}. Then the graph G_{f} of f is a parallel curved surface satisfying P_{xy} ∈ Π_{G_{f}}. If G_{f} is not part of a round sphere, then o is an isolated umbilical point of G_{f} satisfying ind_{o}(G_{f}) = 1. Therefore Theorem 1.3 says that any parallel curved surface with an isolated umbilical point of index one is given by such a function as f.

Example 7.2. Let C_{1} be the curve in P_{xz} defined by z = x^{4} and C_{2} the curve in P_{yz} defined by z = −y^{4}. Then {C_{1}, C_{2}} is a generating pair and S_{C_{1},C_{2}} is homeomorphic to a plane. For each p ∈ S_{C_{1},C_{2}}, we see that each of the closed half spaces of R^{3} determined by T_{p}(S_{C_{1},C_{2}}) does not contain any neighborhood of p in S_{C_{1},C_{2}}. Therefore Gaussian curvature is non-positive on S_{C_{1},C_{2}}. Let f be the real-analytic function on P_{xy} satisfying G_{f} = S_{C_{1},C_{2}}. Then we see that deg(f) = 4 holds and that there exists a real number a_{22} satisfying f^{(4)}(x, y) = x^{4} + a_{22}x^{2}y^{2} − y^{4}. In addition, noticing K_{f} ≦ 0 on P_{xy}, we obtain a_{22} = 0. Then we see that there exists a neighborhood U_{o} of o in P_{xy} such that at each point of U_{o} \ {o}, B_{f} ≠ 0 holds. Therefore we see that o is an isolated umbilical point of S_{C_{1},C_{2}}. We immediately obtain ind_{o}(S_{C_{1},C_{2}}) = 0 and T_{o}(S_{C_{1},C_{2}}) ∉ Π_{S_{C_{1},C_{2}}} .

Proof of Theorem 1.3. Let S be a real-analytic, parallel curved surface in R^{3} with an isolated umbilical point p_{0}. Suppose that T_{p_{0}}(S) ∈ Π_{S} holds. Let (x, y, z) be orthogonal coordinates on R^{3} satisfying p_{0} = o and T_{o}(S) = P_{xy}, and f a real-analytic function on a neighborhood U_{o} of o in P_{xy} such that G_{f} is a neighborhood of o in S. In Section 5 we have shown that f^{(deg(f))} is of one-variable or radial. If f^{(deg(f))} is radial, then Proposition 5.2 says that S is part of a surface of revolution such that o lies on the axis of rotation. Then ind_{o}(S) = 1 holds. Suppose that f^{(deg(f))} is of one-variable. Then we may suppose that f^{(deg(f))}(x, y) = x^{deg(f)} holds. In addition, noticing Proposition 3.1, Corollary 3.2 and Proposition 5.1, we may suppose that B_{f} ≠ 0 holds on U_{o} \ {o}. We represent f as in (2). Suppose that a_{0n} = 0 holds for any n ∈N∪{0}. Then noticing Proposition 5.1, we may suppose that any zero point of f on U_{o} lies on A_{y}. Then f is of one-variable and o is not an isolated umbilical point. Suppose that a_{0n} ≠ 0 holds for some n ∈N∪{0}. The minimum of such integers as n is denoted by n_{0}. Then n_{0} ≧ 3 holds. Since B_{f} ≠ 0 holds on U_{o} \ {o}, we see that deg(f) and n_{0} are even and that a_{0n_{0}} < 0 holds. However these contradict Proposition 5.1. Therfore f^{(deg(f))} is not of one-variable. Hence if T_{p_{0}}(S) ∈ Π_{S}, then ind_{p_{0}}(S) = 1 holds and S is part of a surface of revolution such that p_{0} lies on the axis of rotation.

If p_{0} ∈ Ξ_{S,P} holds for any P ∈ Π_{S}, then by Proposition 3.1 we obtain ind_{p_{0}}(S) = 0.

Hence we have proved Theorem 1.3. □

Acknowledgement. The author is a research fellow of the Japan Society for the Promotion of Science.

References

1. N.Ando, An isolated umbilical point of the graph of a homogeneous polynomial, to appear in Geom.Dedicata.

2. P.Hartman and A.Wintner, Umbilical points and W-surfaces, Amer. J. Math., 76(1954) 502-508.

3. H.Hopf, Lectures on differential geometry in the large, Notes by J. M. Gray, Stanford Univ., 1954. Reprinted in Lecture Notes in Math.,vol.1000, Springer-Verlag.

4. T.Klotz, On Bol’s proof of Carathéodory’s conjecture, Comm. Pure Appl. Math., 12(1959) 277-311.

Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. E-mail address: naoya@mpim-bonn.mpg.de

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguroku, Tokyo 153-8914, Japan. E-mail address: andou@ms.u-tokyo.ac.jp