Lobachevskii Journal of Mathematics Vol.3, 1999, 113–125 \begin{matrix}c \\ ◯ \end{matrix}G. Ishikawa

G. Ishikawa DEVELOPABLE HYPERSURFACES AND HOMOGENEOUS SPACES IN A REAL PROJECTIVE SPACE

(submitted by B.N.Shapukov) Dedicated to Professor Fuichi Uchida on his 60th birthday.

Abstract. We present new examples of non-singular developable hypersurfaces, which are algebraic and homogeneous, in real projective spaces. Moreover we give a characterization of compact homogeneous developable hypersurfaces, using the theory of isoparametric hypersurfaces.

0. Introduction

A C^{∞} hypersurface M in the n-dimensional real projective space RP^{n} is called developable if its Gauss map

γ : M → Gr(n, R^{n+1}) ≅ Gr(1, (R^{n+1})^{∗}) = RP^{n∗}

defined by γ(x) = T_{x}M̂ ⊂ R^{n+1} (x ∈ M) has rank(γ) < dim(M) = n − 1. Here, we mean by M̂ the n-dimensional submanifold of R^{n+1} − {0}, corresponding to M ⊂ RP^{n}, and therefore T_{x}M̂ is a linear hyperplane in R^{n+1}. Moreover we mean by RP^{n∗} the dual projective space, and by rank(γ) the maximum of the rank of differential maps γ_{∗} : T_{x}M → T_{x}RP^{n∗} (x ∈ M) of γ.

Remark that the developability is a notion of projective geometry; the image of a developable hypersurface under a projective transformation is again developable.

In this paper we treat mainly developable hypersurfaces. See [FW][W] for developable submanifolds of arbitrary codimension.

It is well-known, as classical examples of developable surfaces in the three dimensional space, cylinders, cones and tangent developables of space curves [Cay][I]: Among them, only the planes have no singularities in the projective space. Observing the singularities of developable hypersurfaces, we expect, also in the general case, that non-singular compact developable hypersurfaces are strictly restrictive. In fact, it is known that a non-singular complex algebraic developable hypersurface in CP^{n} is necessarily a projective hyperplane ([GH][W][L1]). Also in a real projective space, we have the following restriction, via the geometrical investigation of homogeneous Monge-Ampère equations based on projective duality:

1991 Mathematical Subject Classification.Primary 58C27, 53C30; Secondary 53A05, 53A20, 53C15.

Key words and phrases. Projective duality, Cayley’s octonians, Veronese embedding, Jordan algebra, Monge-Ampère foliation, Severi variety, isoparametric hypersurface.

Theorem 1 ([IM]). For a compact developable C^{∞} hypersurface M in RP^{n}, the maximal rank r = rank(γ) of the Gauss map γ : M → RP^{n∗} is an even integer and satisfies the inequality n < (1/2)r(r + 3), provided r ≠ 0. In particular, if r < 2, then M is necessarily a projective hyperplane of RP^{n}. Any compact developable C^{∞} hypersurfaces in RP^{3} or RP^{5} are projective hyperplanes.

The rank condition appeared in Theorem 1 is essential in the real case; in fact there exist non-trivial examples of compact developable hypersurfaces in real projective spaces:

Proposition 2. For n = 4, 7, 13, 25, there exists a real algebraic cubic nonsingular developable hypersurface in RP^{n}. These developable hypersurfaces have the structure of homogeneous spaces of groups SO(3), SU(3), Sp(3), F_{4}, respectively. Their projective duals are linear projections of Veronese embeddings of projective planes KP^{2}, for K = R, C, H, O (the Cayley’s octonians, octanions or octonions). Each of these real algebraic developable hypersurfaces admits deformations to C^{∞} developable hypersurfaces with 2, 3, 5, 9 functional parameters, or more rigorously, with the space of sections of normal bundles to KP^{2} ⊂ RP^{n} as the infinitesimal space of C^{∞} developable deformations.

Notice that it is classically known that a properly embedded developable hypersurface in R^{n} of rank(γ) ≤ 1 is necessarily a cylinder (Hartman-Nirenberg’s theorem [HN][Ste][Sto]). Similar result is known for C^{n} by Abe [Ab]. For this direction, see the survey [B]. The first example of noncylindrical C^{∞} developable hypersurfaces in R^{4} is given by Sacksteder [Sac]:

M = {(x_{1}, x_{2}, x_{3}, x_{4}) ∈ R^{4} | x_{4} = x_{1} cos x_{3} + x_{2} sin x_{3}}.

DEVELOPABLE HYPERSURFACES AND HOMOGENEOUS SPACES 115

Mori [Mo] gives an example of families of non-cylindrical developable hypersurfaces in R^{4}, in connection with the study of deformable submanifolds. On the other hand, Akivis [Ak] proves the existence of C^{∞} complete developable hypersurfaces in RP^{4} which is not a projective hyperplane, using the theory of differential systems. (See also [AG] Ch. 4, for the method of construction). However notice that it is not given any concrete examples there. Recently, Fischer and Wu ([FW][W]) study developable submanifolds in CP^{n}, C^{n} and R^{n} of higher codimension. In [W], it is introduced an (unpublished) example of non-cylindrical real algebraic developable hypersurfaces in R^{4} by Bourgain:

M = {(x_{1}, x_{2}, x_{3}, x_{4}) ∈ R^{4} | x_{1}x_{4}^{2} + x_{2}(x_{4} − 1) + x_{3}(x_{4} − 2) = 0}.

Then, M is non-singular in R^{4} and even in C^{4} after complexification, while the Zariski closure \overline{M} ⊂ RP^{4} of M has singularities in RP^{4}. (The singular loci is an RP^{2} in the projective hyperplane at infinity).

In general, developable submanifolds has the Monge-Ampère foliation so that the tangent spaces to the submanifold are constant along each leaf. For instance, in the case of F_{4} in Proposition 2, the Gauss mapping is a submersion and the Monge-Ampère foliation is given by the fiberwise Z/2Z quotient of the fibration

OP^{1} ≅ Spin(9)/Spin(8) → F_{4}/Spin(8) → F_{4}/Spin(9) ≅ OP^{2},

arising from the filtration F_{4} ⊃ Spin(9) ⊃ Spin(8). Remark that there exists natural identification OP^{1} ≅ S^{8}, and the antipodal map induces the involution on OP^{1}.

Now we give a characterization of compact homogeneous developable hypersurfaces under some assumption, by means of metric geometry.

We say M and M′ ⊂ RP^{n} are projectively equivalent if there is a projective transformation φ : RP^{n} → RP^{n} with φ(M) = M′.

A C^{∞} hypersurface M ⊂ RP^{n} is called a Cartan hypersurface if, n = 4, 7, 13 or 25, and M is projectively equivalent to one of examples in Proposition 2. Since the developability is projectively invariant, Cartan hypersurfaces are developable.

Remark that a Cartan hypersurface in RP^{n} is a G-orbit for a compact Lie subgroup G ⊂ GL(n+1, R). In general, we call a (projectively) homogeneous submanifold M ⊂ RP^{n} of compact type if M is a G-orbit for a compact Lie subgroup G of GL(n + 1, R), under the action on RP^{n} induced by the natural linear action on R^{n+1}. If M is of compact type, then M is compact.

In this paper we show the following result:

Theorem 3. Let M be a connected homogeneous developable hypersurface of compact type. Then M is a projective hyperplane or a Cartan hypersurface.

By Theorem 3, we know that it is impossible to find other non-trivial examples of developable hypersurfaces than Cartan hypersurfaces, among quotients of compact groups of linear transformations.

In the next section, we recall the notion of projective duality and the second fundamental form of submanifold in a projective space. In §3, we give a direct proof of Proposition 2 within the framework of projective geometry. The main tool for the construction of Proposition 2 is the real projectivecontact geometry [M1][M2] over Jordan algebras. In the last section, we prove Theorem 3 using metric geometry. The proof shows clearly the power of the theory of isoparametric hypersurfaces. However we wish to find alternative proof of Theorem 3 within the framework of projective geometry, since it would be most natural one and may work for just compact projectively homogeneous developable hypersurfaces.

It is interesting to ask the connection between Cartan hypersurfaces and the classification of Severi varieties in the projective spaces over algebraically closed field of characteristic zero, for instance, over C, by Zak [Z] (cf. [FL][LV, p.15]). See also [L1][L2][Kaj] for complex projective geometry on the second fundamental forms and degenerate secant varieties, related to homogeneous spaces and Clifford algebras.

The author would like to thank Professors H. Kaji, S. Kaneyuki, M. Kimura, R. Miyaoka, H. Sato, H. Tazaki, K. Tsukada and K. Yamaguchi, for valuable comments and helpful encouragement. The author is grateful to Professors B. Komrakov and T. Morimoto, for organizing the workshop in Kazan on September 1998.

1. Projective duality and second fundamental forms

Let M ⊂ RP^{n} be a submanifold of dimension m, (m < n). Consider the projective conormal bundle of M:

M̃ = {(p, q) ∈ RP^{n} × RP^{n∗} | p ∈ M, T_{p}M ⊂ q^{∨}},

where q^{∨} is the projective hyperplane of RP^{n} determined by q ∈ RP^{n∗}, and we identify T_{p}M as the corresponding m-dimensional plane through p in RP^{n}. Then we see M̃ is a C^{∞} submanifold in RP^{n} × RP^{n∗} of dimension n − 1. Let ρ : M̃ → RP^{n} (resp. ρ′ : M̃ → RP^{n∗}) denotes the projection to the first (second) component. Then ρ(M̃) = M and ρ′(M̃) = M^{∨} is the projective dual of M.

We call M is developable if the Gauss map γ : M → Gr(m + 1, R^{n+1}), defined by γ(x) = T_{x}M̂ satisfies rank(γ) < dim M.

If M is developable and rank(γ) = r, then there exists an (m − r)- dimensional foliation on Ω = {x ∈ M | rank_{x}(γ) = r}, which we call Monge-Ampère foliation [D]. Moreover in this case, M^{∨} is ruled by r-parameter (n − m − 1)-planes, and rank(ρ′) < dim M̃ = n − 1.

Remark that, if m = n − 1, then ρ is a diffeomorphism onto its image and the Gauss map is decomposed as γ = ρ′ ∘ ρ^{−1}.

Let g : W → RP^{n∗} be an immersion. For x ∈ W, the second fundamental form of g at x is a linear family of quadratic forms (Hessians) on T_{x}W parametrized by conormal vector space N^{∗} = (T_{g(x)}RP^{n∗}/g_{∗}(T_{x}W))^{∗} to g at x:

II^{∗} : N^{∗} → S^{2}(T_{x}^{∗}M) (the symmetric product),

defined as follows: Let u ∈ N^{∗}. Take an affine function v on an affine neighborhood of g(x) such that the corresponding cotangent vector T_{g(x)}RP^{n∗} → R vanishes on g_{∗}(T_{x}W) and represents the conormal vector u. Then we define II^{∗}(u) to be the Hessian at x of the composition v ∘ g. Then II^{∗}(u) is independent of the choice of v and depends only on u (cf. [L1],[L2],[Sas]).

Now we recall the following fundamental result [IM], which we are going to use for showing Proposition 2:

Lemma 4. For an immersed submanifold W of RP^{n∗} of codim ≥ 2, the following conditions are equivalent to each other:

(i) W is a projective dual of a properly immersed hypersurface in RP^{n}. (ii) The second fundamental form at each point of W does not contain any

singular quadratic forms.

(iii) For any projective hyperplane H ⊂ RP^{n∗}, each singular point of the hyperplane section W ∩ H on W is non-degenerate.

Proof. The condition (i) is equivalent to that ρ : W̃ → RP^{n} is an immersion. For a local equation

y_{r+1} = φ_{r+1}(y_{1}, …, y_{r}), …, y_{n} = φ_{n}(y_{1}, …, y_{r})

of W, W̃ is defined by F = ∂F /∂y_{1} = ⋯ = ∂F /∂y_{r} = 0, where

F (X; y_{1}, …, y_{r}) = X_{0}φ_{n} +⋯+X_{n−r−1}φ_{r+1} +X_{n−r}y_{r} +⋯+X_{n+1}y_{1} +X_{n},

for a homogeneous coordinates (X_{0}, X_{1}, …, X_{n}) of RP^{n}. Then ρ is an immersion on W̃ if and only if the second fundamental form

II^{∗}(X_{0}, …, X_{n−r−1}) = ∑_{k=0}^{n−r−1} X_{k} (\frac{∂^{2}φ_{n−k}}{∂y_{i}∂y_{j}} )_{1≤i,j≤r}

does not represent a singular matrix, provided (X_{0}, …, X_{n−r−1}) ≠ (0, …, 0). This condition is equivalent to (ii). The equivalence (ii) ⇔ (iii) is clear.

2. Proof of Proposition 2

First we show the construction of a cubic non-singular developable hypersurface M in RP^{4}. For this, first we construct the projective dual M^{∨}, then M is obtained as the dual of M^{∨}.

Define φ : RP^{2} → RP^{4∗} by

φ([u, v, w]) = [\frac{1}{2}(u^{2} − v^{2}), \frac{1}{2}(v^{2} − w^{2}), uv, vw, wu],

which is an embedding obtained after a linear projection of the Veronese embedding ψ : RP^{2} → RP^{5∗} defined by

ψ([u, v]) = [\frac{1}{2}u^{2}, \frac{1}{2}v^{2}, \frac{1}{2}w^{2}, uv, vw, wu].

(See [Sas] Example 3, and [CDK]). Then we set M^{∨} = φ(RP^{2}). Further we set

F = X_{0} \frac{1}{2}(u^{2} − v^{2}) + X_{1} \frac{1}{2}(v^{2} − w^{2}) + X_{2}uv + X_{3}vw + X_{4}wu.

Then the ρ-projection of the projective conormal bundle M̃^{∨} of M^{∨} is obtained by eliminating u, v, w from

F = \frac{∂F}{∂u} = \frac{∂F}{∂v} = \frac{∂F}{∂w} = 0.

Then we have

| \begin{matrix}X_{0} & X_{2} & X_{4} \\ X_{2} & −X_{0} + X_{1} & X_{3} \\ X_{4} & X_{3} & −X_{1} \end{matrix} | = 0,

which is the equation of required M ⊂ RP^{4}.

In fact, M is the projectivization of the set of real symmetric matrices of determinant zero and of trace zero. Since SO(3) acts on M transitively, we see M is non-singular and M ≅ SO(3)/H, where H is the subgroup of SO(3) of order 8:

( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & −1 \end{matrix} ) , ( \begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) , ( \begin{matrix}−1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} )

( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & −1 \end{matrix} ) , ( \begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & −1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & −1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & −1 \end{matrix} ) .

In general, we set K = R, C, H, O. Then dim_{R} K = 2^{i−1}, i = 1, 2, 3, 4. Consider

J = {A ∈ M_{3}(K) | A^{∗} = A},

the space of “Hermitian” matrices of size 3 ([H][Y]). Each element A of J has the form

A = ( \begin{matrix}ξ_{1} & z_{1} & z_{3} \\ z̄_{1} & ξ_{2} & z_{2} \\ z̄_{3} & z̄_{2} & ξ_{3} \end{matrix} ) , ξ_{j} ∈ R, z_{j} ∈ K, j = 1, 2, 3.

We see

dim_{R} J = 3 · 2^{i−1} + 3 = 6, 9, 15, 27.

For A, B ∈ J, we define the Jordan product

A ∘ B = \frac{1}{2}(AB + BA) ∈ J .

Moreover we set trA = ξ_{1} + ξ_{2} + ξ_{3} ∈ R and

det A = ξ_{1}ξ_{2}ξ_{3} + 2Re((z_{2}z̄_{3})z_{1}) − ξ_{1}z_{2}z̄_{2} − ξ_{2}z_{3}z̄_{3} − ξ_{3}z_{1}z̄_{1} ∈ R,

for A ∈ J. The bilinear form tr(A ∘ B) on the real vector space J is positive definite and induces the isomorphism between J and its dual vector space J^{∗}.

Set

Σ = {A ∈ J | det A = 0}.

Then the projectivization P Σ ⊂ P J = (J − O)/R^{×} ≅ RP^{3·2^{i−1}+2} is a real cubic hypersurface. Setting

J_{0} = {A ∈ J | trA = 0},

we will see

M = P J_{0} ∩ P Σ ⊂ P J_{0} = RP^{4}, RP^{7}, RP^{13}, RP^{25},

is a non-singular real cubic developable hypersurface. The projective dual M^{∨} = KP^{2} is embedded in P J_{0}^{∗} = RP^{4∗}, RP^{7∗}, RP^{13∗}, RP^{25∗}, as a linear projection of the Veronese embedding of KP^{2} in P J ≅ P J^{∗}. Remark that rank(γ) = 2, 4, 8, 16 and the dimension of the Monge-Ampère foliation is 1, 2, 4, 8, respectively.

Recall that the projective plane over K is defined by

KP^{2} = {X ∈ J | X^{2} = X, trX = 1},

= {xx^{∗} | ^{t}x = (x_{1}, x_{2}, x_{3}) ∈ K^{3} − 0, ∥x∥ = 1, x_{1}(x_{2}x_{3}) = (x_{1}x_{2})x_{3}}

which is embedded in P J. The embedding KP^{2} ↪ P J is called the Verenose embedding [F1][F2][H, Lemma 14.90][L2][Z]. This definition fits in the ordinary one in cases K = R, C, H by the correspondence

KP^{2} ∋ [x_{1}, x_{2}, x_{3}] = [^{t}x] ↦ \frac{1}{∥x∥^{2}} xx^{∗}.

In cases K = R, C, H, we set G = O(3), U(3), Sp(3). Then G acts on J by f(A) = P^{−1}AP, (f = P ∈ G). In the case K = O, we take as G the exceptional simple Lie group

F_{4} = {f : J → J , R-linear isomorphism | f(A ∘ B) = f(A) ∘ f(B)}.

Then G preserves the Jordan product, the trace and the determinant, so G naturally acts on P J_{0}, P Σ, so on M = P J_{0} ∩ P Σ, as well as it acts on OP^{2}. Furthermore G acts on M transitively. In fact, for A ∈ J, there exists f ∈ G

such that f(A) = ( \begin{matrix}ξ_{1} & 0 & 0 \\ 0 & ξ_{2} & 0 \\ 0 & 0 & ξ_{3} \end{matrix} ) , for some ξ_{1}, ξ_{2}, ξ_{3} ∈ R. Moreover the

diagonals are permuted freely by an element of G. Then, an A ∈ J_{0} ∩ Σ is transformed into f(A) = ( \begin{matrix}0 & 0 & 0 \\ 0 & ξ & 0 \\ 0 & 0 & −ξ \end{matrix} ) , by some f ∈ G, for some ξ ∈ R.

(See, for K = O, [H] Page 313, [Y] Page 35). Also the action of G on KP^{2} is transitive. ([H] Theorem 14.99, [Y] Theorem 2.21).

Now set

Q = {([A], [B]) ∈ P J × P J | tr(A ∘ B) = 0},

the incidence hypersurface of projective duality ([Sch][IM]). Then G acts on Q naturally by f([A], [B]) = ([f(A)], [f(B)]). Since the action on M is transitive, the action on M̃ is also transitive. Here we remark that M̃ projects diffeomorphically to M by ρ. Then the key fact is the following:

Lemma 5. The projective conormal bundle of KP^{2} ⊂ P J^{∗} is described by

{ \atop KP^{2}} = P T_{KP^{2}}^{∗}P J^{∗} = \{([A], [X]) ∈ Q | X ∈ KP^{2}, A ∘ X = O\}.

Moreover its projection S = ρ(P T_{KP^{2}}^{∗}P J^{∗}) by ρ : P T_{KP^{2}}^{∗}P J^{∗} → P J onto the first component coincides with

P Σ = {[A] ∈ P J | det A = 0}.

Proof. We show for K = O; other cases are treated similarly. Let x = (x_{1}, x_{2}, x_{3}) ∈ K^{3} − 0. Write x_{i} = ∑_{j=0}^{7} x_{ij}e_{j}, i = 1, 2, 3, with the standard basis e_{0} = 1, e_{1}, …, e_{7} and x_{ij} ∈ R. Then the linear subspace T_{x_{0}}OP^{2} ⊂ J of the tangent space to OP^{2} at x_{0} = ^{t}(1, 0, 0) is generated over R by

\frac{∂}{∂x_{10}} = ( \begin{matrix}2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , \frac{∂}{∂x_{20}} = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , \frac{∂}{∂x_{2i}} = ( \begin{matrix}0 & −e_{i} & 0 \\ e_{i} & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) ,

\frac{∂}{∂x_{30}} = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ) , \frac{∂}{∂x_{3i}} = ( \begin{matrix}0 & 0 & −e_{i} \\ 0 & 0 & 0 \\ e_{i} & 0 & 0 \end{matrix} ) , 1 ≤ j ≤ 7,

while \frac{∂}{∂x_{1j}} = O, 1 ≤ j ≤ 7. Set A = ( \begin{matrix}ξ_{1} & w_{1} & w_{3} \\ w̄_{1} & ξ_{2} & w_{2} \\ w̄_{3} & w̄_{2} & ξ_{3} \end{matrix} ). Then the condition

that A annihilates T_{x_{0}}OP^{2} via the inner product tr(A ∘ B), namely that tr(A ∘ \frac{∂}{∂x_{ij}}) = O, i = 1, 2, 3, 0 ≤ j ≤ 7, is equivalent to that ξ_{1} = 0, w_{1} =

0, w_{3} = 0. This is equivalent to that

A ∘ ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) = \frac{1}{2} ( \begin{matrix}2ξ_{1} & w_{1} & w_{3} \\ w̄_{1} & 0 & 0 \\ w̄_{3} & 0 & 0 \end{matrix} )

equals to O. By the transitivity we have the first half. The second half follows from the following Lemma.

Lemma 6. For A ∈ J, (1) A ∘ X = O, for some X ∈ KP^{2}, if and only (2) det A = 0.

Proof. (1) ⇒ (2): Choose f ∈ G such that f(X) = X_{0} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ).

Then, since f(A) ∘ X_{0} = f(A ∘ X) = O, we see det A = det f(A) = 0. (2) ⇒ (1): Take f ∈ G such that

f(A) = ( \begin{matrix}ξ_{1} & 0 & 0 \\ 0 & ξ_{2} & 0 \\ 0 & 0 & ξ_{3} \end{matrix} ) ,

for some ξ_{1}, ξ_{2}, ξ_{3} ∈ R. Then det f(A) = det A = 0, so ξ_{1}ξ_{2}ξ_{3} = 0, thus ξ_{i} = 0, for some i. Changing f if necessary, we may assume ξ_{1} = 0. Then A ∘ f^{−1}(X_{0}) = f(A) ∘ X_{0} = O. □

Thus we see the projective dual of the hyperplane section M = S ∩P J_{0} ⊂ P J_{0} is the linear projection of KP^{2} ⊂ P J ≅ P J^{∗} from the point in P J^{∗} corresponding to the hyperplane P J_{0} ⊂ P J.

Set

A_{0} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) ∈ J_{0} ∩ Σ, and, X_{0} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) ∈ OP^{2} ⊂ J .

Then ([A_{0}], [X_{0}]) ∈ M̃. Let K = O and G = F_{4}. Then the isotropy group for [X_{0}] ∈ P J of the F_{4}-action is isomorphic to Spin(9) ([H] Theorem 14.99,

[Y] Theorem 2.10). Further the isotropy group for A_{0} ∈ J of the F_{4}-action on J is

{f ∈ F_{4} | f(E_{i}) = E_{i}, i = 1, 2, 3},

which is isomorphic to Spin(8). Here E_{i} is the 3×3 matrix with (i, i)-element 1 which is the only non-zero element. (So E_{1} = X_{0}, A_{0} = E_{2}−E_{3}). ([H] Page 313, [Y] Theorem 2.7). Then the isotropy group for [A_{0}] in M is isomorphic to a Z/2Z-extension of Spin(8). Thus we see that the Monge-Ampère foliation is in fact a fibration γ : M → OP^{2} described as in §0.

Similarly we have, in cases K = R, C, H, that the Monge-Ampère foliation is given by the fibration γ : M → KP^{2} which is described as the fiberwise Z/2Z-quotient with respect to the antipodal involution of KP^{1} ≅ S^{2^{i−1}}(i = 1, 2, 3) of the following fibration: For K = R,

\begin{matrix}RP^{1} ≅ O(2)/O(1) × O(1) → O(3)/O(1) × O(1) × O(1) → \\ → O(3)/O(2) × O(1) ≅ RP^{2}, \end{matrix}

For K = C,

\begin{matrix}CP^{1} ≅ U(2)/U(1) × U(1) → U(3)/U(1) × U(1) × U(1) → \\ → U(3)/U(2) × U(1) ≅ CP^{2}, \end{matrix}

and for K = H,

HP^{1} ≅ Sp(2)/Sp(1) × Sp(1) → Sp(3)/Sp(1) × Sp(1) × Sp(1) →
→ Sp(3)/Sp(2) × Sp(1) ≅ HP^{2}.

In particular, M ∈ RP^{n}, (n = 3 · 2^{i−1} + 1, i = 1, 2, 3, 4) is a homogeneous space of SO(3), SU(3), Sp(3) and F_{4}, respectively.

The last statement of Proposition 2 is clear, since the condition (ii) of Lemma 4 is an open condition for immersions KP^{2} → RP^{n∗}.

3. Proof of Theorem 3

Let M be a connected developable homogeneous hypersurface of compact type. By definition, M is an orbit of a compact Lie subgroup G of GL(n + 1, R). We take a G-invariant metric on R^{n+1}. Then, after a projective transformation on RP^{n}, we may assume G ⊂ O(n + 1) and now we use the metric geometry.

Let π : S^{n} → RP^{n} be the natural double covering from the unit sphere S^{n} ⊂ R^{n+1} and endow with RP^{n} the induced metric from S^{n}.

A hypersurface in RP^{n} or S^{n} is called isoparametric if its principal curvatures are constant (see [CR]). The following fact is suggested by K. Tsukada to the author:

Lemma 7. M ⊂ RP^{n} is developable if and only if 0 is a principal curvature of M at each point in M.

Proof. The condition that M is developable is equivalent to that the shape operators are degenerate at any points x ∈ M. The latter is equivalent to that 0 is a principal curvature at each point in M. □

Now M is a G-orbit and G acts on RP^{n} isometrically. So M is a connected compact isoparametric hypersurface.

Actually, we are going to show the following, which completes the proof of Theorem 3.

Proposition 8. Let M ⊂ RP^{n} be projectively equivalent to a connected compact isoparametric hypersurface in RP^{n}. If M is developable, then M is a projective hyperplane or a Cartan hypersurface.

Remark 9: It is known that there exist compact isoparametric hypersurfaces in spheres which are not (isometrically) homogeneous with g = 4 [OT]. Recently Reiko Miyaoka [Mi] has proved that every compact isoparametric hypersurfaces in spheres with g = 6 are necessarily homogeneous, as well as the cases g ≤ 3.

Proof of Proposition 8. Take a connected component of π^{−1}(M) and denote it by M̄. Then M̄ is a connected compact isoparametric hypersurface of S^{n}. Moreover, by Lemma 7, 0 is a principal curvature of M and therefore of M̄.

Let λ_{1}, …, λ_{g} be the disjoint principal curvatures of M̄. Set λ_{i} = cot θ_{i}, 0 < θ_{i} < π, with θ_{1} < θ_{2} < ⋯ < θ_{g}. Moreover denote by m_{i} the multiplicity of the principal curvature λ_{i}. Then, by the general theory of isoparametric hypersurfaces [CR], we have the following:

(a) θ_{i} = θ_{1} + \frac{i − 1}{g}π, 1 ≤ i ≤ g.

(a’) m_{i} = m_{i+2}, (subscripts mod. g).

(b) g = 1, 2, 3, 4, 6, (Münzner).

Now 0 is a principal curvature, so θ_{k} = \frac{π}{2} , for some k, 1 ≤ k ≤ g. Then, by (a), g must be odd and \{θ_{i}\}_{1≤i≤g} are symmetric with respect to \frac{π}{2} . Then, by (a’), all multiplicities are same. Further, we see that the mean curvature of M̄ is identically zero, namely, M̄ is minimal. Besides, by (b), we have g = 1 or 3.

When g = 1, M̄ must be a great sphere in S^{n}. Then M is a projective hypersurface.

Let g = 3. É. Cartan [Car] gave the complete classification of isoparametric hypersurfaces with three distinct principal curvatures (with same multiplicity) in spheres, up to isometry. Those are given by level hypersurfaces of f = det |S^{n} : S^{n} → R. Here det : J_{0} → R is the determinant and S^{n} ⊂ J_{0} is the unit sphere with respect to the metric tr( ·) : J_{0}×J_{0} → R. Among them, minimal hypersurfaces are only f^{−1}(0) ∈ S^{n}. Hence M̄ ⊂ S^{n} is isometric to f^{−1}(0) = {A ∈ S^{n} | det(A) = 0}. Now an isometry of S^{n} is induced by an element O(n + 1) ⊂ GL(n + 1, R). Thus M is projectively equivalent to an example of Proposition 2. Therefore M is a Cartan hypersurface. □

This completes the proof of Theorem 3. □ Remark 10: There are known several alternative proofs of Cartan’s theorem [Car]: See, for instance, [Kar], [KK], [CO]. Remark 11: It is suggested by H. Tazaki to the author, that Theorem 3 can be shown directly, using the concrete classification of (isometrically) homogeneous hypersurfaces in spheres by Hsiang and Lawson [HL], (see also [OT], [Ya]), and the formula of principal curvatures due to Takagi and Takahashi [TT].

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Department of Mathematics, Hokkaido University, Sapporo 060, Japan

E-mail address: ishikawa@math.sci.hokudai.ac.jp

Received July 27, 1999