Lobachevskii Journal of Mathematics Vol.3, 1999, 209–220 \begin{matrix}c \\ ◯ \end{matrix}R. Miyaoka

R. Miyaoka

HYPERSURFACE GEOMETRY AND HAMILTONIAN SYSTEMS OF HYDRODYNAMIC TYPE

(submitted by B.N.Shapukov)

§0. Introduction

Around 1982-1988, Dobrovin and Novikov gave a mathematical formulation of Hamiltonian systems of hydrodynamic type [DuN], which appear in gas dynamics, gas chromatography, etc. In 1991, Ferapontov [F1] showed one to one correspondence between these systems and the hypersurfaces in the (pseudo-)Euclidean space. Using this, he obtained the classification of the integrable non-diagonalizable 3 × 3 Hamiltonian systems of hydrodynamic type [F2] via the classification of Dupin hypersurfaces in E^{4} due to Pinkall [Pi]. Furthermore, he showed that the systems corresponding to the homogeneous isoparametric hypersurfaces in S^{n} with more than two principal curvatures are non-diagonalizable but integrable, by deforming the systems to the n-wave systems which are known to be integrable [F3].

Isoparametric hypersurfaces with less than four principal curvatures are known to be all homogeneous [C1,2,3], while there exist infinitely many nonhomogeneous examples with four principal curvatures [OT, FKM]. S.T. Yau posed a question to classify isoparametric hypersurfaces with four or six principal curvatures [Y], which is the remaining possible cases. Recently, the author proved that isoparametric hypersurfaces with six principal curvatures are homogeneous [M5]. The idea of the proof is a use of “isospectral principle”, which holds for a one-parameter family of (self-adjoint) operators satisfying the Lax equation. This is a basic principle to solve the Cauchy problem of the KdV equation, the non-linear Schrödinger equation, the sine-Gordon equation and so forth, via the inverse scattering method. Now, it turned out that all isoparametric hypersurfaces other than those with four principal curvatures correspond to the integrable systems.

Moreover, Ferapontov showed that the Lie transformation of hypersurfaces corresponds to the “reciprocal transformation” of the Hamiltonian systems, which is a transformation preserving the diagonalizability, integrability and linearly degeneracy of the isoparametric hypersurfaces under the Lie transformations correspond also to the integrable systems. These hypersurfaces belong to the hypersurface class called Dupin. Therefore the classification of Dupin hypersurfaces is also interesting and important.

The author thanks for permission to the editor of the Proceedings of the Third International Conference in Kyungpook University, Korea, where some part of this report was announced.

§1. Isoparametric hypersurfaces

To begin with, we give an intuitive introduction of isoparametric hypersurfaces. As for its historical development, see [Th3].

A deformation of a plane curve is described by the velocity in the normal direction. When this velocity is independent of points, the deformed curves are called parallel curves. A plane curve with constant curvature is a line or a circle. Its parallel curves are again lines or circles, respectively.

Consider this problem on Euclidean surfaces. Here, we replace the constant curvature by constant mean curvature (CMC for short). A CMC surface is a surface which is a critical point of the area integral under the condition that the surrounded volume is constant. There are many compact CMC surfaces in E^{3} of any topological type [K1,2], answering negatively to the Hopf conjecture. Only planes, spheres and circular cylinders are, however, surfaces such that all parallel surfaces are CMC surfaces. This was first considered by Laura [La] and Somigliana [So] in early 20th in geometric optics, and then reproved by Segre [S1] and Levi-Civita [LC]. Segre [S2] also shows in E^{n}, that only E^{n−1}, S^{n−1} and S^{k} × E^{n−k−1} are such hypersurfaces. Note that this holds locally, i.e. a piece of those hypersurfaces are only hypersurfaces of which all parallel hypersurfaces are CMC hypersurfaces.

Finally, E. Cartan investigated this problem systematically in the space form M^{n}(c) and obtained many fundamental results [C1-4], which we explain later. Roughly speaking, an isoparametric hypersurface is a hypersurface foliated by several spheres (including totally geodesic ones) of constant radii.

Another intuitive understanding of isoparametric hypersurfaces may be obtained when we see circles of waves evolving in the pond when we throw a stone into it, or imagine the sound wave coming from a string, cylindrically developing in the space. As is shown by these examples, an important property of isoparametic hypersurfaces is that the whole family of parallel hypersurfaces foliate the ambient space with some singular leaves. Especially in S^{n}, this restricts the topology of the hypersurfaces very strictly.

The “wave” is always a key word. We know the Huygens principle of the developing wave, which causes us to consider a contact structure. In §3, we lift the hypersurface to the Lie contact manifold T_{1}S^{n} as a Legendre submanifold, in order to describe the Lie images of them, which yields Dupin hypersurfaces.

In above setting, isoparametric hypersurfaces are those of which all parallel hypersurfaces have constant mean curvature. Cartan shows that this is equivalent to the hypersurfaces with constant principal curvatures (see also [N2]). When the number of the principal curvatures is g, he shows that in E^{n} and in H^{n}, they are either totally umbilic (g = 1), or a product of an umbilic submanifold with a totally geodesic submanifold (g = 2). In S^{n}, however, he shows the existence of examples with g = 3 and 4, and classified the former completely as tubes around the “Veronese surfaces”.

Isoparametric hypersurfaces are level surfaces of the so-called isoparametric function. This is shown by Cartan, and in S^{n}, Münzner further investigated that this is a restriction of a degree g homogeneous polynomial F on E^{n+1}, satisfying

\{ \begin{matrix}|∇F |^{2} = g^{2}r^{2g−2}, r(x) = |x|, \\ △F = cr^{g−2}, c = \frac{m_{1} − m_{2}}{2} g^{2}. \end{matrix}

For m_{1}, m_{2} see below. Thus isoparametric hypersurfaces are algebraic and uniquely extended to a complete one.

Important examples of isoparametric hypersurfaces are homogeneous hypersurfaces in S^{n}, which are orbits of a subgroup of the isometry group of S^{n}. In 1971, Hsiang and Lawson shows that the homogeneous hypersurfaces in S^{n} are isotropy orbits of rank two symmetric spaces [HL], which are then all classified. In particular for g = 6, these symmetric spaces are G_{2}/SO(4) and G_{2} × G_{2}/G_{2}.

A remarkable progress was made by Ozeki-Takeuchi in 1976 [OT], when they found an infinite number of non-homogeneous isoparametric hypersurfaces with g = 4, by using Clifford algebra to construct Cartan-Münzner functions of degree 4. This was systematically extended by Ferus-Karcher-Münzner [FKM]. All these examples turn out to be tubes of the Clifford-Stiefel manifolds [PT]. On the other hand, Takagi-Takahashi showed that among Hsiang-Lawsons’ classification, appear hypersurfaces with only g = 1, 2, 3, 4, 6 [TT]. Münzner shows generally that these are all possible values taken by g. He uses two singular level sets called the focal submanifolds to show this, applying algebraic topology to the fact that S^{n} is decomposed into two disk bundles over them. Moreover, he shows that the multiplicities of the principal curvatures are common number m when g is odd, and consists of two values m_{1} and m_{2} for even g. Abresch [A] investigated further in detail on m and m_{1}, m_{2}, and showed that when g = 6, m_{1} = m_{2} = m = 1 or 2.

In 1986, Dorfmeister-Neher proved that the Cartan-Münzner function for (g, m) = (6, 1) is unique so that the ypersurfaces are homogeneous, i.e. the isotropy orbits of G_{2}/SO(4) [DN]. The proof is not easy since the Cartan-Münzner function in this ase is a degree 6 homogeneous polynomial in 8 variables. Its extension to the ase (g, m) = (6, 2) does not known.

§2. Recent results

Recently, using a completely new idea, the author proved that the isoparametric hypersurfaces with six principal curvatures are homogeneous, hence are the isotropy orbits of G_{2}/SO(4) and G_{2} × G_{2}/G_{2}. Finally, isoparametric hypersurfaces are all homogeneous unless g = 4.

The idea of the proof is, as described in the introduction, a use of isospectral principle applying to the shape operators of the focal submanifolds. Because these operators are known to be isospectral, they satisfy the Lax equation, and using this, we can show how the eigenvectors, i.e. the principal directions evolve along a geodesic of a curvature surface. We can find a local frame, with respect to which the structure coefficients (i.e. the coefficients of the connection of M) are constant. Here, we remark a global property of isoparametric hypersurface M, “tautness”, that is, the square of the distance function of S^{n} restricted to M is a perfect Morse-Bott function, i.e. the equality holds in the Morse inequality. Because the number of the structure coefficients is 220 in the case (g, m) = (6, 2), we should know the symmetric relations among them, and to do so we have to know the intersection of leaves, which can be done by using tautness*. Moreover, to show the local existence of the frame, we use the Thorbergsson cycles, which assures that sufficiently near two points can be connected by finite number of geodesics of suitable curvature surfaces. When such frame exists, we apply Singer’s strongly curvature-homogeneous theorem [Si, N1], which implies that such hypersurfaces are intrinsically homogeneous. Then applying the rigidity theorem of hypersurfaces with type number (the rank of the shape operator) not less than three, we can show the extrinsic homogeneity of M. This method works for any g if m = 1 or 2, and we have a new proof for homogeneity in the cases g = 3 and 4, the latter is obtained in [Tak] and [OT].

As for the case g = 4, Stolz [St] shows recently that (m_{1}, m_{2}) must be those which are taken by the homogeneous ones or the Clifford examples constructed by Ferus-Karcher-Münzner. This is a remarkable progress in this last classification problem, and even more, since he shows it, in fact, for compact Dupin hypersurfaces.

Isoparametric submanifolds in E^{n} are defined by Terng [Te] as submanifolds having flat normal bundles and satisfying that with respect to a parallel unit normal field, the shape operator has constant eigenvalues. They turned out to be homogeneous if the codimension is not less than three [Th2, O]. The hypersurface case in S^{n} is neglected from this result since the codimension in E^{n+1} is two, though they are isoparametric in Terng’s sense. In the context of integrable systems, Terng and Thorbergsson considered infinite dimensional isoparametric submanifolds in a Hilbert space [TeT]. On the other hand, these are generalized as equi-focal submanifolds in symmetric spaces of compact type. For these subjects and references, see [Th3].

§3. Lie transformations

Consider the space S of all oriented hyperspheres of S^{n}. An oriented hypersphere is determined by its center p ∈ S^{n} and its oriented radius θ, hence S is given by

S = {(p, cos θ, sin θ) ∈ ℝ_{2}^{n+3} | p ∈ S^{n}}/ℤ_{2} = Q^{n+1},

*This is not indeed necessary in isoparametric case, but when we consider Dupin, we need tautness to describe the intersection of leaves.

where ℝ_{2}^{n+3} denotes the real vector space of dimension n + 3 equipped with the indefinite bilinear form ⟨ ⟩ with minus signatures in the last two components. S is identified with the quadric Q^{n+1} = {[k] ∈ P ℝ_{2}^{n+3} | ⟨k, k⟩ = 0} in the real projective space P ℝ_{2}^{n+3}. Two oriented spheres [k_{1}] and [k_{2}] are in oriented contact when ⟨k_{1}, k_{2}⟩ = 0. The Lie transformation group G = P O(n + 1, 2) acts on S preserving the oriented contact. A line in Q^{n+1} consists of [ak_{1} +bk_{2}], [k_{i}] ∈ Q^{n+1} and ⟨k_{1}, k_{2}⟩ = 0, which is a one parameter family of hyperspheres mutually contact in an oriented way at one point. Since a line is determined by the contact point and the unit normal there, the space Λ^{2n−1} of lines in Q^{n+1} is identified with T_{1}S^{n}. It is easy to see that G acts on T_{1}S^{n} transitively so that T_{1}S^{n} = G/K is a homogeneous space with an isotropy subgroup K. The geometric structure modelled by the K-principal bundle P → G/K is called a Lie contact structure, which is a special contact structure on a (2n − 1)-dimensional contact manifold (§6).

Consider a hypersurface f : M → S^{n}. Defining the Legendre map λ : M → T_{1}S^{n} by

λ(p) = (f(p), n_{p})

where n_{p} is the unit normal to M at p, we obtain a Legendre submanifold of T_{1}S^{n}. Then applying φ ∈ G to T_{1}S^{n} followed by the projection π : T_{1}S^{n} → S^{n}, we obtain a new hypersurface f_{φ} = π · φ · λ, possibly with singularities. This is called a Lie image of f. The singularities are mild in the sense that they can be removed by taking a parallel hypersurfaces [Pi]. A good reference to the Lie sphere geometry is [Ce].

§4. Dupin hypersurfaces

Dupin hypersurfaces are extended version of the cyclide of Dupin. Intuitively speaking, they are hypersurfaces foliated by several spheres at each point, of which radi depend on points. They are defined to be the hypersurfaces having a definite number of principal curvatures, which are constant along the principal directions. In contrast to the isoparametric case, Dupin hypersurfaces exist locally for any given number of principal curvatures with any multiplicities [P]. This is easily seen if we start from some sphere in the Euclidean space of codimension greater than one, and take the tubes, cylinders, or rotate it, and repeat this process. Hypersurfaces obtained in this way, or further composed by the stereographic projection to S^{n}, are called reducible Dupin [P]. On this hypersurface, the number of principal curvatures is not necessarily constant, so we obtain non-complete Dupin hypersurfaces.

On the other hand, if we require a global condition such as compactness, they satisfy the same properties with those of isparamatric hypersurfaces in cohomology level [Th1]. Thus Münzner’s argument works, and we obtain similar results on g and the multiplicities.

The Lie images of isoparametric hypersurfaces are examples of compact Dupin hypersurfaces. Now a natural question is whether a compact Dupin hypersurface is a Lie image of an isoparametric hypersurface. This is trivially true if g = 1. For g = 2, Cecil and Ryan showed this in a global version, and Pinkall in a local version. The author proved it for g = 3 [M1]. Then Cecil and Ryan conjectured that this holds for g = 4 and 6. Pinkall and Thorbergsson, however, constructed counterexamples for g = 4 [PT], and independently, Ozawa and the author, constructed counterexamples for g = 4 and 6 [MO]. In both proofs, a Lie invariant called the Lie curvature played an important role. This was found by the author [M2], as the cross ratio of four principal curvatures. The Lie curvature of an isoparametric hypersurface is obviously constant, hence its Lie image must have the same constant Lie curvature. In [M4], we show that the isoparametric hypersurface with (g, m) = (6, 1) is the inverse image of an isoparametric hypersurface N in S^{4} with three principal curvatures by the Hopf map π : S^{7} → S^{4}. The counterexamples in [MO] are obtained by first changing N by a non-isometric conformal transformation, then composing with π^{−1}. This is Dupin with six principal curvatures with Lie curvature different from the isoparametric one. Counterexamples for g = 4 are similarly obtained as the image of the conformally deformed product of spheres in S^{4} under π^{−1}.

Cecil and Jensen showed that an irreducible Dupin hypersurface is a Lie images of an isoparametric hypersurface when g = 3 [CJ1], and so is when (g, m) = (4, 1) if it has the same Lie curvature as isoparametric hypersurfaces [CJ2].

§5. Hamiltonian systems of hydrodynamic type

Consider a phase space given by P = {u(x) = (u^{1}(x), …, u^{n}(x))}, where we demand some decay condition on u at |x| → ∞. This is an infinite dimensional vector space and we always identify T_{u}P ≅ P. The variational derivative ([Pa, 3.4]) of a functional F (u) = ∫ f(u, u_{x}, …)dx on P is given by

\frac{δF}{δu^{i}} = lim_{ε→∞} \frac{dF (u + εv)}{dε} = \frac{∂f}{∂u^{i}} − (\frac{∂f}{∂u_{x}^{i}})_{x} + (\frac{∂f}{∂u_{xx}^{i}})_{xx} − ….  (1)

Using an operator ω^{ij} = ω^{ij}(u, u_{x}, …, d) on C^{∞}(P), where d means a derivative with respect to x, define a product by

\{F, G\} = ∫ \frac{δF}{δu^{i}} ω^{ij} \frac{δG}{δu^{j}} dx, F, G ∈ C^{∞}(P).

In order for { , } to be a Poisson bracket, i.e. to be skew and satisfying the Jacobi identity, ω^{ij} has to satisfy certain condition on the Schouten bracket, but when it is of the form

ω^{ij} = g^{ij}(u)d + b_{k}^{ij}(u)u_{x}^{k}.

Dubrovin-Novikov obtain the following theorem. Let g^{ij} be non-degenerate. Put b_{k}^{ij} = −g^{il}(u)Γ_{lk}^{j}(u). Then observing the transformation rule of g^{ij} and Γ_{lk}^{j} under the coodinate change, we see that Γ_{lk}^{j} defines a connection ∇. Denote its curvature tensor by R.

Theorem [DuN1].

1. { , } is skew symmetric ⇔ g^{ij} = g^{ji}, ∇g = 0.

2. { , } satisfies Jacobi identitiy ⇔ Γ_{lk}^{j} = Γ_{kl}^{j}, R = 0.

Dubrovin-Novikov define also a Hamiltonian of hydrodynamic type by H = ∫ h(u)dx, i.e. if the Hamiltonian density h(u) is independent of u_{x}, u_{xx}, . . .. Calculating its Hamiltonian equation following (1), we obtain

u_{t}^{i} = ω^{ij}(u)\frac{δH}{δu^{j}} = v_{j}^{i}(u)u_{x}^{j},  (2)

where

v_{j}^{i}(u) = ∇^{i}∇_{j}h.

We call the matrix v_{j}^{i} given by the right hand side a local Hamiltonian of hydrodynamic type.

Theorem [Ts1]. A matrix v_{j}^{i} is local Hamiltonian of hydrodynamic type if there exists a non-degenerate metric g_{ij} with flat curvature 0, which satisfies

g_{ik}v_{j}^{k} = g_{jk}v_{i}^{k}, ∇_{k}v_{j}^{i} = ∇_{j}v_{k}^{i}.  (3)

With respect to a flat coordinate, (2) turns out to be

u_{t}^{i} = ε_{i}h_{ij}u_{x}^{j}, h_{ij} = \frac{∂^{2}h}{∂u^{i}∂u^{j}} , ε_{i} ∈ \{±1\}  (2’)

In the following, for simplicity, consider only the case ε_{i} = 1. On the other hand, we call the following operator including the integral operator d^{−1},

ω^{ij} = g^{ij}d − g^{is}Γ_{sk}^{j}u_{x}^{k} + cu_{x}^{i}d^{−1}u_{x}^{j}, c = a non-zero constant,

non-local. In this case, we have

Theorem [FM]. { , } is skew ⇔ g^{ij} = g^{ji}, ∇g = 0.

{ , } satisfies Jacobi identity ⇔ Γ_{sk}^{j} = Γ_{ks}^{j}, R_{kl}^{ij} = c(δ_{k}^{i}δ_{l}^{j} − δ_{l}^{i}δ_{k}^{j}), i.e. g_{ij} is of non-zero constant curvature c.

For Hamiltonian of hydrodynamic type H(u), we have

u_{t}^{i} = (∇^{i}∇_{j}h + cδ_{j}^{i}h)u_{x}^{j}.

Call v_{j}^{i} = (∇^{i}∇_{j}h + cδ_{j}^{i}h) non-local Hamiltonian of hydrodynamic type.

Theorem [FM]. A matrix v_{j}^{i} is non-local Hamiltonian of hydrodynamic type if there exists a non-degenerate metric g_{ij} with non-zero constant curvature, which satisfies

g_{ik}v_{j}^{k} = g_{jk}v_{i}^{k}, ∇_{k}v_{j}^{i} = ∇_{j}v_{k}^{i}.  (3)

The importance is that a non-local system corresponds to a hypersurface in E^{n+1} as follows:

Theorem [F1]. Define three quadratic forms by

\begin{aligned}I & = v_{i}^{k}g_{kl}v_{j}^{l}du^{i}du^{j} \\ II & = v_{i}^{k}g_{kj}du^{i}du^{j} \end{aligned}

III = g_{ij}du^{i}du^{j}.

Then I and II satisfy the Gauss-Coddazi equations, i.e. they correspond to a certain hypersurface in E^{n+1} if and only if g_{ij} is of constant curvature 1 and the matrix v_{j}^{i} satisfies (3).

In fact, the hypersurface has (v_{j}^{i})^{−1} as its Weingarten map. On the other hand, local systems and non-local systems correspond via reciprocal transformation as follows. Thus we could say, a hypersurface corresponds to a local system, too.

A 1-form B(u)dx+A(u)dt is called an integral if on a solution curve of (2), it is closed. Moreover it is of hydrodynamic type if B and A are independent of u_{x}, u_{xx}, . . .. A transformation (x, t) ↦ (X, T) is called reciprocal if

dX = B(u)dx + A(u)dt, dT = N(u)dx + M(u)dt  (4)

are integrals of hydrodynamic type. Then (2) becomes

u_{T}^{i} = V_{j}^{i}(u)u_{X}^{j}, V_{j}^{i} = \frac{Bv − AE}{ME − Nv} , E = (δ_{j}^{i}).  (5)

Under the reciprocal transfomation given by

dX = (∑_{k} \frac{(u^{k})^{2}}{2} + \frac{1}{2})dx + (∑_{k} h_{k}u^{k} − h)dt, dT = dt,

where h_{k} = \frac{dh}{dx^{k}}, (5) becomes

u_{T}^{i} = \{(∑_{k} \frac{(u^{k})^{2}}{2} + \frac{1}{2})h_{ij} − (∑_{k} h_{k}u^{k} − h)δ_{j}^{i})\}u_{X}^{j},

and with respect to a new Hamiltonian h̃(u) = h(u)/(∑_{k} \frac{(u^{k})^{2}}{2} + \frac{1}{2}), this is non-local:

u_{T}^{i} = (∇^{i}∇_{j} + cδ_{j}^{i})h̃u_{X}^{j}.  (6)

This non-local system corresponds to a hypersurface with position vector r and unit normal vector n given by

r = (h_{1} − u^{1} \frac{A}{B} , …, h_{n} − u^{n} \frac{A}{B} , −\frac{A}{B}),
n = (\frac{u^{1}}{B} , …, \frac{u^{n}}{B} , \frac{1}{B} − 1).

Novikov conjectures and Tsarëv [Ts2] proved that when a local system is diagonalizable, i.e. when the so-called Riemann invariants R^{1}, …, R^{n} exist and (2) is written as

R_{t}^{i} = λ^{i}R_{x}^{i},

the system is integrable. On the other hand, few non-diagonalizable integrable examples were known until Ferapontov shows that homogeneous isoparametric hypersurfaces and their Lie images give such examples. Actually, the integrability requires the constantness of eigenvalues of (v_{j}^{i}) along eigen-direction, which is called linealy degeneracy. Thus the corresponding hypersurfaces must be Dupin. Especially for n = 3, we can use the theorem of Pinkall on the 3-dimensional Dupin hypersurfaces in E^{4}, to classify all integrable systems. Furthermore, those corresponding to the homogeneous isoparametric hypersurfaces have Cartan-Münzner functions as the Hamiltonian density, and are shown to be deforemed into the integrable n-wave systems.

Finally, note that the argument in terms of Hamiltonian system is all local, hence global properties of hypersurfaces such as compactness are not effective. It would be better to use local conditions such as irreducibility, when we discuss hypersurfaces from the standpoint of Hamiltonian systems.

§6. Lie contact structures

Let N be a 2n − 1 dimensional contact manifold. When the frame bundle of N is reduced to ρ(K), where ρ : K → GL(2n − 1, ℝ) is the isotropy representation of T_{1}S^{n} = G/K, G = P O(n + 1, 2), we call this reduction a Lie contact structure. This is considered by H. Sato [SY1] and he shows that a typical Lie contact structure exists on the unit tangent bundle of a Riemannian manifold, or on the tangent sphere bundle SM of a manifold with conformal structure. The Lie algebra of the structure group is a simple graded Lie algebra of the second kind. Thus the Tanaka theory [Tan], which extends deeply the Cartan theory of connections, can be applied, and the existence of the normal Cartan connection or Tanaka connection is guaranteed. When its curvature vanishes identically, the structure is called Lie flat. This is the case where the underlying manifold is conformally flat. H. Sato posed a question whether from Lie flatness, the conformal flatness follows. In order to answer to this question, the author constructed the Tanaka connection by deforming the Levi-Civita connection associated with a metric compatible with the conformal structure, and showed that the curvature can be written explicitly by using the Weyl’s conformal curvature tensors [M3]. Thus the answer to Sato’s problem is yes, i.e. SM is Lie flat if and only if M is conformally flat (see also [SY2]).

The restriction of the Lie contact structure on T_{1}S^{n} to the Legendre submanifold obtained from a hypersurface would be helpful on the Lie equivalence problem of hypersurfaces. On the other hand, the Tanaka theory works for some other geometric structures having the structure group whose Lie algebra is a simple graded Lie algebra corresponding to some differential systems. Applications of this theory to some other geometric structures would be important and interesting.

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Department of Mathematics, Sophia University, Chiyoda-ku, Tokyo 1028554, Japan

E-mail address: r-miyaok@hoffman.cc.sophia.ac.jp

Received July 27, 1999