Real hypersurfaces in complex two-plane Grassmannians with parallel shape operator II

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH

PARALLEL SHAPE OPERATOR II

Young Jin Suh

Abstract. In this paper we consider the notion of ξ-invariant or D

-invariant real hypersurfaces in a complex two-plane Grassman- nian G

(C

) and prove that there do not exist such kinds of real hypersurfaces in G

( C

) with parallel second fundamen- tal tensor on a distribution F defined by F = ξ ∪ D

, where D

= Span {ξ

, ξ

, ξ

}.

0.1. Introduction

In the study of real hypersurfaces in complex space forms or in quater- nionic space forms it can be easilily checked that there do not exist any real hypersurfaces with parallel second fundamental tensor by the equa- tion of Codazzi.

Let us denote by G ₂ ( C ^m+2 ) the set of all complex 2-dimensional linear subspaces in C ^m+2 . Then the above situation is not so simple when we consider a real hypersurface in a complex two-plane Grassmannian G ₂ ( C ^m+2 ).

In this paper we study the analogous question in the complex Grass- mann manifold G ₂ ( C ^m+2 ) of all two-dimensional linear subspaces in C ^m+2 . This Riemannian symmetric space has a remarkable geometrical structure. It is the unique compact irreducible Riemannian manifold be- ing equipped with both a K¨ ahler structure J and a quaternionic K¨ ahler structure J not containing J. In other words, G ₂ ( C ^m+2 ) is the unique compact, irreducible, K¨ ahler, quaternionic K¨ ahler manifold which is not a hyperk¨ ahler manifold.

Received May 6, 2003.

2000 Mathematics Subject Classification: Primary 53C40; Secondary 53C15.

Key words and phrases: complex Grassmannians, real hypersurfaces, tubes, shape operator, K¨ ahler structure, quaternionic K¨ ahler structure.

This work was supported by grant Proj. No. R14-2002-003-01001-0 and Proj.

No. R05-2002-000-00047-0 from the Korea Science & Engineering Foundation.

Now let M be a real hypersurfaces in G ₂ ( C ^m+2 ) with a unit normal vector field N of M in G ₂ ( C ^m+2 ). So, in G ₂ ( C ^m+2 ) we have the two natural geometrical conditions for real hypersurfaces that ξ or D ^⊥ = Span {ξ 1 , ξ ₂ , ξ ₃ } is invariant under the shape operator, where ξ = −JN, ξ _i = −J _i N for i = 1, 2, 3, and J _i denotes an element in a quaternionic K¨ ahler structure J.

The main result of this paper is to prove the non-existence of all real hypersurfaces in G ₂ ( C ^m+2 ) with parallel second fundamental tensor on [ξ] ∪D ^⊥ when [ξ] = Span {ξ} or D ^⊥ = Span {ξ 1 , ξ ₂ , ξ ₃ } is invariant under the shape operator of M . From this view point Berndt and the present author [4] have proved the following

Theorem A. Let M be a connected real hypersurface in G ₂ ( C ^m+2 ), m ≥ 3. Then both [ξ] and D ^⊥ are invariant under the shape operator of M if and only if

(1) M is an open part of a tube around a totally geodesic G ₂ ( C ^m+1 ) in G ₂ ( C ^m+2 ), or

(2) m is even, say m = 2n, and M is an open part of a tube around a totally geodesic HP ⁿ in G ₂ ( C ^m+2 ).

In the proof of Theorem A we have proved that the one-dimensional distribution [ξ] is contained in either the 3-dimensional distribution D ^⊥ or in the orthogonal complement D such that T _x M = D⊕D ^⊥ . The case (2) in Theorem A is just the case that the one dimensional distribution [ξ] is contained in D ^⊥ .

We have mentioned that there do not exist any real hypersurfaces with parallel second fundamental form in complex space forms and in quater- nionic space forms. But when we consider its situation in G ₂ ( C ⁿ⁺² ), it is not so simple to prove it. From this point of a view the present author [11] has proved the following.

Theorem B. There do not exist any real hypersurface M in G ₂ ( C ^m+2 ) with parallel second fundamental form, that is, ∇A = 0.

On the other hand, Kimura and Maeda [8] have proved that a real

hypersurface M in a complex projective space CP ⁿ satisfying ∇ ξ A = 0

is locally congruent to a real hypersurface of type A ₁ , A ₂ , that is, a tube

over a totally geodesic complex submanifold CP ^k with radius 0 < r < ^π ₂ .

In a quaternionic projective space HP ^m P´ erez [9] has considered the

notion of ∇ ξ

A = 0, i = 1, 2, 3, for real hypersurfaces in HP ^m and

classified that M is locally congruent to of A ₁ , A ₂ -type, that is, a tube

over HP ^k with radius 0 < r < ^π ₄ . Moreover, in a paper [10] due to P´ erez

and the present author we have considered the notion of ∇ ξ

R = 0, i = 1, 2, 3, where R denotes the curvature tensor of a real hypersurface M in HP ^m , and proved that M is locally congruent to a tube of radius

π 4 over HP ^k .

But if we consider such notions in a complex two-plane Grassmannian G ₂ ( C ^m+2 ), then its situations are quite different from the above ones.

Now in this paper let us consider a distribution F combined by [ξ]

and D ^⊥ in such a way that F = [ξ] ∪ D ^⊥ for a real hypersurface M in G ₂ ( C ^m+1 ). Then naturally as in above classifications we can restrict the condition ∇A = 0 to F, which means that the second fundamental tensor of M in G ₂ ( C ^m+2 ) is parallel on F. Of course, this condition is much more weaker than ∇A = 0 as in Theorem B. Then by virtue of Theorem A we assert the following remarkable facts :

Theorem 1. There do not exist any real hypersurface M in G ₂ ( C ^m+2 ) with the parallel second fundamental tensor on F when ξ is invariant by the shape operator of M .

When the structure vector field [ξ] of a real hypersurface M in G ₂ ( C ^m+2 ) is invariant by the shape operator, that is, Aξ = αξ, a real hypersurface M is said to be a Hopf hypersurface. When the distribution D ^⊥ of a hypersurface M in G ₂ ( C ^m+2 ) is invariant by the shape operator, we say M a D ^⊥ - invariant hypersurface.

Theorem 1 means that in the class of Hopf real hypersurfaces in G ₂ ( C ^m+2 ) there do not exist any real hypersurface with parallel sec- ond fundamental tensor on F. Now we also assert the following.

Theorem 2. There do not exist any real hypersurface M in G ₂ ( C ^m+2 ) with the parallel second fundamental tensor on F when D ^⊥ is invariant by the shape operator of M .

The condition A[ξ] ⊂ [ξ] in Theorem 1 appears to be rather natural, and in fact there is a well-established theory for such hypersurfaces. Any tube around a complex submanifold in CP ^m satisfies this geometrical condition. Cecil and Ryan proved in [6] that these tubes are essentially characterized by this feature. Here the word essentially refers to some additional condition on the focal map. So, roughly speaking, the theory of real hypersurfaces in CP ^m with A[ξ] ⊂ [ξ] is the theory of tubes around complex submanifolds in CP ^m .

The analogous question in quaternionic projective space HP ^m leads

to a surprise. The corresponding geometrical feature in Theorem 2 is

that the three-dimensional distribution D ^⊥ = Span {ξ 1 , ξ ₂ , ξ ₃ } on M,

which is obtained by applying the almost Hermitian structures in the quaternionic K¨ ahler structure J of HP ^m to a unit normal N , is invariant under A. In fact, every tube around a quaternionic submanifold of HP ^m has this kind of geometrical feature. (Note that by a result of Alekseevskii [1] such a quaternionic submanifold is necessarily totally geodesic.) But the converse is not true. Berndt proved in [3] that also every tube around a totally geodesic CP ^m in HP ^m satisfies AD ^⊥ ⊂ D ^⊥ , and that there are no other ones. So the real hypersurfaces in HP ^m with A[ξ] ⊂ [ξ] are precisely the tubes around totally geodesic HP ^k , k ∈ {0, . . . , m − 1}, and CP ^m .

Any tube around G ₂ ( C ^m+1 ) has four distinct constant principal cur- vatures and might also be regarded as a tube around the focal set of G ₂ ( C ^m+1 ) in G ₂ ( C ^m+2 ), which is a totally geodesic CP ^m . Any tube around HP ⁿ has five distinct constant principal curvatures, and the other focal set of the tube is a complex hypersurface in G ₂ ( C ^m+2 ) which is a Riemannian homogeneous space isomorphic to Sp(n + 1)/(U (1) × Sp(1) × Sp(n − 1)). The two families of tubes together with their fo- cal sets are just the orbits of the isometric actions of the subgroups SU (m + 1) and Sp(n + 1) of SU (m + 2), respectively.

In Section 0.2 we recall Riemannian geometry of two dimensional com- plex Grassmannian G ₂ ( C ^m+2 ) and in Section 1 we will show the equation of Codazzi for real hypersurfaces in G ₂ ( C ^m+2 ) explicitly. Then in Sec- tion 2 by the equation of Codazzi we find some fundamental formulas, which will be useful to prove Theorems 1 and 2, for real hypersurfaces in G ₂ ( C ^m+2 ) satisfying ∇ ξ A = 0 and ∇ ξ

A = 0, i = 1, 2, 3.

In Section 3 by using Theorem A and some formulas in previous sections we prove Theorem 1 which is a non-existence theorem for the class of Hopf real hypersurfaces in G ₂ ( C ^m+2 ). Finally in Sections 4 and 5 we will prove Theorem 2 which is another non-existence theorem for real hypersurfaces satisfying g(AD, D ^⊥ ) = 0 in G ₂ ( C ^m+2 ) with the parallel second fundamental tensor on F.

The present author would like to express his sincere gratitude to the referee for his careful reading of the manuscript and valuable comments to develop the first version of it.

0.2. Riemannian geometry of G ₂ ( C ^m+2 )

In this section we summarize basic material about G ₂ ( C ^m+2 ), for de-

tails we refer to [3],[4] and [5]. By G ₂ ( C ^m+2 ) we denote the set of all

complex two-dimensional linear subspaces in C ^m+2 . The special unitary

group G = SU (m + 2) acts transitively on G ₂ ( C ^m+2 ) with stabilizer

isomorphic to K = S(U (2) × U(m)) ⊂ G. Then G 2 ( C ^m+2 ) can be identified with the homogeneous space G/K, which we equip with the unique analytic structure for which the natural action of G on G ₂ ( C ^m+2 ) becomes analytic. Denote by g and k the Lie algebra of G and K, re- spectively, and by m the orthogonal complement of k in g with respect to the Cartan-Killing form B of g. Then g = k ⊕ m is an Ad(K)-invariant reductive decomposition of g. We put o = eK and identify T _o G ₂ ( C ^m+2 ) with m in the usual manner. Since B is negative definite on g, its neg- ative restricted to m × m yields a positive definite inner product on m.

By Ad(K)-invariance of B this inner product can be extended to a G- invariant Riemannian metric g on G ₂ ( C ^m+2 ). In this way G ₂ ( C ^m+2 ) becomes a Riemannian homogeneous space, even a Riemannian sym- metric space. For computational reasons we normalize g such that the maximal sectional curvature of (G ₂ ( C ^m+2 ), g) is eight. Since G ₂ ( C ³ ) is isometric to the three-dimensional complex projective space CP ³ with constant holomorphic sectional curvature eight we will assume m ≥ 2 from now on. Note that the isomorphism Spin(6)  SU(4) yields an isometry between G ₂ ( C ⁴ ) and the real Grassmann manifold G ⁺ ₂ ( R ⁶ ) of oriented two-dimensional linear subspaces of R ⁶ .

The Lie algebra k has the direct sum decomposition k = su(m) ⊕ su(2) ⊕ R, where R is the center of k. Viewing k as the holonomy algebra of G ₂ ( C ^m+2 ), the center R induces a K¨ahler structure J and the su(2)-part a quaternionic K¨ahler structure J on G ₂ ( C ^m+2 ). If J ₁ is any almost Hermitian structure in J, then JJ ₁ = J ₁ J , and J J ₁ is a symmetric endomorphism with (J J ₁ ) ² = I and tr(J J ₁ ) = 0. This fact will be used frequently throughout this paper.

A canonical local basis J ₁ , J ₂ , J ₃ of J consists of three local almost Hermitian structures J _ν in J such that J _ν J _ν+1 = J _ν+2 = −J ν+1 J _ν , where the index is taken module three. Since J is parallel with respect to the Riemannian connection ¯ ∇ of (G 2 ( C ^m+2 ), g), there exist for any canonical local basis J ₁ , J ₂ , J ₃ of J three local one-forms q ₁ , q ₂ , q ₃ such that

∇ ¯ X J _ν = q _ν+2 (X)J _ν+1 − q ν+1 (X)J _ν+2

for all vector fields X on G ₂ ( C ^m+2 ). Also this fact will be used fre- quently.

Let p ∈ G 2 ( C ^m+2 ) and W a subspace of T _p G ₂ ( C ^m+2 ). We say that

W is a quaternionic subspace of T _p G ₂ ( C ^m+2 ) if J W ⊂ W for all J ∈ J p .

And we say that W is a totally complex subspace of T _p G ₂ ( C ^m+2 ) if

there exists a one-dimensional subspace V of J _p such that J W ⊂ W for all J ∈ V and JW ⊥ W for all J ∈ V ^⊥ ⊂ J p . Here, the orthogonal complement of V in J _p is taken with respect to the bundle metric and orientation on J for which any local oriented orthonormal frame field of J is a canonical local basis of J. A quaternionic (resp. totally complex) submanifold of G ₂ ( C ^m+2 ) is a submanifold all of whose tangent spaces are quaternionic (resp. totally complex) subspaces of the corresponding tangent spaces of G ₂ ( C ^m+2 ).

The Riemannian curvature tensor ¯ R of G ₂ ( C ^m+2 ) is locally given by R(X, Y )Z = g(Y, Z)X ¯ − g(X, Z)Y + g(JY, Z)JX

− g(JX, Z)JY − 2g(JX, Y )JZ +

3 ν=1

{g(J ν Y, Z)J _ν X − g(J ν X, Z)J _ν Y − 2g(J ν X, Y )J _ν Z }

+

3 ν=1

{g(J ν J Y, Z)J _ν J X − g(J ν J X, Z)J _ν J Y },

where J ₁ , J ₂ , J ₃ is any canonical local basis of J. A non-zero tangent vector X of G ₂ ( C ^m+2 ) is said to be singular if X is tangent to more than one flat of G ₂ ( C ^m+2 ). In G ₂ ( C ^m+2 ) there are two types of singular tangent vectors X which are characterized by the properties J X ⊥ JX and J X ∈ JX. We will have to compute explicitly Jacobi vector fields along geodesics whose tangent vectors are all singular. For this we need the eigenvalues and eigenspaces of the Jacobi operator ¯ R _X := ¯ R(., X)X.

Let X be a unit vector tangent to G ₂ ( C ^m+2 ). If J X ⊥ JX then the eigenvalues and eigenspaces of ¯ R _X are

0 RX ⊕ JJX 1 ( HCX) ^⊥ 4 RJX ⊕ JX ,

where HCX = RX ⊕ RJX ⊕ JX ⊕ JJX. If JX ∈ JX, there exists an almost Hermitian structure J ₁ in J such that JX = J ₁ X. Then the eigenvalues and eigenspaces of ¯ R _X are

0 RX ⊕ {Y | Y ⊥ HX, JY = −J 1 Y }, 2 C ^⊥ X ⊕ {Y | Y ⊥ HX, JY = J 1 Y },

8 RJX ,

where CX and HX denote the complex and quaternionic span of X,

respectively, and C ^⊥ X is the orthogonal complement of CX in HX.

1. The Codazzi equation for real hypersurfaces in G ₂ ( C ^m+2 ) In this section we derive some basic formulae from the Codazzi equa- tion for a real hypersurface in a complex two-plane Grassmannian G ₂ ( C ^m+2 ).

Let M be a real hypersurface of G ₂ ( C ^m+2 ), that is, a hypersurface of G ₂ ( C ^m+2 ) with real codimension one. The induced Riemannian met- ric on M will also be denoted by g, and ∇ denotes the Riemannian connection of (M, g). Let N be a local unit normal field of M and A the shape operator of M with respect to N . The K¨ ahler structure J of G ₂ ( C ^m+2 ) induces on M an almost contact metric structure (φ, ξ, η, g).

Furthermore, let J ₁ , J ₂ , J ₃ be a canonical local basis of J. Then each J _ν induces an almost contact metric structure (φ _ν , ξ _ν , η _ν , g) on M . Using the above expression for ¯ R, the Codazzi equation becomes

( ∇ X A)Y − (∇ Y A)X

= η(X)φY − η(Y )φX − 2g(φX, Y )ξ +

3 ν=1

{η ν (X)φ _ν Y − η ν (Y )φ _ν X − 2g(φ ν X, Y )ξ _ν }

+

3 ν1

{η ν (φX)φ _ν φY − η ν (φY )φ _ν φX }

+

3 ν=1

{η(X)η ν (φY ) − η(Y )η ν (φX) }ξ ν .

The following identities can be proved in a straightforward method and will be used frequently in subsequent calculations:

φ _ν+1 ξ _ν = −ξ ν+2 , φ _ν ξ _ν+1 = ξ _ν+2 , φξ _ν = φ _ν ξ, η _ν (φX) = η(φ _ν X), φ _ν φ _ν+1 X = φ _ν+2 X + η _ν+1 (X)ξ _ν , φ _ν+1 φ _ν X = −φ ν+2 X + η _ν (X)ξ _ν+1 . (1.1)

Then in this section let us give some basic formulas which will be used in the later.

Now let us put

J X = φX + η(X)N, J _ν X = φ _ν X + η _ν (X)N

for any tangent vector X of a real hypersurface M in G ₂ ( C ^m+2 ), where N denotes a normal vector of M in G ₂ ( C ^m+2 ). Then from this and the formulas in Section 0.2 we have the following:

(1.2) ( ∇ X φ)Y = η(Y )AX − g(AX, Y )ξ, ∇ X ξ = φAX,

(1.3) ∇ X ξ _ν = q _ν+2 (X)ξ _ν+1 − q ν+1 (X)ξ _ν+2 + φ _ν AX,

( ∇ X φ _ν )Y = − q ν+1 (X)φ _ν+2 Y + q _ν+2 (X)φ _ν+1 Y + η _ν (Y )AX

− g(AX, Y )ξ _ν . (1.4)

Summing up these formulas, we know that

∇ _X (φ _ν ξ) = ∇ _X (φξ _ν )

= ( ∇ X φ)ξ _ν + φ( ∇ X ξ _ν )

= q _ν+2 (X)φ _ν+1 ξ − q ν+1 (X)φ _ν+2 ξ + φ _ν φAX

− g(AX, ξ)ξ ν + η(ξ _ν )AX.

(1.5)

Moreover, from J J _ν = J _ν J , ν = 1, 2, 3, it follows that (1.6) φφ _ν X = φ _ν φX + η _ν (X)ξ − η(X)ξ ν .

2. Some fundamental formulas

Let M be a real hypersurface in G ₂ ( C ^m+2 ) satisfying ∇ ξ A = 0 and

∇ ξ

A = 0, where we have put D ^⊥ = Span {ξ 1 , ξ ₂ , ξ ₃ }. Then in this section let us give some basic formulas which will be used in the later.

If we put Y = ξ into the equation of Codazzi, then by the assumption of ∇ ξ A = 0, we have

(2.1) ( ∇ X A)ξ = −φX + ΨX,

where we have put Ψ(X) =
₃

ν=1 {η ν (X)φ _ν ξ − η ν (ξ)φ _ν X − 2g(φ ν X, ξ)ξ _ν }

−
3

ν=1 η _ν (φX)ξ _ν .

From this it follows that

∇ Y Aξ = ( ∇ Y A)ξ + A ∇ Y ξ

= ( ∇ _Y A)ξ + AφAY

= − φY + ψY + AφAY.

Now differentiating this formula one more time, we have

∇ X ∇ Y Aξ − ∇ ∇

Y Aξ

= − (∇ X φ)Y + ( ∇ X Ψ)Y

+ ( ∇ X A)φAY + A( ∇ X φ)AY + Aφ( ∇ X A)Y

= − η(Y )AX + g(AX, Y )ξ + (∇ X Ψ)Y + ( ∇ X A)φAY + A {η(AY )AX − g(AX, AY )ξ} + Aφ(∇ X A)Y.

Then we have the following

R(X, Y )Aξ = ( ∇ X ∇ Y − ∇ Y ∇ X − ∇ [X,Y ] )Aξ

= ( ∇ X A)φAY − (∇ Y A)φAX + η(AY )A ² X − η(AX)A ² Y

− η(Y )AX + η(X)AY + (∇ X Ψ)Y − (∇ Y Ψ)X + Aφ {(∇ X A)Y − (∇ Y A)X }.

(2.2)

Here the derivative of Ψ is given by

( ∇ Y Ψ)X =
3

ν=1 {(∇ Y η _ν )Xφ _ν ξ + η _ν (X) ∇ Y (φ _ν ξ)

− Y (η ν (ξ))φ _ν X − η ν (ξ)( ∇ Y φ _ν )X

− 2g((∇ Y φ _ν )X, ξ)ξ _ν − 2g(φ ν X, ∇ Y ξ)ξ _ν

− 2g(φ ν X, ξ) ∇ Y ξ _ν } −
3

ν=1 {(∇ Y η _ν )φX + η _ν (( ∇ _Y φ)X) }ξ _ν −
3

ν=1 η _ν (φX) ∇ _Y ξ _ν . (2.3)

On the other hand, from (1.2), (1.3) and (1.5) we have the following formulas

( ∇ _Y η _ν )X = Y (η _ν (X)) − η _ν ( ∇ _Y X)

= q _ν+2 (Y )η _ν+1 (X) − q ν+2 (Y )η _ν+2 (X) + g(φ _ν AY, X),

η _ν (( ∇ Y φ)X) = η(X)η _ν (AY ) − g(AY, X)η ν (ξ).

Substituting these formulas into (2.3) and taking an inner product with ξ, we have

g(( ∇ X Ψ)Y, ξ)

=

ν

 η _ν (Y )η(φ _ν φAX) − X(η _ν (ξ))η(φ _ν Y ) 

− 3
η _ν (ξ) 

− q ν+1 (X)η(φ _ν+2 ) + q _ν+2 (X)η(φ _ν+1 Y ) + η _ν (Y )η(AX)

− g(AX, Y )η ν (ξ) 

− 2

g(φ _ν Y, φAX)η _ν (ξ)

− 3

g(φ _ν Y, ξ) 

q _ν+2 (X)η _ν+1 (ξ) − q _ν+1 (X)η _ν+2 (ξ) + η(φ _ν AX) 

−

η _ν (ξ) 

q _ν+2 (X)η _ν+1 (φY ) − q ν+1 (X)η _ν+2 (φY ) + g(φ _ν AX, φY )η(Y )η _ν (AX) − g(AX, Y )η ν (ξ) 

. (2.4)

On the other hand, by (1.1), (1.2) and (1.4) we know X(η _ν (ξ)) = g( ∇ X ξ, ξ _ν ) + g(ξ, ∇ X ξ _ν )

= η _ν (φAX) + q _ν+2 (X)η(ξ _ν+1 ) − q ν+1 (X)η(ξ _ν+2 ) + η(φ _ν AX),

η(φ _ν φAX) = −η ν (AX) + η(AX)η _ν (ξ),

g(φ _ν Y, φAX) = g(φ _ν AX, φY ) − η ν (Y )η(AX) + η(Y )η _ν (AX).

Substituting these formulas into (2.4), we have g(( ∇ _X Ψ)Y, ξ)

= −

ν η _ν (Y )η _ν (AX) −

ν {2η ν (φAX) + q _ν+2 (X)η(ξ _ν+1 )

− q ν+1 (X)η(ξ _ν+2 ) }η(φ ν Y )

− 4

ν η _ν (ξ) {−q ν+1 (X)η(φ _ν+2 Y ) + q _ν+2 (X)η(φ _ν+1 Y )

− g(AX, Y )η ν (ξ) }

− 3

ν η _ν (ξ)η(Y )η _ν (AX) − 3

ν g(φ _ν AX, φY )η _ν (ξ).

(2.5)

From this, taking skew-symmetric part, we have g(( ∇ X Ψ)Y − (∇ Y Ψ)X, ξ)

= −

ν {η ν (AX)η _ν (Y ) − η ν (AY )η _ν (X) }

− 2

ν {η ν (φAX)η(φ _ν Y ) − η ν (φAY )η(φ _ν X) }

(2.6)

+ 4

ν η _ν (ξ) {q ν+1 (X)η(φ _ν+2 Y ) − q ν+1 (Y )η(φ _ν+2 X) }

− 4

ν η _ν (ξ) {q ν+2 (X)η(φ _ν+1 Y ) − q ν+2 (Y )η(φ _ν+1 X) }

− 3

ν η _ν (ξ) {η(Y )η ν (AX) − η(X)η ν (AY )

− g(φ ν AX, φY ) + g(φ _ν AY, φX) }

−

ν η(ξ _ν+1 ) {q ν+2 (X)η(φ _ν Y ) − q ν+2 (Y )η(φ _ν X) }

+

ν η(ξ _ν+2 ) {q ν+1 (X)η(φ _ν Y ) − q ν+1 (Y )η(φ _ν X) }.

Now let us take an inner product (2.2) with ξ, we have g(R(X, Y )Aξ, ξ)

= g(( ∇ X A)ξ, φAY ) − g((∇ Y A)ξ, φAX) + η(A ² X)η(AY )

− η(AX)η(A ² Y ) − η(Y )η(AX) + η(X)η(AY ) + g(( ∇ _X Ψ)Y − (∇ _Y Ψ)X, ξ)

+ g(Aφ {(∇ X A)Y − (∇ Y A)X }, ξ).

(2.7)

The first term of the right side of (2.7) gives g(( ∇ X A)ξ, φAY )

= − g(AX, Y ) + η(X)η(AY ) +

ν {η ν (X)g(φ _ν ξ, φAY )

− η ν (ξ)g(φ _ν X, φAY ) − 3g(φ ν X, ξ)η _ν (φAY ) }.

So it follows

g(( ∇ X A)ξ, φAY ) − g(∇ Y A)ξ, φAX)

= η(X)η(AY ) − η(Y )η(AX)

+

ν {η ν (X)g(φ _ν ξ, φAY ) − η ν (Y )g(φ _ν , φAX) }

−

ν η _ν (ξ) {g(φ _ν X, φAY ) − g(φ _ν Y, φAX) }

− 3

ν {η(φ ν X)η _ν (φAY ) − η(φ ν Y )η _ν (φAX) }.

(2.8)

Moreover, the last term of (2.7) gives g(Aφ {(∇ X A)Y − (∇ Y A)X }, ξ)

= − g((∇ X A)Y − (∇ Y A)X, φAξ)

= − η(X)η(AY ) + η(Y )η(AX) −

ν {η ν (X)g(φ _ν Y, φAξ)

(2.9)

− η ν (Y )g(φ _ν X, φAξ) − 2g(φ ν X, Y )g(ξ _ν , φAξ) }

−

ν {η ν (φX)η(Aφ _ν Y ) − η ν (φY )η(Aφ _ν X) }.

Now substituting (2.6), (2.8) and (2.9) into (2.7), we have

g(R(X, Y )Aξ, ξ)

=

ν {η _ν (X)g(φ _ν ξ, φAY ) − η _ν (Y )g(φ _ν , φAX) }

−

ν η _ν (ξ) {g(φ ν X, φAY ) − g(φ ν Y, φAX) }

−

ν {η(φ ν X)η _ν (φAY ) − η(φ ν Y )η _ν (φAX) }

+ η(A ² X)η(AY ) − η(AX)η(A ² Y ) − η(Y )η(AX) + η(X)η(AY )

−

ν {η ν (X)g(φ _ν Y, φAξ) − η ν (Y )g(φ _ν X, φAξ)

− 2g(φ ν X, Y )g(ξ _ν , φAξ) }

−

ν {η ν (φX)η(Aφ _ν Y ) − η ν (φY )η(Aφ _ν X) }

−

ν {η ν (AX)η _ν (Y ) − η ν (AY )η _ν (X) } + 4

η _ν (ξ) {q ν+1 (X)η(φ _ν+2 Y )

− q ν+1 (Y )η(φ _ν+2 X) − q ν+2 (X)η(φ _ν+1 Y ) + q _ν+2 (Y )η(φ _ν+1 X) }

− 3

η _ν (ξ) {η(Y )η ν (AX) − η(X)η ν (AY ) − g(φ ν AX, φY ) + g(φ _ν AY, φX) } −

η(ξ _ν+1 ) {q ν+2 (X)η(φ _ν Y ) − q ν+2 (Y )η(φ _ν X) }

+

η(ξ _ν+2 ) {q ν+1 (X)η(φ _ν Y ) − q ν+1 (Y )η(φ _ν X) }.

(2.10)

On the other hand, by the equation of Gauss, the curvature tensor R(X, Y )Z of a real hypersurface M in G ₂ ( C ^m+2 ) is given by

R(X, Y )Z

= g(Y, Z)X − g(X, Z)Y + g(φY, Z)φX − g(φX, Z)φY

− 2g(φX, Y )φZ

+

ν {g(φ ν Y, Z)φ _ν X − g(φ ν X, Z)φ _ν Y − 2g(φ ν X, Y )φ _ν Z }

+

ν {g(φ ν φY, Z)φ _ν φX − g(φ ν φX, Z)φ _ν φY }

−

ν {η(Y )η ν (Z)φ _ν φX − η(X)η ν (Z)φ _ν φY }

−

ν {η(X)g(φ ν φY, Z) − η(Y )g(φ ν φX, Z) }ξ ν

+ g(AY, Z)AX − g(AX, Z)AY.

From this together with the formula

η(φ _ν φX)g(φ _ν φY, Aξ) = {−η ν (X) + η(X)η _ν (ξ) }g(φ ν φY, Aξ) we have

g(R(X, Y )Aξ, ξ)

= η(X)η(AY ) − η(Y )η(AX)

+

ν {η(φ ν X)g(φ _ν Y, Aξ) − η(φ ν Y )g(φ _ν X, Aξ)

− 2η(φ ν Aξ)g(φ _ν X, Y ) }

−

ν {η ν (X)g(φ _ν φY, Aξ) − η ν (Y )g(φ _ν φX, Aξ) }

+

ν {η(Y )η ν (X)η _ν (Aξ) − η(X)η ν (Y )η _ν (Aξ) } + η(A ² Y )η(AX) − η(A ² X)η(AY ).

(2.11)

From this, if we use the following formulas such that

g(φ _ν φY, Aξ) = − g(φ ν Y, φAξ) + η(Y )η _ν (Aξ) − η ν (Y )η(Aξ) g(φ _ν ξ, φAY ) = g(φξ _ν , φAY ) = η _ν (AY ) − η(AY )η(ξ ν ),

g(φ _ν X, φAY ) = − g(φφ ν X, AY ) = −g(φ ν φX, AY ) + η _ν (X)η(AY )

− η(X)η _ν (AY )

and compare with (2.10), then we have finally as follows:

0 = 2

ν η _ν (ξ) {g(φ ν AY, φX) − g(φ ν AX, φY ) }

− 2

ν η _ν (X)η(AY )η(ξ _ν ) + 2

ν η _ν (Y )η(AX)η(ξ _ν ) + 2η(A ² X)η(AY ) − 2η(AX)η(A ² Y )

− 2

ν {η ν (X)g(φ _ν Y, φAξ) − η ν (Y )g(φ _ν X, φAξ)

− 2g(φ ν X, Y )g(ξ _ν , φAξ) }

(2.12)

− 2

ν {η ν (φX)η _ν (Aφ _ν Y ) − η ν (φY )η(Aφ _ν X) }

− 2

ν {η ν (AX)η _ν (Y ) − η ν (AY )η _ν (X) } + 4

ν η _ν (ξ) {q ν+1 (X)η(φ _ν+2 Y ) − q ν+1 (Y )η(φ _ν+2 X) }

− 4

ν η _ν (ξ) {q _ν+2 (X)η(φ _ν+1 Y ) − q _ν+2 (Y )η(φ _ν+1 X) }

− 4

ν η _ν (ξ) {η(Y )η ν (AX) − η(X)η ν (AY ) }

−

ν {η ν (φX)η _ν (φAY ) − η ν (φY )η _ν (φAX) }

−

ν η(ξ _ν+1 ) {q ν+2 (X)η(φ _ν Y ) − q ν+2 (Y )η(φ _ν X) }

+

ν η(ξ _ν+2 ) {q ν+1 (X)η(φ _ν Y ) − q ν+1 (Y )η(φ _ν X) }.

3. Hopf real hypersurfaces in G ₂ ( C ⁿ⁺² )

Let M be a real hypersurface in G ₂ ( C ^m+2 ) satisfying ∇ ξ A = 0 and

∇ _ξ

A = 0, where we have put D ^⊥ = Span {ξ ₁ , ξ ₂ , ξ ₃ }. Then from the equation of Codazzi we have

( ∇ X A)ξ _i = η(X)φξ _i − η(ξ i )φX − 2g(φX, ξ i )ξ +
3

ν=1 {η ν (X)φ _ν ξ _i − η ν (ξ _i )φ _ν X − 2g(φ ν X, ξ _i )ξ _ν } +
3

ν=1 {η ν (φX)φ _ν φξ _i − η ν (φξ _i )φ _ν φX } +
3

ν=1 {η(X)η ν (φξ _i ) − η(ξ i )η _ν (φX) }ξ ν .

From this, putting X = ξ and using the assumption of ∇ ξ A = 0, we have

0 =( ∇ ξ A)ξ _i

=φξ _i +
3

ν=1 {η ν (ξ)φ _ν ξ _i − η ν (ξ _i )φ _ν ξ − 2g(φ ν ξ, ξ _i )ξ _ν } +
3

ν=1 η _ν (φξ _i )ξ _ν . (3.1)

Thus for i = 1 in (3.1) we have

(3.2) 0 = −2g(ξ, ξ 3 )ξ ₂ + 2g(ξ, ξ ₂ )ξ ₃ .

This implies ξ is orthogonal to ξ ₂ and ξ ₃ . Moreover, by putting i = 2 in

(3.1), we can assert ξ is orthogonal to ξ ₁ and ξ ₃ . Thus we have

Lemma 3.1. Let M be a real hypersurface of G ₂ ( C ^m+2 ) satisfying

∇ ξ A = 0 and ∇ ξ

A = 0 for i = 1, 2, 3. Then ξ ∈D.

Thus by Lemma 3.1 we have known that ξ ∈D. Then the formula (2.12) reduces to

0 = 2η(A ² X)η(AY ) − 2η(AX)η(A ² Y )

− 2

ν {η ν (X)g(φ _ν Y, φAξ) − η ν (Y )g(φ _ν X, φAξ)

− 2g(φ _ν X, Y )g(ξ _ν , φAξ) }

− 2

ν {η ν (φX)η(Aφ _ν Y ) − η ν (φY )η(Aφ _ν X) }

− 2

ν {η ν (AX)η _ν (Y ) − η ν (AY )η _ν (X) }

−

ν {η ν (φX)η _ν (φAY ) − η ν (φY )η _ν (φAX) }.

(3.3)

Let us put Aξ = αξ + βU , where α = η(Aξ). Then putting X = ξ in (3.3), we have

0 =2η(A ² ξ)η(AY ) − 2η(Aξ)η(A ² Y ) + 2 {

ν η _ν (Y )g(φ _ν ξ, φAξ) + 2

ν g(φ _ν ξ, Y )g(ξ _ν , φAξ) } + 2

ν η _ν (φY )η(Aφ _ν ξ) − 2

ν η _ν (Aξ)η _ν (Y )

+

ν η _ν (φY )η _ν (φAξ).

From this, if we substitute the formulas

η(Aφ _ν ξ) = −βg(φU, ξ ν ), η _ν (φAξ) = βg(φU, ξ _ν ), and

η _ν (Aξ) = βg(U, ξ _ν ), then

(3.4) 2η(Aξ)A ² ξ = 2η(A ² ξ)Aξ + 5β

ν η _ν (φU )φξ _ν .

Hereafter in this section let us consider ξ- invariant hypersurface in

G ₂ ( C ⁿ⁺² ), that is, Aξ = αξ, where α = η(Aξ, ξ). In such a case a real

hypersurface M is said to be a Hopf hypersurface in G 2 ( C ^m+2 ).

Now we have assumed that M is a Hopf hypersurface. Then by definition its structure vector ξ is principal. So from this fact together with Lemma 3.1 we know that η _ν (Aξ) = 0, ν = 1, 2, 3.

Putting X = ξ ₁ in (3.3) and using Lemma 3.1, we have

0 = − 2{g(φ 1 Y, φAξ) + η ₂ (Y )g(ξ ₃ , φAξ) − η 3 (Y )η ₂ (φAξ) + 2η ₃ (Y )η ₂ (φAξ) − 2η 2 (Y )η ₃ (φAξ) }

+ 2 {η 2 (φY )η(Aξ ₃ ) + η ₃ (φY )η(Aξ ₂ ) }

− 2{η 1 (Aξ ₁ )η ₁ (Y ) + η ₂ (Aξ ₁ )η ₂ (Y ) − η 3 (Aξ ₁ )η ₃ (Y ) − η 1 (AY ) } + {η 1 (φY )η ₁ (φAξ ₁ ) + η ₂ (φY )η ₂ (φAξ ₁ ) + η ₃ (φY )η ₃ (φAξ ₁ ) }.

(3.5)

Now we are going to prove the following

Lemma 3.2. Let M be a Hopf hypersurface in G ₂ (C ^m+1 ) satisfying

∇ ξ

A = 0 and ∇ ξ A = 0, i = 1, 2, 3. Then g(AD, D ^⊥ ) = 0.

Proof. By Lemma 3.1, we know ξ∈D. From this together with (3.5) and ξ is principal we have

0 = − 2{η 1 (Aξ ₁ )η ₁ (Y ) + η ₂ (Aξ ₁ )η ₂ (Y ) − η 3 (Aξ ₁ )η ₃ (Y ) − η 1 (AY ) } + {η 1 (φY )η ₁ (φAξ ₁ ) + η ₂ (φY )η ₂ (φAξ ₁ ) + η ₃ (φY )η ₃ (φAξ ₁ ) }.

(3.6)

From this, putting Y = ξ ₃ , we have

4η ₃ (Aξ ₁ ) = 0.

Similarly, we can assert η _i (Aξ _j ) = 0 for any distinct i and j. So in order to verify the above assertion let us put

Aξ _i = α _i ξ _i + β _i X _i for some X _i ∈D. Then

η ₁ (φAξ ₁ ) = β ₁ g(ξ ₁ , φX ₁ ), η ₂ (φAξ ₁ ) = β ₁ g(ξ ₂ , φX ₁ ), η ₃ (φAξ ₁ )

= β ₁ g(ξ ₃ , φX ₁ ).

Let us construct an open set U = {p∈M|β 1 (p) =0}. Then on this open U if we substitute the above formulas into (3.6), we have for some X ₁ ∈D

0 = − 2{η 1 (Aξ ₁ )η ₁ (Y ) − η 1 (AY ) }

+ β ₁ {η 1 (φY )η ₁ (φX ₁ ) + η ₂ (φY )η ₂ (φX ₁ ) + η ₃ (φY )η ₃ (φX ₁ ) }.

From this, putting Y = X ₁ , we have

2η ₁ (AX ₁ ) + β ₁ {η 1 (φX ₁ ) ² + η ₂ (φX ₁ ) ² + η ₃ (φX ₁ ) ² } = 0.

Since η ₁ (AX ₁ ) = β ₁ =0, the above equation implies 

i η _i (φX _i ) ² = −2, which makes a contradiction. That is, there does not exist such an open set. So we should have β ₁ = 0. Similarly, if there exists any open set V = {p∈M|β 2 (p) =0} or W = {p∈M|β 3 (p) =0}, we can also make a contradiction. From this we conclude β _i = 0, i.e, Aξ _i = α _i ξ _i , that is,

g(AD, D ^⊥ ) = 0.

Then by Lemmas 3.1 and 3.2 we are able to introduce a Proposition given in the paper of Berndt and the present author ([4]) as follows:

Proposition 3.1. Let M be a connected real hypersurface of G ₂ ( C ^m+2 ). Suppose that A D ⊂ D, Aξ = αξ, and ξ is tangent to D. Then the quaternionic dimension m of G ₂ ( C ^m+2 ) is even, say m = 2n, and M has five distinct constant principal curvatures

α = −2 tan(2r) , β = 2 cot(2r) , γ = 0 , λ = cot(r) , µ = − tan(r) with some r ∈ (0, π/4). The corresponding multiplicities are

m(α) = 1 , m(β) = 3 = m(γ) , m(λ) = 4n − 4 = m(µ) and the corresponding eigenspaces are

T _α = Rξ , T β = JJξ , T _γ = Jξ , T _λ , T _µ , where

T _λ ⊕ T µ = ( HCξ) ^⊥ , JT _λ = T _λ , JT _µ = T _µ , J T _λ = T _µ .

Now let us consider a unit eigenvector Y ∈T λ . Then by Proposition 3.1 we write the following formulas:

AY = cot rY, AφY = − tan rφY, φAY = cot rφY,

AφAY = cot rAφY = −φY.

So from these formulas and the equation of Codazzi it follows that for any Y ∈T λ

0 = ( ∇ ξ A)Y

= ( ∇ Y A)ξ + η(ξ)φY

= (αI − A)φAY + φY

= αφAY − AφAY + φY

= {−2 tan 2r· cot r + 2}φY where in the second equality we have used the fact that

JT _λ = T _λ , JT _µ = T _µ .

Then it gives tan ² r + 1 = 0, which makes a contradiction. Finally we conclude that there do not exist any Hopf real hypersurfaces in G ₂ ( C ⁿ⁺² ) with parallel second fundamental tensor along the distribution F = ξ⊕D ^⊥ . This completes the proof of our Theorem 1.

4. D ^⊥ -invariant real hypersurfaces in G ₂ ( C ^m+2 )

Now as a classification problem concerned with curvature adapted real hypersurface in a complex two-plane Grassmannian G ₂ ( C ^m+2 ) we assert the following

Lemma 4.1. Let M be a D ^⊥ -invariant real hypersurface in G ₂ ( C ^m+2 ) satisfying ∇ _F A = 0, F = ξ⊕D ^⊥ . Then φU ∈D ^⊥ or ξ is principal.

By virtue of Lemma 4.1, hereafter unless otherwise sated, in this section we always refer that φU ∈D ^⊥ when the structure vector ξ is not principal.

Proof. From the hypothesis we know that by Lemma 3.1

Aξ _i = α _i ξ _i , and ξ, φξ _i , Aξ, A ² ξ, U ∈D.

From these formulas, if we put X = ξ ₁ in (3.3), we have

0 = − 2{g(φ 1 Y, φAξ) + η ₂ (Y )g(ξ ₃ , φAξ) − η 3 (Y )η ₂ (φAξ)

+ 2η ₃ (Y )η ₂ (φAξ) − 2η 2 (Y )η ₃ (φAξ) }.

Now we assume ξ is not principal. Then if we put Aξ = αξ + βU in the above formula, then the function β =0 implies

g(φ ₁ Y, φU ) + η ₃ (Y )η ₂ (φU ) − η 2 (Y )η ₃ (φU ) = 0.

From this, replacing Y by φ ₁ Y , Y ∈D, we have g(Y, φU) = 0. So

φU ∈D ^⊥ .

Now next let us prove the following

Theorem 4.2. Let M be a D ^⊥ -invariant real hypersurface in G ₂ ( C ^m+2 ) satisfying ∇ ξ

A = 0 and ∇ ξ A = 0, then its structure vector ξ is principal.

Proof. By Lemma 3.1 we know ξ∈D. Now let us put Aξ = αξ + βU,

where α = η(Aξ). Then under the same situation as in (3.4) we find the formula

2η(Aξ)A ² ξ = 2η(A ² ξ)Aξ + 5β

ν η _ν (φU )φξ _ν .

Now in order to prove this Theorem 4.2 we consider two cases such that η(Aξ) = 0 and η(Aξ) =0. For the case where η(Aξ) =0 in Theorem 4.2 we will prove it in Section 5.

Now in this section we only consider the case η(Aξ) = 0. Then in this case let us consider the following two subcases:

Case I: β = 0.

Then in this case Aξ = 0 implies our assertion.

Case II: β =0.

Then in this case we know

ξ, φξ ₁ , φξ ₂ , φξ ₃ ∈D.

Bearing this in mind, (3.4) implies 0 = 2β ² Aξ + 5β

ν η _ν (φU )φξ _ν

= 2β ³ U + 5β

ν η _ν (φU )φξ _ν . (*)

From this, if we apply the operator φ, we have

0 = β(2β ² − 5)η ν (φU ).

Thus for a case where β ² = ⁵ ₂ we know φU ∈D. This makes a contradiction to φU ∈D ^⊥ in Lemma 4.1.

Now hereafter unless otherwise stated, in this Case II let us only consider the case β ² = ⁵ ₂ . Then also in this situation we will show a contradiction. Let us prove this fact as the following. From ∇ ξ A = 0 together with Aξ = βU , where β = const, it follows that

( ∇ X A)ξ = ∇ X (Aξ) − A∇ X ξ

=β ∇ X U − AφAX

= − φX +
3

ν=1 {η ν (X)φ _ν ξ − 2g(φ ν X, ξ)ξ _ν }

−
3

ν=1 η _ν (φX)ξ _ν ,

where in the third equality we have used the equation of Codazzi. From this, if we take an inner product with ξ, then we have for any X in T _x M

βg( ∇ X U, ξ) = g(AφAX, ξ) = βg(φAX, U ).

On the other hand, we have

βg( ∇ _X U, ξ) = −βg(U, ∇ _X ξ) = −βg(U, φAX).

From this and β =0, we know

(4.1) AφU = 0.

On the other hand, from (*) together with the fact that 2β ² = 5 we know

(4.2) U +

ν η _ν (φU )φξ _ν = 0.

Thus the formulas AφU = 0 and (4.2) imply 0 = AφU =

η _ν (φU )Aξ _ν =

α _ν η _ν (φU )ξ _ν . That is,

(4.3) α ₁ η ₁ (φU ) = α ₂ η ₂ (φU ) = α ₃ η ₃ (φU ) = 0.

Sub II.1: α ₁ , α ₂ , α ₃ =0.

Then (4.3) implies

η _i (φU ) = 0 for i = 1, 2, 3.

This contradicts the fact  ₃

i=1 η _i (φU ) ² = 1.

Sub II.2: α ₁ = 0, α ₂ , α ₃ =0.

That is, one of α ₁ , α ₂ , α ₃ is vanishing. Then (4.3) implies η ₂ (φU ) = η ₃ (φU ) = 0.

So η ₁ (φU ) = ±1. That is, we can put φU = ξ 1 . Then by the equation of Codazzi and Aξ ₁ = 0, we have

η(X)φξ ₁ − η(ξ 1 )φX − 2g(φX, ξ 1 )ξ +
3

ν=1 {η _ν (X)φ _ν ξ ₁ − η _ν (ξ ₁ )φ _ν X − 2g(φ _ν X, ξ ₁ )ξ _ν } +
3

ν=1 {η ν (φX)φ _ν φξ ₁ − η ν (φξ ₁ )φ _ν φX } +
3

ν=1 {η(X)η ν (φξ ₁ ) − η(ξ 1 )η _ν (φX) }ξ ν

= ( ∇ X A)ξ ₁ = −A(∇ X ξ ₁ )

= − A{q 3 (X)ξ ₂ − q 2 (X)ξ ₃ + φ ₁ AX }

= − α 2 q ₃ (X)ξ ₂ + α ₃ q ₂ (X)ξ ₃ − Aφ 1 AX.

(4.4)

On the other hand, from the assumption g(AD, D ^⊥ ) = 0 i.e., Aξ _i = α _i ξ _i and the equation of Codazzi it follows

(α _i − α j )q _k (X) = 0 for any X in D. This means

(4.5)

⎧ ⎪

⎨

⎪ ⎩

α ₁ = α ₂ or q ₃ | _D = 0, α ₂ = α ₃ or q ₁ | D = 0, α ₃ = α ₁ or q ₂ | D = 0.

Thus by (4.5), if we consider for any X ∈D, we know that q 2 | D = q ₃ | D = 0 in Sub II.2. So (4.4) implies

−Aφ 1 AX = ( ∇ X A)ξ ₁

= η(X)φξ ₁ − 2g(φX, ξ 1 )ξ − φ 1 X +
3

ν=1 η _ν (φX)φ _ν φξ ₁ .

(4.6)

From this, putting X = ξ in D, we have

(4.7) −βAφ 1 U = φξ ₁ − φ 1 ξ = 0.

On the other hand, the formula (4.2) implies 0 =φ ₁ U +

ν η _ν (φU )φ ₁ φξ _ν

=φ ₁ U + η ₁ (ξ ₁ )φ ₁ φξ ₁

=φ ₁ U + φ ² ₁ ξ,

because we have put φU = ξ ₁ . So it follows

(4.8) φ ₁ U = ξ.

Then from (4.7) and (4.8) we have

0 = Aφ ₁ U = Aξ = βU, which makes a contradiction.

Sub. II.3: α ₁ = α ₂ = 0 and α ₃ =0.

Now we apply (4.4) and (4.5) for any tangent vector X ∈M and for i = 1, 2. Then we have

−Aφ i AX = η(X)φξ _i − 2g(φX, ξ i )ξ +
₃

ν=1 {η ν (X)φ _ν ξ _i

− η ν (ξ _i )φ _ν X − 2g(φ ν X, ξ _i )ξ _ν } +

ν η _ν (φX)φ _ν φξ _i . From this, putting X = ξ and also using φξ _ν = φ _ν ξ, we have

Aφ ₁ U = Aφ ₂ U = 0.

On the other hand, (4.2) and (4.3) imply

φ ₁ U − η 1 (φU )ξ + η ₂ (φU )φ ₃ ξ = 0, φ ₂ U − η 1 (φU )φ ₃ ξ − η 2 (φU )ξ = 0.

(4.9)

Multiplying η ₁ (φU ) to the first of (4.9) and η ₂ (φU ) to the second and summing up, then in such a case by (4.2) we have

ξ = η ₁ (φU )φ ₁ U + η ₂ (φU )φ ₂ U.

This implies

βU = Aξ = η ₁ (φU )Aφ ₁ U + η ₂ (φU )Aφ ₂ U = 0.

So this case also can not occur.

Sub. II.4: Aξ _i = 0, i = 1, 2, 3.

Then we are able to apply the formula (4.4) for i = 1, 2, 3 as follows:

−Aφ i AX = − A(∇ X ξ _i ) = ( ∇ X A)ξ _i

= η(X)φξ _i − 2g(φX, ξ i )ξ +
3

ν=1 {η ν (X)φ _ν ξ _i

− η ν (ξ _i )φ _ν X − 2g(φ ν X, ξ _i )ξ _ν } +
3

ν=1 η _ν (φX)φ _ν φξ _i , (4.10)

where we have used ξ ∈D and φξ ν = φ _ν ξ for any i = 1, 2, 3. From this, putting X = ξ and also using φξ _ν = φ _ν ξ, we have respectively for any i = 1, 2, 3

(4.11) Aφ ₁ U = 0, Aφ ₂ U = 0, and Aφ ₃ U = 0.

On the other hand, (4.2) implies

φ ₁ U − η 1 (φU )ξ + η ₂ (φU )φξ ₃ − η 3 (φU )φξ ₂ = 0, φ ₂ U − η ₁ (φU )φ ₃ ξ − η ₂ (φU )ξ + η ₃ (φU )φξ ₁ = 0, φ ₃ U + η ₁ (φU )φ ₂ ξ − η 2 (φU )φ ₁ ξ − η 3 (φU )ξ = 0.

(4.12)

From this, multiplying η ₁ (φU ), η ₂ (φU ) and η ₃ (φU ) respectively to the first, the second and the third and summing up all of these formulas and using (4.2), we have

ξ =

ν η _ν (φU )φ _ν U.

Then from this together with (4.11) we have βU = Aξ =

ν η _ν (φU )Aφ _ν U = 0.

So it follows β = 0. This contradicts 2β ² = 5.