Hopf hypersurfaces in nonflat complex space forms

Proceedings of The Sixteenth International Workshop on Diff. Geom. 16(2012) 25-34 Hopf hypersurfaces in nonflat complex space forms Makoto Kimura Department of Mathematics, Ibaraki University, Mito, Ibaraki 310-8512, Japan e-mail : mkimura@math.shimane-u.ac.jp (2010 Mathematics Subject Classification : 53C40.) Abstract. We study Hopf hypersurfaces in non-flat complex space forms by using some Gauss maps from a real hypersurface to complex (either positive definite or indefinite) 2-plane Grassmannian and quaternionic (para-)kähler structures. 1 Introduction Among real hypersurfaces in complex space forms, Hopf hypersurfaces have good geometric properties and studied widely. A real hypersurface M in a complex space form M(c) (of constant holomorphic sectional curvature 4c) is called Hopf if the structure vector ξ = JN is an eigenvector of the shape operator A of M. We say that the (constant) eigenvalue µ with Aξ = µξ a Hopf curvature of Hopf hypersurface M. When c > 0, Cecil and Ryan [4] proved that a Hopf hypersurface M with Hopf curvature µ = 2 cot 2r (0 < r < π/2) lies on the tube of radius r over a complex submanifold in CP n, provided that the focal map φ r : M CP n, φ r (x) = exp x (rn x ) (x M, N x Tx M, N x = 1) has constant rank on M. After that Montiel [15] proved similar result for Hopf hypersurfaces with large Hopf curvature, (i.e., µ > 2) in complex hyperbolic space CH n (with c = 1). On the other hand, Ivey [7] and Ivey-Ryan [8] proved that a Hopf hypersurface with µ < 2 in CH n may be constructed from an arbitrary pair of Legendrian submanifolds in S 2n 1. Here we will study (Hopf) hypersurfaces M in CP n (resp. CH n ) by using some Gauss map G (resp. G) from M to complex 2-plane Grassmannian G2 (C n+1 ) (resp. Grassmannian G 1,1 (C n+1 1 ) of 2-planes with signature (1, 1) in C n+1 1 ), which maps each point of M to the 2-plane spanned by position vector and structure vector Key words and phrases: * This work was supported by grant Proj. No. 24540080 from Grants-in-Aid for Scientific Research, the Ministry of Education, Science, Sports and Culture, Japan. 25

26 Makoto Kimura at the point in M. We see that: If M is a Hopf hypersurface in CP n, then the Gauss image G(M) is a real (2n 2)-dimensional totally complex submanifold with respect to quaternionic Kähler structure in G 2 (C n+1 ) (Theorem 4.1). If M is a Hopf hypersurface in CH n with (constant) Hopf curvature µ, then the Gauss image G(M) is a real (2n 2)-dimensional submanifold and invariant under a quaternionic (para-)kähler structure I of G 1,1 (C n+1 1 ) satisfying 1. I 2 = 1 provided µ > 2, 2. I 2 = 1 provided µ < 2, 3. I 2 = 0 provided µ = 2 (Theorem 6.1). Detailed descriptions will appear elsewhere. The author would like to thank Professor Young Jin Suh for inviting me to this international workshop. 2 Real hypersurfaces in CP n and CH n First we recall the Fubini-Study metric on the complex projective space CP n (cf. [4, 11]). The Euclidean metric, on C n+1 is given by z, w = Re( t z w) for z, w C n+1. The unit sphere S 2n+1 in C n+1 is the principal fiber bundle over CP n with the structure group S 1 and the Hopf fibration π : S 2n+1 CP n. The tangent space of S 2n+1 at a point z is T z S 2n+1 = {w C n+1 z, w = 0}. Let T z = {w C n+1 z, w = iz, w = 0}. With respect to the principal fiber bundle S 2n+1 (CP n, S 1 ), there is a connection such that T z is the horizontal subspace at z. Then the Fubini-Study metric g of constant holomorphic sectional curvature 4 is given by g(x, Y ) = X, Y, where X, Y T x CP n, and X, Y are respectively their horizontal lifts at a point z with π(z) = x. The complex structure on T defined by multiplication by 1 induces a canonical complex structure J on CP n through π. Next we recall the metric of negative constant holomorphic sectional curvature on the complex hyperbolic space CH n (cf. [15, 11]). The indefinite metric, of index 2 on C n+1 is given by ( ) n z, w = Re z 0 w 0 + z k w k for z = (z 0, z 1,..., z n ), w = (w 0, w 1,..., w n ) C n+1. The anti de Sitter space is defined by H 2n+1 1 = {z C n+1 z, z = 1}. k=1

Hopf hypersurfaces in nonflat compelx space forms 27 H 2n+1 1 is the principal fiber bundle over CH n with the structure group S 1 and the fibration π : H1 2n+1 CH n. The tangent space of H1 2n+1 at a point z is Let T z H 2n+1 1 = {w C n+1 z, w = 0}. T z = {w C n+1 z, w = iz, w = 0}. We observe that the restriction of, to T z is positive-definite. The distribution T z defines a connection in the principal fiber bundle H1 2n+1 (CH n, S 1 ), because T z is complementary to the subspace {iz} tangent to the fibre through z, and invariant under the S 1 -action. Then the metric g of constant holomorphic sectional curvature 4 is given by g(x, Y ) = X, Y, where X, Y T x CH n, and X, Y are respectively their horizontal lifts at a point z with π(z) = x. The complex structure on T defined by multiplication by 1 induces a canonical complex structure J on CH n through π. We recall some facts about real hypersurfaces in complex space forms. Let M n (c) be a space of constant holomorphic sectional curvature 4c with real dimension 2n. Let J : T M M be the complex structure with properties J 2 = 1, J = 0, and JX, JY = X, Y, where is the Levi-Civita connection of M. Let M 2n 1 be a real hypersurface in M(c) and let N be a local unit normal vector field of M in M. We define structure vector of M as ξ = JN. If ξ is a principal vector, i.e., Aξ = µξ holds for the shape operator of M in M, M is called a Hopf hypersurface and µ is called Hopf curvature. It is well known ([13], [12]) that Hopf curvature µ of Hopf hypersurface M is constant for non flat complex space forms. We note that µ is not constant for Hopf hypersurfaces in C n+1 in general ([16], Theorem 2.1). Fundamental result for Hopf hypersurfaces in complex projective space CP n was proved by Cecil-Ryan [4]: Theorem 2.1. Let M be a connected, orientable Hopf hypersurface of CP n with Hopf curvature µ = 2 cot 2r (0 < r < π/2). Suppose the focal map φ r (x) = exp x (rn x ) (x M, N x T x M, N x = 1) has constant rank q on M. Then: 1. q is even and every point x 0 M has a neighborhood U such that φ r U is an embedded complex (q/2)-dimensional submanifold of CP n. 2. For each point x in such a neighborhood U, the leaf of the foliation T 0 through x intersects U in an open subset of a geodesic hypersphere in the totally geodesic CP (m q/2) orthogonal to T p (φ r U) at p = φ r (x). Thus U lies on the tube of radius r over φ r U. Similar result for Hopf hypersurfaces with large Hopf curvature in complex hyperbolic space CH n was proved by Montiel [15]:

28 Makoto Kimura Theorem 2.2. Let M be an orientable Hopf hypersurface of CH n and we assume that φ r (r > 0) has constant rank q on M. Then, if µ = 2 coth 2r (hence µ > 2), for every x 0 M there exists an open neighbourhood W of x 0 such that φ r W is a (q/2)-dimensional complex submanifold embedded in CH n. Moreover W lies in a tube of radius r over φ r W. On the other hand, with respect to Hopf hypersurfaces with small Hopf curvature in CH n, Ivey [7] and Ivey-Ryan [8] proved that a Hopf hypersurface with µ < 2 in CH n may be constructed from an arbitrary pair of Legendrian submanifolds in S 2n 1. 3 Totally complex submanifolds in Quaternionic Kähler manifolds We recall the basic definitions and facts on totally complex submanifolds of a quaternionic Kähler manifold (cf. [18]) Let ( M 4m, g, Q) be a quaternionic Kähler manifold with the quaternionic Kähler structure ( g, Q), that is, g is the Riemannian metric on M and Q is a rank 3 subbundle of EndT M which satisfies the following conditions: 1. For each p M, there is a neighborhood U of p over which there exists a local frame field {Ĩ1, Ĩ2, Ĩ3} of Q satisfying Ĩ 2 1 = Ĩ2 2 = Ĩ2 3 = 1, Ĩ 1 Ĩ 2 = Ĩ2Ĩ1 = Ĩ3, Ĩ 2 Ĩ 3 = Ĩ3Ĩ2 = Ĩ1, Ĩ 3 Ĩ 1 = Ĩ1Ĩ3 = Ĩ2. 2. For any element L Q p, g p is invariant by L, i.e., g p (LX, Y )+ g p (X, LY ) = 0 for X, Y T p M, p M. 3. The vector bundle Q is parallel in End T M with respect to the Riemannian connection associated with g. A submanifold M 2m of M is said to be almost Hermitian if there exists a section Ĩ of the bundle Q M such that (1) Ĩ2 = 1, (2) ĨT M = T M (cf. D.V. Alekseevsky and S. Marchiafava [1]). We denote by I the almost complex structure on M induced from Ĩ. Evidently (M, I) with the induced metric g is an almost Hermitian manifold. If (M, g, I) is Kähler, we call it a Kähler submanifold of a quaternionic Kähler manifold M. An almost Hermitian submanifold M together with a section Ĩ of Q M is said to be totally complex if at each point p M we have LT p M T p M, for each L Q p with g(l, Ĩp) = 0 (cf. S. Funabashi [5]). It is known that a 2m (m 2)-dimensional almost Hermitian submanifold M 2m is Kähler if and only if it is totally complex ([1] Theorem 1.12).

Hopf hypersurfaces in nonflat compelx space forms 29 4 Gauss map of real hypersurfaces in CP n We recall complex Grassmann manifold G 2 (C n+1 ) according to [11], Example 10.8. Let M = M (n + 1, 2; C) be the space of complex matrices Z with n + 1 rows and 2 columns such that t ZZ = E2 (or, equivalently, the 2 column vectors are orthonormal with respect to the standard inner product in C n+1 ). M is identified with complex Stiefel manifold V 2 (C n+1 ). The group U(2) acts freely on M on the right: Z ZB, where B U(2). We may consider complex 2-plane Grassmannian G 2 (C n+1 ) as the base space of the principal fiber bundle M with group U(2). For each Z M, the tangent space T Z (M ) is T Z (M ) = {W M(n + 1, 2; C) t W Z + t ZW = 0}. In T Z (M ) we have an inner product g(w 1, W 2 ) = Re trace t W 2 W 1. Let T Z = {ZA A u(2)} T Z (M ) and let T Z be the orthogonal complement of T Z in T Z (M ) with respect to g. The subspace T Z admits a complex structure W iw. We see that π : M G 2 (C n+1 ) induces a linear isomorphism of T Z onto T π(z) (G 2 (C n+1 )). By transferring the complex structure i and the inner product g on T Z by π we get the usual complex structure J on G 2 (C n+1 ) and the Hermitian metric on G 2 (C n+1 ), respectively. Note that G 2 (C n+1 ) is a complex 2n 2-dimensional Hermitian symmetric space. Let U be an open subset in G 2 (C n+1 ) and let Z : U π 1 (U) be a cross section with respect to the fibration π : M G 2 (C n+1 ). Then for each p U, the subspace T Z(p) admits 3 linear endomorphisms I 1, I 2, I 3 : ( ) ( ) i 0 0 1 I 1 : W W, I 0 i 2 : W W, 1 0 ( ) 0 i I 3 : W W. i 0 By transferring I 1, I 2, I 3 on T Z(p) (p U) by π we get local basis Ĩ1, Ĩ2, Ĩ3 of quaternionic Kähler structure Q p on U G 2 (C n+1 ) (cf. [2, 19]). Let M 2n 1 be a real hypersurface in complex projective space CP n and let w S 2n+1 with π(w) = x for x M. We denote ξ w as the horizontal lift of the structure vector ξ at x to T w with respect to the Hopf fibration π : S 2n+1 CP n. Then the Gauss map G : M 2n 1 G 2 (C n+1 ) of M is defined by (4.1) G(x) = π(w, ξ w), where π : M = V 2 (C n+1 ) G 2 (C n+1 ) given as above. It is seen that G is well defined. Theorem 4.1. Let M be a real hypersurface in CP n and let G : M 2n 1 G 2 (C n+1 ) be the Gauss map defined by (4.1). Suppose M is a Hopf hypersurface. Then the image G(M) is a real (2n 2)-dimensional totally complex submanifold of G 2 (C n+1 ).

30 Makoto Kimura 5 Totally (para-)complex submanifolds in Quaternionic Para-Kähler manifolds We recall real Clifford algebras (cf. [6, 14]) and (quaternionic) para-complex structure (cf. [14]). Let (V = R(p, q),, ) be a real symmetric inner product space of signature p, q. The Clifford algebra C(p, q) is the quotient V/I(V ), where I(V ) is the two-sided ideal in V generated by all elements: Examples of Clifford algebras are: x x + x, x with x V. 1. C = C(0, 1), Complex numbers: z = x + iy, i 2 = 1, z 2 = x 2 + y 2, there is no zero divisors. 2. C = C(1, 0), Para-complex numbers: z = x + jy, j 2 = 1, z 2 = x 2 y 2, there exists zero divisors. 3. H = C(0, 2), Quaternions: q = q 0 +iq 1 +jq 2 +kq 3, i 2 = j 2 = k 2 = 1, q 2 = q q = q 2 0 + q 2 1 + q 2 2 + q 2 3, there is no zero divisors, H = C 2, q = z 1 (q) + jz 2 (q). 4. H = C(2, 0) = C(1, 1), Para-quaternions: q = q0 +iq 1 +jq 2 +kq 3, i 2 = 1, j 2 = k 2 = 1, q 2 = q 2 0 + q 2 1 q 2 2 q 2 3, there exists zero divisors, H = C 2, q = z 1 (q) + jz 2 (q). There is a natural correspondence between complex numbers C and Euclidean plane R 2, and para-complex numbers C are naturally identified with real symmetric inner product space R(1, 1) of signature (1, 1). Also there is a natural correspondence between quaternions H and Euclidean 4-space R 4, and para-quaternions H are naturally identified with real symmetric inner product space R(2, 2) of signature (2, 2). Let M be a differentiable manifold. We consider the following structures on a real vector space V = T x M, x M, where one assumes that I, J, K are given endomorphisms: 1. Complex: J 2 = 1, V C = V + J 2. Para-complex: J 2 = 1, V C = V + V J, V ± J = {u V C Ju = ±iu}, J V J, V ± J = {u V C Ju = ±iu}, dim V + J = dim V J, 3. Hypercomplex: H = (I, J, K), I 2 = J 2 = K 2 = 1, IJ = JI = K (JK = KJ = I, KI = IK = J), 4. Para-hypercomplex: H = (I, J, K), I 2 = 1, J 2 = K 2 = 1, IJ = JI = K (JK = KJ = I, KI = IK = J). It is known that the space of (oriented) geodesics (i.e., great circles S 1 ) in round sphere S n is naturally identified with (oriented) real 2-plane Grassmannian G 2 (R n+1 ), and it has complex structure. On the other hand, the space of (oriented) geodesics (i.e., hyperbolic lines H 1 ) in hyperbolic space H n is naturally identified with (oriented) Grassmannian G 1,1 (R n+1 1 ) of 2-plane with signature (1, 1) in Minkowski space R n+1 1, and it has para-complex structure [3, 9, 10]. A (para-)hypercomplex structure on V generates respectively a structure:

Hopf hypersurfaces in nonflat compelx space forms 31 1. Quaternionic, generated by H = (I, J, K): Q = H = RI + RJ + RK, (I, J, K) determined up to A SO(3), S(Q) = {L Q L 2 = 1}, 2. Para-quaternionic, generated by H = (I, J, K): Q = H = RI +RJ +RK, (I, J, K) determined up to A SO(2, 1), S( Q) = S + ( Q) S ( Q), S + (Q) = {L Q L 2 = 1}, S (Q) = {L Q L 2 = 1}. To treat the quaternionic and para-quaternionic case simultaneously, we set η = 1 or η = 1 and (ɛ 1, ɛ 2, ɛ 3 ) = ( 1, 1, 1) or (ɛ 1, ɛ 2, ɛ 3 ) = ( 1, 1, 1) respectively for hypercomplex or para-hypercomplex structure (I 1, I 2, I 3 ): then the multiplication table looks I 2 α = ɛ α 1, I α I β = I β I α = ηɛ γ I γ (α = 1, 2, 3) where (α, β, γ) is any circular permutation of (1, 2, 3). An almost (para-)quaternionic structure on a differentiable manifold M 4n is a rank 3 subbundle Q End(T M), which is locally spanned by a field H = (I 1, I 2, I 3 ) of (para-)hypercomplex structures. Such a locally defined triple H = (I α ), α = 1, 2, 3, will be called a (local) admissible basis of Q. An almost (para-)quaternionic connection is a linear connection which preserves Q; equivalently, for any local admissible basis H = (I α ) of Q, one has I α = ɛ β ω γ I β + ɛ γ ω β I γ (P.C.) where (P.C.) means that (α, β, γ) is any circular permutation of (1, 2, 3). An almost (para-)quaternionic structure Q is called a (para-)quaternionic structure if M admits a (para-)quaternionic connection, i.e. a torsion-free almost (para- )quaternionic connection. An (almost) (para-)quaternionic manifold (M 4n, Q) is a manifold M 4n endowed with an (almost) (para-)quaternionic structure Q. An almost (para-)quaternionic Hermitian manifold (M, Q, g) is an almost (para- )quaternionic manifold (M, Q) endowed with a pseudo Riemannian metric g which is Q-Hermitian, i.e. any endomorphism of Q is g-skew-symmetric. (M, Q, g), (n > 1), is called a (para-)quaternionic Kähler manifold if the Levi-Civita connection of g preserves Q, i.e. g is a (para-)quaternionic connection. Let ( M 4m, g, Q) be a para-quaternionic Kähler manifold with the paraquaternionic Kähler structure ( g, Q). A submanifold M 2m of M is said to be almost Hermitian (resp. almost para-hermitian) if there exists a section Ĩ of the bundle Q M such that (1) Ĩ2 = 1 (resp. Ĩ 2 = 1), (2) ĨT M = T M. An almost (para- )Hermitian submanifold M together with a section Ĩ of Q M is said to be totally (para-)complex if at each point p M we have LT p M T p M, for each L Q p with g(l, Ĩp) = 0.

32 Makoto Kimura 6 Gauss map of real hypersurfaces in CH n We consider Grassmann manifold G 1,1 (C n+1 1 ) of 2-planes with signature (1, 1) in C n+1 1. We put n n matrix J n as 1 0 0 0 1 0 J n =....... 0 0 1 Let M = M (n + 1, 2; C) be the space of complex matrices Z with n + 1 rows and 2 columns such that t ZJn+1 Z = J 2 (or, equivalently, the 2 column vectors are orthonormal with respect to the inner product in C n+1 1 ). M is identified with complex (indefinite) Stiefel manifold V 1,1 (C n+1 1 ). The group U(1, 1) acts freely on M on the right: Z ZB, where B U(1, 1). We may consider complex (1, 1)- plane Grassmannian G 1,1 (C n+1 1 ) as the base space of the principal fiber bundle M with group U(1, 1). For each Z M, the tangent space T Z ( M ) is T Z ( M ) = {W M(n + 1, 2; C) t W J n+1 Z + t ZJ n+1 W = 0}. In T Z (M ) we have an inner product g(w 1, W 2 ) = Re trace t W 2 J n+1 W 1. Let T Z = {ZA A u(1, 1)} T Z (M ) and let T Z be the orthogonal complement of T Z in T Z (M ) with respect to g. The subspace T Z admits a complex structure W iw. We see that π : M G 1,1 (C n+1 1 ) induces a linear isomorphism of T Z onto T π(z) (G 1,1 (C n+1 1 )). By transferring the complex structure i and the inner product g on T Z by π we get the complex structure J on G 1,1 (C n+1 1 ) and the (indefinite) Hermitian metric on G 1,1 (C n+1 1 ), respectively. Let U be an open subset in G 1,1 (C n+1 1 ) and let Z : U π 1 (U) be a cross section with respect to the fibration π : M G 1,1 (C n+1 1 ). Then for each p U, the subspace T Z(p) admits 3 linear endomorphisms I 1, I 2, I 3 : I 1 : W W ( ) i 0, I 0 i 2 : W W ( ) 0 i I 3 : W W. i 0 ( ) 0 1, 1 0 By transferring I 1, I 2, I 3 on T Z(p) (p U) by π we get local basis Ĩ1, Ĩ2, Ĩ3 of quaternionic para-kähler structure Q p on U G 1,1 (C n+1 1 ). Let M 2n 1 be a real hypersurface in complex hyperbolic space CH n and let w H1 2n+1 with π(w) = x for x M. We denote ξ w as the horizontal lift of the structure vector ξ at x to T w with respect to the fibration π : H1 2n+1 CH n. Then the Gauss map G : M 2n 1 G 1,1 (C n+1 1 ) of M is defined by (6.1) G(x) = π(w, ξ w ),

Hopf hypersurfaces in nonflat compelx space forms 33 where π : M = V 1,1 (C n+1 1 ) G 1,1 (C n+1 1 ) given as above. It is seen that G is well defined. Theorem 6.1. Let M 2n 1 be a real hypersurface in CH n and let G : M G 1,1 (C n+1 1 ) be the Gauss map defined by (6.1). Suppose M is a Hopf hypersurface with Hopf curvature µ. Then the image G(M) is a real (2n 2)-dimensional submanifold of G 1,1 (C n+1 1 ) such that 1. G(M) is totally complex provided µ > 2, 2. G(M) is totally para-complex provided µ < 2, 3. There exists a section Ĩ of the bundle Q G(M) satisfying Ĩ2 = 0 and ĨT G(M) = T G(M) provided µ = 2. References [1] D. V. Alekseevsky and S. Marchiafava, Hermitian and Kahler submanifolds of a quaternionic Kahler manifold, Osaka J. Math. 38 (2001), no. 4, 869 904. [2] J. Berndt, Riemannian geometry of complex two-plane Grassmannians, Rend. Sem. Mat. Univ. Politec. Torino 55 (1997), no. 1, 19 83. [3] V. Cruceanu, P. Fortuny and P. M. Gadea, A survey on paracomplex geometry, Rocky Mountain J. 26 (1996), 83-115. [4] T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), no. 2, 481 499. [5] S. Funabashi, Totally complex submanifolds of a quaternionic Kaehlerian manifold, Kodai Math. J. 2 (1979), no. 3, 314 336. [6] F. R. Harvey, Spinors and calibrations, Perspectives in Mathematics, 9. Academic Press, Inc., Boston, MA, 1990. xiv+323 pp. ISBN: 0-12-329650-1 [7] T. E. Ivey, A d Alembert formula for Hopf hypersurfaces, Results Math. 60 (2011), no. 1-4, 293 309. [8] T. E. Ivey and P. J. Ryan, Hopf hypersurfaces of small Hopf principal curvature in CH 2, Geom. Dedicata 141 (2009), 147 161. [9] M. Kimura, Space of geodesics in hyperbolic spaces and Lorentz numbers, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci. 36 (2003), 61 67. [10] S. Kaneyuki and M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math. 8 (1985), 81-98.

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