The parallelism of shape operator related to the generalized Tanaka-Webster connection on real hypersurfaces in complex two-plane Grassmannians

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1 Proceedings of The Fifteenth International Workshop on Diff. Geom. 15(2011) The parallelism of shape operator related to the generalized Tanaka-Webster connection on real hypersurfaces in complex two-plane Grassmannians Imsoon Jeong Department of Mathematics, Kyungpook National University, Taegu , Korea imsoon.jeong@gmail.com Hyunjin Lee Graduate School of Electrical Engineering and Computer Science, Kyungpook National University, Taegu , Korea lhjibis@hanmail.net Young Jin Suh Department of Mathematics, Kyungpook National University, Taegu , Korea yjsuh@knu.ac.kr (2010 Mathematics Subject Classification : 53C40, 53C15.) Abstract. In this article we shall provide a survey on our paper [9]. In a paper due to [8] we have shown that there does not exist a hypersurface in G 2(C m+2 ) with parallel shape operator in the generalized Tanaka-Webster connection (see [13], [14]). In this paper, we introduce the notion of -parallel for a hypersurface M in G 2(C m+2 ) and prove that M is an open part of a tube around a totally geodesic G 2(C m+1 ) in G 2(C m+2 ). 1 Introduction The generalized Tanaka-Webster connection (in short, the g-tanaka-webster connection) for contact metric manifolds has been introduced by Tanno [14] as a Key words and phrases: Real hypersurfaces, Complex two-plane Grassmannians, Hopf hypersurface, Generalized Tanaka-Webster connection, parallel shape operator. * The first author is supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology( ) and the second and third authors are supported by grant Proj. No. BSRP from National Research Foundation of Korea. 183

2 184 Imsoon Jeong, Hyunjin Lee & Young Jin Suh generalization of the well-known connection defined by Tanaka in [13] and, independently, by Webster in [15]. This connection coincides with Tanaka-Webster connection if the associated CR-structure is integrable. Tanaka-Webster connection is defined as the canonical affine connection on a non-degenerate, pseudo-hermitian CR-manifold. For a real hypersurface in a Kähler manifold with almost contact metric structure (φ,, η, g), Cho defined the g-tanaka-webster connection for a non-zero real number k (see [5], [6] and [7]). In particular, if a real hypersurface satisfies φa + Aφ = 2kφ, then the g-tanaka-webster connection coincides with the Tanaka-Webster connection (see Proposition 7 in [7]). Using the g-tanaka-webster connection, many geometers have studied some characterizations of real hypersurfaces in complex space form M n (c) with constant holomorphic sectional curvature c. For instance, when c > 0, that is, M n (c) is a complex projective space CP n, Cho [5] proved that if the shape operator A of M in CP n is -parallel (it means that the shape operator A satisfies A = 0), then the Reeb vector field becomes a Hopf vector field and M is locally congruent to a real hypersurface of Type (A) or of Type (B). (In fact, he also gave a complete classification for real hypersurfaces in a complex hyperbolic space (c < 0) or a complex Euclidean space (c = 0) under the assumption that -parallel shape operator [5]). Moreover, in [6] he gave the classification theorem of Levi-parallel Hopf hypersurface in M n (c), c 0. Here, a real hypersurface of M n (c) is called Leviparallel if its Levi form is parallel with respect to the g-tanaka-webster connection. Now let us denote by G 2 (C m+2 ) the set of all complex two-dimensional linear subspaces in C m+2. This Riemannian symmetric space G 2 (C m+2 ) has a remarkable geometric structure. It is the unique compact irreducible Riemannian manifold being equipped with both a Kähler structure J and a quaternionic Kähler structure J not containing J. In other words, G 2 (C m+2 ) is the unique compact, irreducible, Kähler, quaternionic Kähler manifold which is not a hyper-kähler manifold. So, in G 2 (C m+2 ) we have the two natural geometric conditions for real hypersurfaces M that the 1-dimensional distribution [] = Span{ and the 3-dimensional distribution D = Span{ 1, 2, 3 are invariant under the shape operator A of M. Here the almost contact structure vector field defined by = JN is said to be a Reeb vector field, where N denotes a local unit normal vector field of M in G 2 (C m+2 ). The almost contact 3-structure vector fields { 1, 2, 3 for the 3- dimensional distribution D of M in G 2 (C m+2 ) are defined by ν = J ν N (ν = 1, 2, 3), where J ν denotes a canonical local basis of a quaternionic Kähler structure J, such that T x M = D D, x M. By using these two invariant conditions and the result in Alekseevskii [1], Berndt and Suh [3] proved the following: Theorem A. Let M be a connected real hypersurface in G 2 (C m+2 ), m 3. Then both [] and D are invariant under the shape operator of M if and only if

3 Parallelism of shape operator w.r.t. the g-tanaka-webster connection 185 (A) M is an open part of a tube around a totally geodesic G 2 (C m+1 ) in G 2 (C m+2 ), or (B) 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 ). Furthermore, the Reeb vector field is said to be Hopf if it is invariant under the shape operator A. The 1-dimensional foliation of M by the integral curves of the Reeb vector field is said to be a Hopf foliation of M. We say that M is a Hopf hypersurface in G 2 (C m+2 ) if and only if the Hopf foliation of M is totally geodesic. By the almost contact metric structure (φ,, η, g) and the formula X = φax for any X T M in [12], it can be easily checked that M is Hopf if and only if the Reeb vector field is Hopf. Moreover, we say that the Reeb flow on M in G 2 (C m+2 ) is isometric, when the Reeb vector field on M is Killing. In [4], Berndt and Suh gave some equivalent conditions of isometric Reeb flow. They gave a characterization of real hypersurfaces of Type (A) in Theorem A in terms of the Reeb flow on M as follows: Theorem B. Let M be a connected orientable real hypersurface in G 2 (C m+2 ), m 3. Then the Reeb flow on M is isometric if and only if M is an open part of a tube around a totally geodesic G 2 (C m+1 ) in G 2 (C m+2 ). In addition, in [4] it is known that the Reeb flow on M is isometric if and only if the shape operator A commutes with the structure tensor field φ, that is, Aφ = φa. Recently, Lee and Suh [11] gave a new characterization of real hypersurfaces of Type (B) in G 2 (C m+2 ) in terms of the Reeb vector field. On the other hand, Jeong, Lee and Suh [8] considered a generalized Tanaka- Webster parallel (in short, g-tanaka-webster parallel) for real hypersurfaces in complex two-plane Grassmannian G 2 (C m+2 ). In other words, the shape operator A is called g-tanaka-webster parallel if the shape operator A satisfies ( X A)Y = 0 for any vector fields X, Y on M. Using this notion, the authors [8] gave a non-existence theorem for Hopf hypersurfaces in G 2 (C m+2 ) as follows: Theorem C. There does not exist any Hopf hypersurface, α 2k, in complex two-plane Grassmannians G 2 (C m+2 ), m 3 with parallel shape operator in the generalized Tanaka-Webster connection. Now in this paper, we consider new notion of parallel shape operator related to the g-tanaka-webster connection of M in G 2 (C m+2 ), namely, generalized Tanaka- Webster -parallel (in short, g-tanaka-webster -parallel) shape operator. When the shape operator A of M in G 2 (C m+2 ) satisfies ( A)Y = 0 for any vector field

4 186 Imsoon Jeong, Hyunjin Lee & Young Jin Suh Y T M, we say that the shape operator A is g-tanaka-webster -parallel. Naturally, the notion of g-tanaka-webster -parallel is weaker than g-tanaka-webster parallel. Using such a notion, we give a characterization of real hypersurface of Type (A) in G 2 (C m+2 ) in Theorem A as follows: Main Theorem. Let M be a connected orientable Hopf hypersurface, α 2k, in G 2 (C m+2 ), m 3. If the shape operator A is the generalized Tanaka-Webster -parallel, then M is locally congruent to an open part of a tube around a totally geodesic G 2 (C m+1 ) in G 2 (C m+2 ). Moreover, we consider the notion of g-tanaka-webster F-parallel shape operator of M in G 2 (C m+2 ). It means that the shape operator A of M in G 2 (C m+2 ) satisfies ( X A)Y = 0 for any vector fields X F and Y T M. Here the distribution F is defined by F = [] D where [] = Span{ and D = Span{ ν ν = 1, 2, 3. Naturally, we know that the g-tanaka-webster F-parallel shape operator is weaker than the g-tanaka-webster parallel. But it becomes a condition stronger than the g-tanaka-webster -parallel. From such a view point, we assert the following: Corollary. There does not exist any Hopf hypersurface, α 2k, in complex twoplane Grassmannians G 2 (C m+2 ), m 3, with F-parallel shape operator in the generalized Tanaka-Webster connection. 2 Preliminaries Before going to give our assertions, let us summarize basic material about complex two-plane Grassmannians G 2 (C m+2 ), for details we refer to [2], [3] and [4]. 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, respectively, 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 negative 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 symmetric space. For computational reasons we normalize g such that the maximal sectional curvature of (G 2 (C m+2 ), g) is eight.

5 Parallelism of shape operator w.r.t. the g-tanaka-webster connection 187 Now let us consider a real hypersurface M on G 2 (C m+2 ), that is, a hypersurface of G 2 (C m+2 ) with real codimension one. The induced Riemannian metric on M will also be denoted by g, and denotes the Riemannian connection of (M, g). Let N be a local unit normal vector field of M and A the shape operator of M with respect to N. Using the expression for the curvature tensor R of G 2 (C m+2 ) given in Berndt and Suh [4], the equation of Codazzi becomes (2.1) ( X A)Y ( Y A)X = η(x)φy η(y )φx 2g(φX, Y ) + {η ν (X)φ ν Y η ν (Y )φ ν X 2g(φ ν X, Y ) ν + + { η ν (φx)φ ν φy η ν (φy )φ ν φx { η(x)η ν (φy ) η(y )η ν (φx) ν. As mentioned above, the Tanaka-Webster connection is the canonical affine connection defined on a non-degenerate pseudo-hermitian CR-manifold (see [13], [15]). In [14], Tanno defined the g-tanaka-webster connection for contact metric manifolds by the canonical connection which coincides with the Tanaka-Webster connection if the associated CR-structure is integrable. Now let us recall the g-tanaka-webster connection defined by Cho for real hypersurfaces of Kähler manifolds as follows: (2.2) X Y = XY + g(φax, Y ) η(y )φax kη(x)φy for a non-zero real number k (see [5], [6] and [7]) (Note that is invariant under the choice of the orientation. Namely, we may take k instead of k in (2.2) for the opposite orientation N). 3 Key Lemmas Let M be a Hopf hypersurface in G 2 (C m+2 ) with -parallel shape operator A in generalized Tanaka-Webster connection. In other words, the shape operator A of a Hopf hypersurface M satisfies the following: (*) ( A)X = 0 for any tangent vector field X on M. In this section, we will prove that the Reeb vector field of M belongs to either the distribution D or the distribution D, under our assumptions.

6 188 Imsoon Jeong, Hyunjin Lee & Young Jin Suh First of all, we give the fundamental equation which is induced from (*). From the definition of the g-tanaka-webster connection (2.2), we have (3.1) ( X A)Y = X (AY ) A( X Y ) = ( X A)Y + g(φax, AY ) η(ay )φax kη(x)φay g(φax, Y )A + η(y )AφAX + kη(x)aφy for any tangent vector fields X and Y on M. Under our assumptions, ( A)X = 0 and A = α, it follows that (3.2) ( A)X kφax + kaφx = 0 for any tangent vector field X on M. By using the equation of Codazzi (2.1), we have (3.3) ( A)X = ( X A) + φx + {η ν ()φ ν X η ν (X)φ ν 3g(φ ν, X) ν. On the other hand, using the notion of Hopf, that is, A = α, we have (3.4) ( X A) = AφAX + (Xα) + αφax for any vector field X tangent to M. Thus from (3.3) and (3.4), we have (3.5) ( A)X = AφAX + (Xα) + αφax + φx + {η ν ()φ ν X η ν (X)φ ν 3g(φ ν, X) ν. Now substituting (3.5) into (3.2), the assumption of g-tanaka-webster -parallel is equivalent to the following equation (3.6) AφAX + (Xα) + (α k)φax + kaφx + φx + {η ν ()φ ν X η ν (X)φ ν + 3η ν (φx) ν = 0 for any vector field X tangent to M. Replacing X by in (3.6) and using M is Hopf, we can assert: Lemma 3.1. Let M be a Hopf hypersurface in G 2 (C m+2 ). If M has the g-tanaka- Webster -parallel shape operator, then the principal curvature α = g(a, ) is constant along the direction of, that is, α = 0. Now we introduce a Lemma which is given in [10] as follows:

7 Parallelism of shape operator w.r.t. the g-tanaka-webster connection 189 Lemma A. Let M be a Hopf hypersurface in G 2 (C m+2 ). If the principal curvature α = g(a, ) is constant along the direction of, then the distribution D or D component of the Reeb vector field is invariant by the shape operator. On the other hand, using the notion of the geodesic Reeb flow, Berndt and Suh [4] gave the following: Lemma B. If M is a connected orientable real hypersurface in G 2 (C m+2 ) with geodesic Reeb flow, then we have the following equation: (3.7) αg ( (Aφ + φa)x, Y ) 2g(AφAX, Y ) + 2g(φX, Y ) = 2 { η ν (X)η ν (φy ) η ν (Y )η ν (φx) g(φ ν X, Y )η ν () 2η(X)η ν (φy )η ν () + 2η(Y )η ν (φx)η ν () for any tangent vector fields X and Y on M. Moreover, in [3], Berndt and Suh also gave the following: Lemma C. Under the assumption M is Hopf, the principal curvature α of the Reeb vector field satisfies (3.8) Y α = (α)η(y ) 4 η ν ()η ν (φy ) for any tangent vector field Y on M. Now we give a Key Lemma as follows: Lemma 3.2. Let M be a Hopf hypersurface in G 2 (C m+2 ), m 3. If M has the g-tanaka-webster -parallel shape operator, then the Reeb vector field belongs to either the distribution D or the distribution D. 4 The proof of Main Theorem and Corollary From now on, let us assume that M is a Hopf hypersurface in G 2 (C m+2 ) with -parallel shape operator in g-tanaka-webster connection. Then by Lemma 3.2, we can consider two cases, that is, D or D, respectively. First, we consider the case D. From this, we put = 1.

8 190 Imsoon Jeong, Hyunjin Lee & Young Jin Suh Lemma 4.1. Let M be a Hopf hypersurface, α 2k, in G 2 (C m+2 ), m 3 with g-tanaka-webster -parallel shape operator. If the Reeb vector belongs to the distribution D, then the shape operator A commutes with the structure tensor φ. As mentioned in the Introduction, the structure tensor field φ commutes with the shape operator A of M if and only if the Reeb flow on M is isometric (see [4]). Thus, from Lemma 4.1 and Theorem B we have the following remark. Remark 4.2. Let M be a Hopf hypersurface, α 2k, in G 2 (C m+2 ), m 3 with g-tanaka-webster -parallel shape operator. If the Reeb vector belongs to the distribution D, then M is locally congruent to a real hypersurface of Type (A) in Theorem A. Now let us consider the converse problem of Remark 4.2. In other words, if M is locally congruent to a real hypersurface of Type (A) in G 2 (C m+2 ), let us check whether the shape operator A of M is g-tanaka-webster -parallel or not. In order to solve this problem, we introduce a proposition due to Berndt and Suh [3] concerned with a tube of Type (A) as follows : Proposition A. Let M be a connected real hypersurface of G 2 (C m+2 ). Suppose that AD D, A = α, and is tangent to D. Let J 1 J be the almost Hermitian structure such that JN = J 1 N. Then M has three (if r = π/2 8) or four (otherwise) distinct constant principal curvatures α = 8 cot( 8r), β = 2 cot( 2r), λ = 2 tan( 2r), µ = 0 with some r (0, π/ 8). The corresponding multiplicities are m(α) = 1, m(β) = 2, m(λ) = 2m 2 = m(µ), and the corresponding eigenspaces are T α = R = RJN = R 1 = Span { = Span { 1, T β = C = C N = R 2 R 3 = Span { 2, 3, T λ = {X X H, JX = J 1 X, T µ = {X X H, JX = J 1 X where R, C and H respectively denotes real, complex and quaternionic span of the structure vector field and C denotes the orthogonal complement of C in H.

9 Parallelism of shape operator w.r.t. the g-tanaka-webster connection 191 From the above proposition, we naturally see that if M is a real hypersurface of Type (A) in G 2 (C m+2 ), then M is Hopf and D. Thus, taking the covariant derivative along -direction with respect to the g-tanaka-webster connection to a shape operator A, we have for any tangent vector field X on M (3.1) ( A)X = AφAX + (Xα) + (α k)φax + kaφx + φx + φ 1 X { + η ν (X)φ ν + 3η ν (φx) ν = AφAX + (α k)φax + kaφx + φx + φ 1 X 2η 2 (X) 3 + 2η 3 (X) 3, where we have used that the principal curvature α is constant and = 1 in the second equality. Using this equation (3.1), let us check our converse problem for each eigenspace, T α, T β, T λ and T µ, of real hypersurfaces of Type (A) in G 2 (C m+2 ). Case 1 : X = (= 1 ) T α. By putting X = into (3.1), we know that the shape operator A is -parallel in g-tanaka-webster connection for X T α. Case 2 : X T β where T β = Span{ 2, 3. Since T β is spanned by 2 and 3, we can put X = 2 and X = 3 in (3.1). Then we have ( A) 2 = (β 2 αβ 2) 3, and ( A) 3 = (β 2 αβ 2) 2, respectively. On the other hand, we know that β 2 αβ 2 = 0, because α = 8 cot( 8r) and β = 2 cot( 2r) in Proposition A. So we can assert the shape operator of M also satisfies ( A)X = 0 for any eigenvector X T β. Case 3 : X T λ = {X X ν, JX = J 1 X. When M is a real hypersurface of Type (A) in G 2 (C m+2 ), we naturally see that if X T λ then φx = φ 1 X. Moreover, the vector φx also belong to the eigenspace T λ for any X T λ, that is, φt λ T λ. By putting X T λ in (3.1) and together with these facts, we obtain ( A)X = (λ 2 αλ 2)φX. But in Proposition A, since the principal curvatures α and λ are given by α = 8 cot( 8r) and λ = 2 tan( 2r) for r (0, π/ 8), respectively, we get (3.2) λ 2 αλ 2 = 0. It implies that ( A)X = 0 for any tangent vector field X T λ.

10 192 Imsoon Jeong, Hyunjin Lee & Young Jin Suh Case 4 : X T µ = {X X ν, JX = J 1 X. In this case, if X T µ then φx = φ 1 X. Moreover, we see φt µ T µ. So, the equation (3.1) could be changed by ( A)X = 0, because µ = 0. Summing up all cases mentioned above, we can assert that: Remark 4.3. The shape operator A of real hypersurfaces of Type (A) in G 2 (C m+2 ) is g-tanaka-webster -parallel. From Remarks 4.2 and 4.3, we assert the following: Theorem 4.4. Let M be a Hopf hypersurface, α 2k, in G 2 (C m+2 ), m 3 with g-tanaka-webster -parallel shape operator. The Reeb vector field belongs to the distribution D if and only if M is locally congruent to an open part of a tube around a totally geodesic G 2 (C m+1 ) in G 2 (C m+2 ). Now we introduce the following proposition which is given by Berndt and Suh in [3] as follows: Proposition B. Let M be a connected real hypersurface in G 2 (C m+2 ). Suppose that AD D, A = α, and 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 = R = Span {, T β = JJ = Span { ν ν = 1, 2, 3, T γ = J = Span { φ ν ν = 1, 2, 3, T λ, T µ, where T λ T µ = (HC), JT λ = T λ, JT µ = T µ, JT λ = T µ.

11 Parallelism of shape operator w.r.t. the g-tanaka-webster connection 193 The distribution (HC) is the orthogonal complement of HC where HC = R RJ J JJ. Using this proposition, we will check whether the shape operator of real hypersurfaces of Type (B) in G 2 (C m+2 ) is g-tanaka-webster -parallel or not. In order to do this, we suppose that M is a real hypersurface of Type (B) in G 2 (C m+2 ) with g-tanaka-webster -parallel. From the equation (3.6), we see that (3.3) ( A)X = 0 AφAX + (α k)φax + kaφx + φx { + η ν (X)φ ν + 3η ν (φx) ν = 0 for any tangent vector field X on M, because M has constant principal curvatures with D. When M becomes a real hypersurface of Type (B), the tangent space T M can be decomposed with five distinct eigenspaces T α, T β, T γ, T λ, and T µ, in such a way that, T M = T α T β T γ T λ T µ. Now, if we put X = 1 in (3.3), then it follows β(α k)φ 1 = 0, because γ = 0 and D. Since β = 0 can not occur for some r (0, π/4), we see that α = k where k is non-zero constant. Substituting α = k into (3.3), we have (3.4) AφAX + αaφx + φx + for any tangent vector field X on M. By putting X T λ in (3.4), we have { η ν (X)φ ν + 3η ν (φx) ν = 0 ( λµ + αµ + 1)φX = 0, since if X T λ then X ν, φ ν (ν = 1.2.3) and φt λ T µ. From this, we see that λµ αµ 1 = 0. By using the properties in Proposition B, we have 1 + tan 2 r = 0 for some r (0, π/4). This gives a contradiction. So this case also can not be occurred. Thus, summing up, we give the following remark. Remark 4.5. Real hypersurfaces of Type (B) in G 2 (C m+2 ) do not have the g- Tanaka-Webster -parallel shape operator.

12 194 Imsoon Jeong, Hyunjin Lee & Young Jin Suh On the other hand, by Main Theorem due to Lee and Suh [11] and Remark 4.5, we assert the following: Theorem 4.6. There does not exist any Hopf hypersurface in G 2 (C m+2 ), m 3, with the g-tanaka-webster -parallel shape operator if the Reeb vector field belongs to the distribution D. Hence summing up Lemma 3.2, together with Theorems 4.4 and 4.6, we give a complete proof of our Main Theorem in the introduction. Now we give the proof of Corollary in the Introduction. From now on, let us suppose that M is Hopf hypersurfaces in G 2 (C m+2 ), (m 3) with g-tanaka- Webster F-parallel shape operator. By definitions which are explained in the Introduction, we naturally see that the notion of g-tanaka-webster F-parallel shape operator is stronger than -parallel shape operator in g-tanaka-webster connection. Thus, from Lemma 3.2, Main Theorem in [11] and Remark 4.5, we assert the following: Fact I. If the Reeb vector belong to the distribution D, then there do not exist Hopf hypersurfaces with g-tanaka-webster F-parallel shape operator in G 2 (C m+2 ). Moreover, using the relation between F-parallel and -parallel in g-tanaka- Webster connection, and together with Lemma 3.2 and Remark 4.2, we also have Fact II. Let M be a Hopf hypersurface, α 2k, in G 2 (C m+2 ), (m 3) with g- Tanaka-Webster F-parallel shape operator. If the Reeb vector field belongs to the distribution D, then M is locally congruent to a tube of Type (A) in Theorem A. Finally, we consider the converse problem of Fact II. That is, if M is a real hypersurface of Type (A) in Theorem A, let us check whether the shape operator A of M satisfies F-parallel in g-tanaka-webster connection or not. Now, since M is a real hypersurface of Type (A) in G 2 (C m+2 ), we put = 1. In order to solve this problem, suppose that M has the g-tanaka-webster F-parallel shape operator. From this assumption, by putting X = 2 and Y T λ in (3.1), we obtain λ 2 Y A( 2 Y ) = 0. It means that the vector 2 Y becomes a principal vector with corresponding eigenvalue λ, that is, 2 Y T λ for any Y T λ. From this, we see that (3.5) φ( 2 Y ) = φ 1 ( 2 Y ) for any vector Y T λ. Moreover, since Y T λ, we see φy = φ 1 Y. It follows that (3.6) φ( 2 Y ) = 2 (φy ) ( 2 φ)y = 2 (φ 1 Y ) ( 2 φ)y = ( 2 φ 1 )Y + φ 1 ( 2 Y ) ( 2 φ)y.

13 Parallelism of shape operator w.r.t. the g-tanaka-webster connection 195 From (3.5) and (3.6), we have ( 2 φ 1 )Y ( 2 φ)y = 0. Using the formulas obtained by the covariant derivative to the structure tensors φ and φ 1 (see Section 2 in [8] and [11]), we have (3.7) q 2 ( 2 )φ 3 Y + q 3 ( 2 )φ 2 Y = 0. On the other hand, taking a covariant derivative along any direction X to our assumption = 1, we obtain φax = q 3 (X) 2 q 2 (X) 3 + φ 1 AX. From this, applying an inner product with 2 and 3 to above equation, we have respectively. Then it follows that q 2 (X) = 2g(AX, 2 ) and q 3 (X) = 2g(AX, 3 ), q 2 ( 2 ) = 2g(A 2, 2 ) = 2β and q 3 ( 2 ) = 2g(A 3, 2 ) = 0, because 2 and 3 belong to the eigenspace T β. Then from (3.7), we have consequently β = 0. But actually, the principal curvature β is given by β = 2 cot( 2r) where 0 < r < π 2 (see Proposition A) and it is not vanish. From such a point of view, 2 we can assert that Fact III. The shape operator A of real hypersurfaces of Type (A) in G 2 (C m+2 ) does not satisfies the g-tanaka-webster F-parallel. Summing up above Facts I, II, and III, we give a complete proof of our Corollary in the Introduction. Acknowledgment This article is a brief survey in the purpose of 2011 NIMS Hot Topic Workshop titled The 15th International Workshop on Differential Geometry and related fields. The detailed proof of lemmas and theorems in this manuscript was given in a paper [9] due to I. Jeong, H. Lee and Y.J. Suh. References [1] D.V. Alekseevskii, Compact quaternion spaces, Funct. Anal. Appl. 2 (1968), [2] J. Berndt, Riemannian geometry of complex two-plane Grassmannian, Rend. Sem. Mat. Univ. Politec. Torino 55 (1997),

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