Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 2, 413-421.

Lagrangian Submanifolds of Quaternion Kaehlerian Manifolds

Satisfying Chen’s Equality

Dedicated to Professor Qi Keng Lu on the occasion of his seventieth birthday

Yi Hong* Chorng Shi Houh

Department of Applied Mathematics, South China Institute of Technology Guangzhou, China Department of Mathematics, Wayne State University Detroit, MI 48202, U.S.A

Abstract. Lagrangian submanifolds satisfying Chen’s equality in a quaternionspace-form are investigated. In particular, we prove that (1) every Lagrangian submanifold of a quaternion-space-form satisfying Chen’s equality must be minimal and (2) the only Lagrangian isometric immersion from a Riemannian n-sphere into a quaternionic projective space P^{n}(H) satisfying Chen’s equality is the totally geodesic one.

1. Preliminary

Let M̃^{m} be a 4m-dimensional Riemannian manifold with metric g. M̃^{m} is called a quaternionic Kaehlerian manifold if there exists a 3-dimensional vector space V of tensors of type (1, 1) with local basis of almost Hermitian structure I, J and K such that

(a) IJ = −JI = K, JK = −KJ = I, KI = −IK = J, I^{2} = J^{2} = K^{2} = −1,

(b) for any local cross-section φ of V, ˜∇_{X}φ is also a cross-section of V, where X is an arbitrary vector field on M̃^{m} and ˜∇ the Riemannian connection on M̃^{m}.

Condition (b) is equivalent to the following condition: (b’) there exist local 1-forms p, q and r such that

(1.1)  \begin{matrix}˜∇_{X}I = & r(X)J − q(X)K, \\ ˜∇_{X}J = & − r(X)I  + p(X)K, \\ ˜∇_{X}K = & q(x)I − p(X)J. \end{matrix}

* The first author was partially supported by project 19771039, NSFC.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1998 Heldermann Verlag

414 Y. Hong, C. S. Houh: Lagrangian Submanifolds of Quaternion Kaehlerian …

Let X be a unit vector on M̃^{m}. Then X, IX, JX and KX form an orthonormal frame on M̃^{m}. We denote by Q(X) the 4-plane spanned by them. For any two orthonormal vectors X, Y on M̃^{m}, if Q(X) and Q(Y) are orthogonal, the plane π(X, Y) spanned by X, Y is called a totally real plane. Any 2-plane in a Q(X) is called a quaternionic plane. The sectional curvature of a quaternionic plane π is called the quaternionic sectional curvature of π. A quaternionic Kaehlerian manifold is a quaternion-space-form if its quaternionic sectional curvatures are equal to a constant, say 4c. We denote such a 4m-dimensional quaternion-space-form by M̃^{m}(4c). A quaternionic projective space, denoted by P^{m}(H), is a quaternionic Kaehlerian manifold of constant quaternionic sectional curvature.

It is known that a quaternionic Kaehlerian manifold M̃^{m} is a quaternion-space-form if and only if its curvature tensor R̃ is of the following form [7]:

(1.2)  \begin{matrix}R̃(X, Y)Z = & c\{g(Y, Z)X − g(X, Z)Y \\ + g(IY, Z)IX − g(IX, Z)IY + 2g(X, IY)IZ \\ + g(JY, Z)JX − g(JX, Z)JY + 2g(X, JY)JZ \\ + g(KY, Z)KX − g(KX, Z)KY + 2g(X, KY)KZ\} \end{matrix}

for vectors X, Y, Z tangent to M̃^{m}.

Let N be an n-dimensional Riemannian manifold and x : N → M̃^{m}(4c) be an isometric immersion of N into a quaternion-space-form M̃^{m}(4c). We call N a totally real submanifold of M̃^{m}(4c) if each 2-plane of N is mapped by x into a totally real plane in M̃^{m}(4c). Consequently, if N is a totally real submanifold of M̃^{m}(4c), then φ(T N) ⊂ T^{⊥}N for φ = I, J or K, where T^{⊥}N is the normal bundle of N in M̃^{m}(4c).

An n-dimensional totally real submanifold of a quaternion-space-form M̃^{m}(4c) is called a Lagrangian submanifold when n = m.

Assume that N is a totally real submanifold of M̃^{m}(4c). For any orthonormal vectors X, Y in N, the plane π(X, Y) spanned by X and Y is totally real in M̃^{m}(4c), Q(X) and Q(Y) are orthogonal and g(X, φY) = g(ψX, Y) = 0 for φ, ψ = I, J or K.

By (1.2) we have

R̃(X, Y)Z = c{g(Y, Z)X − g(X, Z)Y }, X, Y, Z ∈ T N.

By the Gauss formula the curvature tensor R of N satisfies

(1.3)  \begin{matrix}g(R(X, Y)Z, W) = & c\{g(Y, Z)g(X, W) − g(X, Z)g(Y, W)\} \\ + g(h(Y, Z), h(X, W)) − g(h(X, Z), h(Y, W)), \end{matrix}

where h is the second fundamental form of N.

Since dim N = n, and N is totally real in M̃^{m}(4c), we have n ≤ m. Let p = 4m − n. In M̃^{m}(4c) we can choose a local field of orthonormal frames:

(1.4)  \begin{matrix}e_{1}, …, e_{n}, e_{n+1}, …, e_{m}; e_{I(1)} = Ie_{1}, …, e_{I(m)} = Ie_{m}; \\ e_{J(1)} = Je_{1}, …, e_{J(m)} = Je_{m}; e_{K(1)} = Ke_{1}, …, e_{K(m)} = Ke_{m}, \end{matrix}

in such a way that, restricting to N, e_{1}, e_{2}, …, e_{n} are tangent to N.

We shall use the following convention in the range of indices unless mentioned otherwise:

A, B, C, D, … = 1, …, m, I(1), …, I(m), J(1), …, J(m), K(1), …, K(m),
i, j, k, l, … = 1, …, n,
r, s, t, … = n + 1, …, m, I(1), …, K(m),
u, v, … = n + 1, …, m,
φ, ψ, … = I, J, K.

By (1.3) the Ricci tensor of N is given by

\begin{aligned}S(X, Y) & = ∑_{i} g(R(e_{i}, X)Y, e_{i}) \\ & = (n − 1)cg(X, Y) + ∑_{r}\{(T rA_{r})g(A_{r}X, Y) − (g(A_{r}X, A_{r}Y))^{2}\}, \end{aligned}

where A_{r} = A_{e_{r}}, −A_{e_{r}}X is the tangential component of ˜∇_{X}e_{r} (= −A_{r}X + De_{r}), and it relates to the second fundamental form h by

g(h(X, Y), e_{r}) = g(A_{r}X, Y).

2. Chen’s inequality

The scalar curvature τ of N is given by

(2.1)  τ = n^{2}|H|^{2} − |h|^{2} + n(n − 1)c,

where H denotes the mean curvature vector of N and |h|^{2} the square length of the second fundamental form. Put

(2.2)  η = τ − \frac{n^{2}(n − 2)}{n − 1}|H|^{2} − (n + 1)(n − 2)c.

Substituting (2.1) in (2.2) we have η = \frac{n^{2}}{n−1}|H|^{2} − |h|^{2} + 2c, or equivalently,

(2.3)  n^{2}|H|^{2} = (n − 1)(|h|^{2} + (η − 2c)).

Let

\begin{matrix}ω^{1}, …, & ω^{n}, ω^{n+1}, …, ω^{m}, ω^{I(1)}, …, ω^{I(m)}, …, \\ ω^{J(1)}, …, ω^{J(m)}, ω^{K(1)}, …, ω^{K(m)} \end{matrix}

be the dual frame of the frame given by (1.4).

Since N is totally real, Q(e_{i}) and Q(e_{j}), i ≠ j, are orthogonal. Thus g(φ(e_{i}), ψ(e_{j})) = 0 when i ≠ j. From the structure equations:

dω^{A} = −Σω_{B}^{A} ∧ ω^{B}, ω_{B}^{A} + ω_{A}^{B} = 0

and (1.1), we obtain

(2.4)  \begin{matrix}ω_{j}^{i} = ω_{φ(j)}^{φ(i)}, ω_{j}^{φ(i)} = ω_{i}^{φ(j)}, ω_{u}^{i} = ω_{φ(u)}^{φ(i)}, \\ ω_{u}^{φ(i)} = ω_{i}^{φ(u)}, ω_{v}^{u} = ω_{φ(v)}^{φ(u)}, ω_{v}^{φ(u)} = ω_{u}^{φ(v)}. \end{matrix}

If we write ω_{i}^{r} = ∑_{j} h_{ij}^{r}ω^{j} then h_{ij}^{r} = h_{ji}^{r}, h_{ij}^{r} = g(A_{r}e_{i}, e_{j}) and the mean curvature vector of N is H = \frac{1}{n} ∑_{i,r} h_{ii}^{r}e_{r}.

From (2.4) we have

(2.5)  h_{jk}^{φ(i)} = h_{ik}^{φ(j)} = h_{ji}^{φ(k)}.

Let us choose e_{n+1} to be the unit vector in the direction H, H = |H|e_{n+1}, then (2.3) can be rewritten as

(2.6) ( ∑ h_{ii}^{n+1})^{2} = (n − 1){ ∑(h_{ii}^{n+1})^{2} + ∑_{i≠j}(h_{ij}^{n+1})^{2} + ∑_{r=n+2}^{4m} ∑_{i,j=1}^{n}(h_{ij}^{r})^{2} + η − 2c}.

By (2.6) and using an algebraic lemma given in [2], B.-Y. Chen proved in [2] the following inequality:

(2.7)  2h_{11}^{n+1}h_{22}^{n+1} ≥ ∑_{i≠j}(h_{ij}^{n+1})^{2} + ∑_{r=n+2}^{4m} ∑_{i,j=1}^{n}(h_{ij}^{r})^{2} + η − 2c

with the equality holding if and only if

(2.8)  h_{11}^{n+1} + h_{22}^{n+1} = h_{33}^{n+1} = ⋯ = h_{nn}^{n+1}.

Let π ⊂ T_{p}N be a plane section. If we choose orthonormal vectors e_{1}, e_{2} which span π, then, by (1.3), the sectional curvature K(π) is given by

\begin{matrix}K(π) = & c + ∑_{r}\{g(A_{r}e_{1}, e_{1})g(A_{r}e_{2}, e_{2}) − (g(A_{r}e_{1}, e_{2}))^{2}\} \\ = & ∑_{r=n+1}^{4m} h_{11}^{r}h_{22}^{r} − ∑_{r=n+1}^{4m} h_{12}^{r}h_{12}^{r} + c. \end{matrix}

From (2.7), we find

K(π) ≥\frac{1}{2} ∑_{i≠j}(h_{ij}^{n+1})^{2} + \frac{1}{2} ∑_{r=n+2}^{4m} ∑_{i,j}(h_{ij}^{r})^{2} + \frac{η}{2} + ∑_{r=n+2}^{4m} h_{11}^{r}h_{22}^{r} − ∑_{r=n+1}^{4m} h_{12}^{r}h_{12}^{r}

= ∑_{r=n+1}^{4m} ∑_{j>2}\{(h_{1j}^{r})^{2} + (h_{2j}^{r})^{2}\} + \frac{1}{2} ∑_{i,j>2}^{i≠j}(h_{ij}^{r})^{2}

+ \frac{1}{2} ∑_{r=n+2}^{4m} ∑_{i,j>2}(h_{ij}^{r})^{2} + \frac{1}{2} ∑_{r=n+2}^{4m}(h_{11}^{r} + h_{22}^{r})^{2} + \frac{η}{2} .

Hence we obtain from (2.2) that

K(π) ≥ \frac{η}{2} = \frac{1}{2}\{τ − \frac{n^{2}(n − 2)}{n − 1}|H|^{2} − (n + 1)(n − 2)c\},

with the equality holding if and only if

h_{1j}^{n+1} = 0, h_{2j}^{n+1} = 0, j > 2,
h_{ij}^{n+1} = 0, i ≠ j > 2,
h_{1j}^{r} = h_{2j}^{r} = h_{ij}^{r} = 0, r = n + 2, …, 4m, i, j = 3, …, n,
h_{11}^{r} + h_{22}^{r} = 0, r ≥ n + 2,
h_{11}^{n+1} + h_{22}^{n+1} = h_{33}^{n+3} = ⋯ = h_{nn}^{n+1}.

Consequently, we have the following.

Proposition 1. Let N be an n-dimensional (n ≥ 2) totally real submanifold of a quaternion-space-form M̃^{m}(4c). Then

(2.9)  δ ≤ \frac{n^{2}(n − 2)}{2(n − 1)}|H|^{2} + \frac{1}{2}(n + 1)(n − 2)c,

where δ is the Chen invariant defined by δ = \frac{1}{2}τ − inf K.

The equality case of (2.9) holds at a point p ∈ N if and only if, with respect to a suitable orthonormal basis {e_{1}, …, e_{n}, e_{n+1}, …, e_{m}} at p, the operators A_{r} = A_{e_{r}} take the following forms:

A_{n+1} = ( \begin{matrix}a & 0 & · & · & · & 0 \\ 0 & b & 0 & · & · & 0 \\ 0 & 0 & μ & 0 & · & 0 \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ 0 & · & · & · & · & μ \end{matrix} ) , a + b = μ;

A_{r} = ( \begin{matrix}h_{11}^{r} & h_{12}^{r} & 0 & · & 0 \\ h_{12}^{r} & −h_{11}^{r} & 0 & · & 0 \\ 0 & 0 & · & · & 0 \\ · & · & · & · & · \\ · & · & · & · & · \\ 0 & 0 & · & · & 0 \end{matrix} ) , n + 2 ≤ r ≤ 4m.

Corollary. Let N be an n-dimensional totally real submanifold of a quaternion-space-form M̃^{m}(4c). If the equality in (2.9) holds and ξ is a normal vector at a point p of N, then the operator A_{ξ} has at most 3 eigenspaces and the dimension of one of the eigenspaces is at least n − 2. Moreover, referring to the entries in the proposition, one of the following cases must occur:

(1) A_{ξ} = 0, this occurs when a = b = 0 or h_{11}^{r} = h_{12}^{r} = 0 for some r where ξ = e_{n+1} or e_{r}.

(2) A_{ξ} has a two-dimensional eigenspace with a non-zero real number α as the eigenvalue and an (n − 2)-dimensional eigenspace with eigenvalue 2α. This occurs only when a = b = α, μ = 2α, ξ = e_{n+1}.

(3) A_{ξ} has a one-dimensional eigenspace with eigenvalue zero and an (n − 1)-dimensional eigenspace with nonzero eigenvalue α. This occurs when α = μ, b = 0, ξ = e_{n+1}.

(4) A_{ξ} has an eigenspace with eigenvalue, say α, another eigenspace with eigenvalue, say β, both of them are one-dimensional, and an (n − 2)-dimensional eigenspace with eigenvalue α + β. This occurs when ξ = e_{n+1}, α ≠ β or ξ = e_{r} for some r with α = −β = ±\sqrt{(h_{11}^{r})^{2} + (h_{12}^{r})^{2}}.

Inequality (2.9) is known as Chen’s inequality. The submanifold N is said to satisfy Chen’s equality if the equality case of (2.9) holds identically.

We will prove the following theorems in the next sections.

Theorem 1. Let N be a Lagrangian submanifold of dimension > 2 in a quaternion-space-form M̃^{m}(4c). If N satisfies Chen’s equality, then N is minimal.

There exist many non-totally geodesic Lagrangian submanifolds in quaternionic projective spaces satisfying Chen’s equality. In contrast, we have the following uniqueness result for spheres.

Theorem 2. The only Lagrangian isometric immersion from a Riemannian n-sphere into a quaternionic projective space P^{n}(H) which satisfies Chen’s equality is the totally geodesic one.

3. Some Lemmas

Let N be a totally real submanifold of M̃^{m}(4c), {ϕ_{r}, ϕ_{s}, ϕ_{t}} be the set {I, J, K} or a set of a circular permutation of the three elements I, J, and K.

Lemma 1. [6] For any X, Y, Z, U and V in T N, we have

(a) R̃(U, V, ϕ_{r}X, ϕ_{r}Y) = R̃(U, V, X, Y),

(b) < h(X, Y), ϕ_{r}Z >=< h(Z, Y), ϕ_{r}X >, r = 1, 2, 3.

The following Lemma follows immediately from Lemma 1 (b).

Lemma 2. If N is a Lagrangian submanifold, then

h(X, Y) = IA_{IX}Y + JA_{JX}Y + KA_{KX}Y.

Lemma 3. For X ∈ T N and ξ ∈ I(T N) ⊕ J(T N) ⊕ K(T N), we have

A_{IX}(Iξ) = A_{JX}(Jξ) = A_{KX}(Kξ) = −A_{ξ}X, A_{ϕ_{r}X}(ϕ_{s}ξ) = 0 (r ≠ s).

Proof. It is clear that ϕ_{r}ξ ∈ T N. For any Y ∈ T N, Lemma 1 (b) yields

\begin{aligned}< A_{ϕ_{r}X}(ϕ_{r}ξ), Y > & =< h(ϕ_{r}ξ, Y), ϕ_{r}X >=< h(X, Y), ϕ_{r}^{2}ξ > \\ & = − < h(X, Y), ξ >= − < A_{ξ}X, Y > . \end{aligned}

This proved the first relation. The second relation follows from

\begin{matrix}< & A_{ϕ_{r}X}(ϕ_{s}ξ), Y >=< h(ϕ_{s}ξ, Y), ϕ_{r}X > \\ =< h(X, Y), ϕ_{r}(ϕ_{s}ξ) >=< h(X, Y), ϕ_{t}ξ >= 0 \end{matrix}

by virtue of Lemma 1 (b).

4. Proof of Theorems

Let N be a Lagrangian submanifold of dimension n > 2 in a quaternion-space-form M̃^{m}(4c). Assume that N satisfies Chen’s equality. Let p be a given point in N and ξ a normal vector to N at p.

Suppose that A_{ξ} has a one-dimensional eigenspace spanned by an eigenvector v with a non-zero eigenvalue. Then by case 4 in the corollary of Proposition 1, for any normal vector η to N, {A_{η}v : η ∈ T^{⊥}N} is at most two-dimensional and is invariant under any A_{η}. Furthermore, assume A_{ξ} has a one-dimensional eigenspace with an eigenvalue α, a one-dimensional eigenspace with eigenvalue β and an (n − 2)-dimensional eigenspace with eigenvalue α+β. Let the (n−2)-dimensional eigenspace be W and the frame {e_{1}, e_{2}, …, e_{n}} so chosen that W = {e_{3}, …, e_{n}}.

For any w ∈ W if A_{ξ}w = μw for some μ, then A_{η}w = 0 for any normal vector orthogonal to ξ. Also for an normal vector η orthogonal to ξ if A_{η}w = 0, then A_{ξ}w = μw for some μ. In any case for any w ∈ W there is a real number μ such that A_{η}w = μw for any normal vector to N. We sum up this as the following.

Lemma 4. Suppose that N is a Lagrangian submanifold of dimension > 2 in M̃^{n}(4c) satisfying Chen’s equality. Let p ∈ N. Then we have

(1) If there exists a normal vector ξ at p such that A_{ξ} has a one-dimensional eigenspace with nonzero eigenvalue and a unit eigenvector v, then the vector space spanned by vectors A_{η}v is at most two-dimensional, invariant under all A_{η} and contains v, where η is any normal vector at p.

(2) If A_{ξ} has eigenvalue α, β and α + β and W the eigenspace associated with the eigenvalue α + β with dim W = n − 2. Then, for any normal vector η at p, there is a real number μ such that A_{η}w = μw for w ∈ W.

At a given point p ∈ N let UN_{p} = {v ∈ T_{p}N : < v, v >= 1}. Then UN_{p} is a compact set. Let ϕ be one of the I, J, K and consider the function f : UN_{p} → R associated with ϕ defined by u ↦ f(u) =< h(u, u), ϕu >. Following the same argument as in [4] we can derive the following. When f has a maximum at v ∈ T_{p}N, then v is an eigenvector of A_{ϕv}, A_{ϕv}v = λ_{1}v with eigenvalue λ_{1} > 0. Choose a frame {e_{1}, e_{2}, …, e_{n}} of T_{p}N so that e_{1} = v and e_{i} (i ≥ 2) be an eigenvector of A_{ϕe_{1}} with eigenvalue λ_{i}. The function f_{i}(t) = f(cos t e_{1} + sin t e_{2}) has relative maximum at t = 0, f′′_{i}(0) ≤ 0. This will lead to the inequality λ_{1} ≥ 2λ_{i}. Since λ_{1} > 0 we have λ_{i} ≠ λ_{1} for all i ≥ 2. By Lemma 1, A_{ϕe_{i}}e_{1} = A_{ϕe_{1}}e_{i} = λ_{i}e_{i}.

According to Proposition 1 we conclude that λ_{1} + λ_{2} = 0, λ_{3} = λ_{4} = ⋯ = λ_{n} = 0 for eigenvalues of A_{ϕe_{1}} and

(4.1)  A_{ϕe_{1}}e_{1} = λ_{1}e_{1}, A_{ϕe_{1}}e_{2} = −λ_{1}e_{2}, A_{ϕe_{1}}e_{i} = 0, i > 2.

Since e_{1} spans a one-dimensional eigenspace of A_{ϕe_{1}} with non-zero eigenvalue λ_{1} and A_{ϕe_{1}}e_{2} = −λ_{1}e_{2}, the space {e_{1}, e_{2}} spanned by e_{1}, e_{2} is invariant under A_{η} for an arbitrary normal vector η by Lemma 4.

For the normal vector ϕe_{2}, using Lemma 1 we find

A_{ϕe_{2}}e_{1} = −λ_{1}e_{2}, A_{ϕe_{2}}e_{2} = −λ_{1}e_{1} + ρe_{2}.

This concludes that A_{ϕe_{2}} takes the form of the second matrix in Proposition 1 and ρ = 0. Consequently, we have

(4.2)  A_{ϕe_{2}}e_{1} = −λ_{1}e_{2}, A_{ϕe_{2}}e_{2} = −λ_{1}e_{1}, A_{ϕe_{2}}e_{i} = 0, i ≥ 3.

For A_{ϕe_{k}}, k ≥ 3, since its eigenspace spanned by e_{1}, e_{2} has eigenvalue 0, A_{ϕe_{k}}e_{1} = A_{ϕe_{1}}e_{k} = 0, and A_{ϕe_{k}}e_{2} = A_{ϕe_{2}}e_{k} = 0, we have

(4.3)  A_{ϕe_{k}} = 0, k ≥ 3.

From (4.1), (4.2) and (4.3), A_{ϕe_{i}} has one of the following forms for i = 1, …, n:

(4.4) A_{ϕe_{1}} = ( \begin{matrix}λ_{1} & 0 & · & 0 \\ 0 & −λ_{1} & · & 0 \\ · & · & · & · \\ 0 & 0 & · & 0 \end{matrix} ) , A_{ϕe_{2}} = ( \begin{matrix}0 & −λ_{1} & · & 0 \\ −λ_{1} & 0 & · & 0 \\ · & · & · & · \\ 0 & 0 & · & 0 \end{matrix} ) , A_{ϕe_{i}} = 0 i ≥ 3.

Similarly, for φ ∈ \{I, J, K\} other than ϕ, there also exists a frame \{\overline{e}_{1}, \overline{e}_{2}, …, \overline{e}_{n}\} on T N such that with respect to this frame we have

A_{φ\overline{e}_{1}} = ( \begin{matrix}μ_{1} & 0 & · & 0 \\ 0 & −μ_{1} & · & 0 \\ · & · & · & · \\ 0 & 0 & · & 0 \end{matrix} ) , A_{φ\overline{e}_{2}} = ( \begin{matrix}0 & −μ_{1} & · & 0 \\ −μ_{1} & 0 & · & 0 \\ · & · & · & · \\ 0 & 0 & · & 0 \end{matrix} ) , A_{φ\overline{e}_{i}} = 0, i ≥ 3,

for some μ_{1}. Therefore, the submanifold N is minimal in M̃^{n}(4c). This proves Theorem 1.

Theorem 2 follows immediately from Theorem 1 and Theorem 9 of [6].

Acknowledgement. The authors would like to express their thanks to the referee who made suggestions and corrected some mistakes, especially on the formulation of Theorem 2.

References

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[3] Chen, B.-Y.: A Riemannian invariant and its applications to submanifold theory. Results Math. 27 (1995), 17–26.

[4] Chen, B.-Y.; Dillen, F.; Verstraelen, L.; Vrancken, L.: Totally real submanifolds of CP^{n} satisfying a basic equality. Arch. Math. 63 (1994), 553–564.

[5] Chen, B.-Y.; Dillen, F.; Verstraelen, L.; Vrancken, L.: An exotic totally real minimal immersion of S^{3} in CP^{3} and its characterization. Proc. Royal Soc. Edinburgh, Sect. A, Math. 126 (1996), 153–165.

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Received October 1, 1997