On four dimensional Dupin hypersurfaces in Euclidean space

On four dimensional Dupin hypersurfaces in Euclidean space

CARLOS M. C. RIVEROS; KETI TENENBLAT

Member of Academia Brasileira de Ciências

Departamento de Matemática, IE.UNB, 70910-900 Brasília, Brazil

^{Correspondence}

ABSTRACT

Dupin hypersurfaces in five dimensional Euclidean space parametrized by lines of curvature, with four distinct principal curvatures, are considered. A generic family of such hypersurfaces is locally characterized in terms of the principal curvatures and four vector valued functions of one variable. These functions are invariant by inversions and homotheties.

Key words: Dupin hypersurfaces, Laplace invariants, lines of curvature, principal curvatures.

RESUMO

Consideramos hipersuperfícies de Dupin no espaço Euclideano de dimensão cinco, parametrizadas por linhas de curvaturas, com quatro curvaturas principais distintas. Uma família genérica de tais hipersuperfícies é localmente caracterizada em termos das curvaturas principais e quatro funções vetoriais de uma variável. Essas funções são invariantes por inversões e homotetias.

Palavras-chave: hipersuperfícies de Dupin, invariantes de Laplace, linhas de curvatura, curvaturas principais.

1. INTRODUCTION

Dupin surfaces were first studied by Dupin in 1822 and more recently by many authors (Riveros and Tenenblat preprint, Cecil and Chern 1989, Cecil and Jensen 1998, 2000, Miyaoka 1984, Niebergall 1992, Pinkall 1985a,b, Stolz 1999 and Thorbergsson 1983), which studied several aspects of Dupin hypersurfaces. The class of Dupin hypersurfaces is invariant under conformal transformations. Moreover, the Dupin property is invariant under Lie transformations (Pinkall 1985b). Therefore, the classification of Dupin hypersurfaces is considered up to these transformations. The local classification of Dupin hypersurfaces is considered up to these transformations. The local classification of Dupin surfaces in

³ is well known. Pinkall (1985a) gave a complete classification up to Lie equivalence for Dupin hypersurfaces M³Ì ⁴. However, the classification of Dupin hypersurfaces for higher dimensions is far from complete. Therefore, it is important to characterize such submanifolds. Our main result, Theorem 5, provides a local characterization of generic Dupin hypersurfaces in ⁵, with four distinct principal curvatures. The details of the proof can be found in (Riveros and Tenenblat preprint) and they will appear elsewhere. The proof is based on the theory of higher-dimensional Laplace invariants, which we recall in section 2, and the properties of Dupin hypersurfaces with distinct principal curvatures, given in section 3.

2. THE HIGHER-DIMENSIONAL LAPLACE INVARIANTS

The results in this section were obtained by Kamram and Tenenblat (1996, 1998).

We consider linear systems of second-order partial differential equations, of the form

where Y is a scalar function of the independent variables x₁, x₂,..., x_n, Y_{, l} denotes the derivative of Y with respect to x_l and the coefficients a and c are smooth functions of x₁, x₂,..., x_n which are symmetric in the pair of lower indices and satisfy certain compatibility conditions. The general form of the system (1) is preserved under admissible transformations

where j is smooth and non-vanishing and the 's are smooth and have non-vanishing derivatives. It is easily verified that under an admissible transformation, the coefficients a and c transform according to,

and the system (1) is

The higher-dimensional Laplace invariants of (1) are defined to be the n(n - 1)² functions given by

for all ordered pairs (i, j), 1 i j n.

LEMMA 1. The higher-dimensional Laplace invariants of a compatible system (1) satisfy the following relations:

for 1

i, j, k, l n, i, j, k, l distinct.

The functions m_ij, m_ijk are invariant under pure rescalings (2). The expression of a system (1) in terms of its higher-dimensional Laplace invariants is established in the following Theorem. For proof and details we refer also to (Tenenblat 1998).

THEOREM 2. Given any collection of n(n - 1)², n 3, smooth functions of x₁, x₂,..., x_nm_ij, m_ijk, 1 i, j, k n, i, j, k distinct, satisfying the constraints (4), there exists a linear system (1) whose higher-dimensional Laplace invariants are the given functions m_ij, m_ijk. Any such system is defined up to rescaling (2). A representative is given by

where (i, j) is a fixed (ordered) pair, 1 i, j, k, l n are distinct and A is a function which satisfies the following:

3. DUPIN HYPERSURFACES WITH DISTINCT PRINCIPAL CURVATURES

DEFINITION 3. An immersion X

Ì ⁿ

^{n + 1} is called a Dupin hypersurface if along each curvature line the corresponding principal curvature is constant.

Let X

Ì ⁿ

^{n + 1} be a Dupin hypersurface parametrized by lines of curvature, with distinct principal curvatures l_i, 1 i n and le N

Ì ⁿ

^{n + 1} be a unit vector field normal to X. Then

Moreover

where

are the Christoffel symbols.

As a consequence of the relations (3) and Lemma 1, we obtain the following expressions for the higher-dimensional Laplace invariants

Moreover, for 1

i, j, k, l n , i, j, k, l distinct, one has

Let X

Ì ⁿ

^{n + 1}, n 3, be a Dupin hypersurface parametrized by lines of curvature, with distinct principal curvatures. Consider a homothety

and assuming 0 Ï X() an inversion

Let Y = Iⁿ⁺¹(X) and = D(X). Then the higher-dimensional Laplace invariants of Y and are those of X, since inversions and homotheties are rescalings (2).

The next result is an application of Theorem 2.

LEMMA 4. Let X

Ì ⁿ

R^{n + 1}, n 3, be a Dupin hypersurface parametrized by lines of curvature and l _r, 1 r n, distinct principal curvatures of each point. For i, j, k fixed, 1 i j k n, the transformation

transforms the system (5) into

where l r are distinct from i, j,

and

4. A CHARACTERIZATION OF DUPIN HYPERSURFACES IN

⁵

We can now state our main result which provides a local characterization for generic Dupin hypersurfaces in

⁵, with four distinct curvatures. Considering the functions m_ijk satisfying (6), we introduce the following functions admitting that m₂₁₃ 0, m₂₁₄ 0 and m₃₁₄ 0,

THEOREM 5. Let X

Ì ⁴

⁵, be a Dupin hypersurface parametrized by lines of curvature, with principal curvatures, l_i, 1 i 4, distinct at each point. Suppose m₂₃₄ 0, m₄₂₃ 0, P

0, for all i, 1 i 3, then

where

P , 1 i 3, are defined by (10), G_i(x_i), 1 i 4, are vector valued functions of ⁵, A = – m_{231, 1}dx₃ and

Moreover, considering

where M = (- 1)^{i + 1}B , the functions G_i(x_i) satisfy the following properties in , for all 1 i, j 4:

a) aⁱ 0,

b) áaⁱ, a^jñ = 0, i j,

c) l_i = -

Conversely, let l_i

Ì ⁴

, 1 i 4 be real functions distinct at each point, such that l_i,i = 0, and the functions m_ijk, defined by

satisfy (6). Then for any vector valued functions G_i(x_i) satisfying the properties a)b)c) above, where aⁱ are defined by (14), the application given by (11) describe a Dupin hypersurface parametrized by lines of curvature whose principal curvatures are l_i.

SKETCH OF THE PROOF

Let X be a Dupin hypersurface as in Theorem 5 then it follows from Lemma 4 that

where V is given by (12) and the vector valued functions W³(x₁, x₂, x₄) and W²(x₁, x₃, x₄) satisfy systems of differential equations of Laplace type. The solutions of the systems are given by

It can be shown that

which proves (11).

The conditions a), b) of Theorem 5 are obtained considering that the vectors X_{, i} are orthogonal and non-vanishing, condition c) is equivalent to requiring l_i to be principal curvature of X. The converse is a straightforward calculation.

REMARK. One can show that the vector valued functions G_i(x_i) in Theorem 5 are invariant by inversions and homotheties of the corresponding Dupin hypersurfaces in ⁵. We conclude by mentioning that a characterization of a non-generic family of Dupin hypersurface MⁿÌ ^{n + 1}, n 3 whose principal curvatures are distinct and m_ijk = 0, i, j, k distinct has been obtained and it will appear elsewhere.

5. ACKNOWLEDGMENTS

The authors were partially supported by CNPq.

Manuscript received on August 8, 2002;

accepted for publication on October 28, 2002;

contributed by KETI TENENBLAT

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Publication Dates

• Publication in this collection
  17 Apr 2003
• Date of issue
  Mar 2003

History

• Received
  08 Aug 2002
• Accepted
  28 Oct 2002