Abstracts
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.
Dupin hypersurfaces; Laplace invariants; lines of curvature; principal curvatures
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.
hipersuperfícies de Dupin; invariantes de Laplace; linhas de curvatura; curvaturas principais
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 Correspondence to Keti Tenenblat E-mail: keti@mat.unb.br
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,..., xn, Y, l denotes the derivative of Y with respect to xl and the coefficients a and c are smooth functions of x1, x2,..., xn 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 mij, mijk 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,..., xnmij, mijk, 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 mij, mijk. 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 nn + 1 is called a Dupin hypersurface if along each curvature line the corresponding principal curvature is constant.Let X
Ì nn + 1 be a Dupin hypersurface parametrized by lines of curvature, with distinct principal curvatures li, 1 i n and le NÌ nn + 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 nn + 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 nRn + 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
5We can now state our main result which provides a local characterization for generic Dupin hypersurfaces in
5, with four distinct curvatures. Considering the functions mijk satisfying (6), we introduce the following functions admitting that m213 0, m214 0 and m314 0,
THEOREM 5. Let X
Ì 45, be a Dupin hypersurface parametrized by lines of curvature, with principal curvatures, li, 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), Gi(xi), 1 i 4, are vector valued functions of 5, A = m231, 1dx3 and
Moreover, considering
where M = (- 1)i + 1B , the functions Gi(xi) satisfy the following properties in , for all 1 i, j 4:
a) ai 0,
b) áai, ajñ = 0, i j,
c) li = -
Conversely, let li
Ì 4, 1 i 4 be real functions distinct at each point, such that li,i = 0, and the functions mijk, defined by
satisfy (6). Then for any vector valued functions Gi(xi) 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 li.
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 li to be principal curvature of X. The converse is a straightforward calculation.
REMARK. One can show that the vector valued functions Gi(xi) 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 mijk = 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