Real hypersurfaces in the complex quadric with certain condition of normal Jacobi operator

07/11/2017
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We introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric Qm. The normal Jacobi operator of Codazzi type implies that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a non-existence theorem of real hypersurfaces in Qm with normal Jacobi operator of Codazzi type.

Real hypersurfaces in the complex quadric with certain condition of normal Jacobi operator

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application/pdf Real hypersurfaces in the complex quadric with certain condition of normal Jacobi operator Imsoon Jeong, Gyu Jong Kim, Young Jin Suh
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contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
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We introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric Qm. The normal Jacobi operator of Codazzi type implies that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a non-existence theorem of real hypersurfaces in Qm with normal Jacobi operator of Codazzi type.
Real hypersurfaces in the complex quadric with certain condition of normal Jacobi operator

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        <resourceType resourceTypeGeneral="Text">Text</resourceType><subjects><subject>normal Jacobi operator of Codazzi type</subject><subject>A-isotropic</subject><subject>A- principal</subject><subject>complex conjugation</subject><subject>complex quadric</subject><subject>Kähler structure</subject></subjects><dates>
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Real hypersurfaces in the complex quadric with certain condition of normal Jacobi operator Imsoon Jeong1 , Gyu Jong Kim2 , and Young Jin Suh2 1 Pai Chai University, Daejeon, 35345, Republic of Korea, 2 Kyungpook National University, Daegu 702-701, Republic of Korea Abstract. We introduce the notion of normal Jacobi operator of Co- dazzi type for real hypersurfaces in the complex quadric Qm . The normal Jacobi operator of Codazzi type implies that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a non-existence theorem of real hypersurfaces in Qm with normal Jacobi operator of Codazzi type. Keywords: normal Jacobi operator of Codazzi type, A-isotropic, A- principal, Kähler structure, complex conjugation, complex quadric 1 Introduction In the complex projective space CPm+1 and the quaternionic projective space QPm+1 some classifications related to the Ricci tensor and the structure Jacobi operator were investigated by Kimura [10], [11], Pérez and Suh [17], [18], Pérez and Santos [14], and Pérez, Santos and Suh [15], [16], respectively. Some examples of Hermitian symmetric space of rank 2 are G2(Cm+2 ) = SUm+2/S(U2Um) and G∗ 2(Cm+2 ) = SU2,m/S(U2Um), which are said to be complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians, respectively (see [23], [24], and [25]). These are viewed as Hermitian symmetric spaces and quaternionic Kähler symmetric spaces equipped with the Kähler structure J and the quaternionic Kähler structure J. The classification problems of real hypersurfaces in the complex two-plane Grass- mannian SUm+2/S(U2Um) with certain geometric conditions were mainly inves- tigated by Jeong, Kim and Suh [6], Jeong, Machado, Pérez and Suh [7], [8] and Suh [23], [24], [25], where the classification of commuting and parallel Jacobi op- erator, contact hypersurfaces, parallel Ricci tensor, and harmonic curvature for real hypersurfaces in complex two-plane Grassmannians were extensively stud- ied. Moreover, in [26] we have asserted that the Reeb flow on a real hypersurface in SU2,m/S(U2Um) is isometric if and only if M is an open part of a tube around a totally geodesic SU2,m−1/S(U2Um−1) ⊂ SU2,m/S(U2Um) . As another kind of Hermitian symmetric space with rank 2 of compact type different from the above ones, we can give the example of complex quadric Qm = SOm+2/SOmSO2, which is a complex hypersurface in complex projective space CPm+1 (see Klein [9], and Smyth [21]). The complex quadric can also be regarded as a kind of real Grassmann manifold of compact type with rank 2 (see Kobayashi and Nomizu [12]). Accordingly, the complex quadric admits two im- portant geometric structures, a complex conjugation structure A and a Kähler structure J, which anti-commute with each other, that is, AJ = −JA. Then for m≥2 the triple (Qm , J, g) is a Hermitian symmetric space of compact type with rank 2 and its maximal sectional curvature is equal to 4 (see Klein [9] and Reckziegel [20]). In addition to the complex structure J there is another distinguished geometric structure on Qm , namely a parallel rank two vector bundle A which contains an S1 -bundle of real structures, that is, complex conjugations A on the tan- gent spaces of Qm . This geometric structure determines a maximal A-invariant subbundle Q of the tangent bundle TM of a real hypersurface M in Qm as follows: Q = {X∈TzM|AX∈TzM for all A∈A}. Moreover, the derivative of the complex conjugation A on Qm is defined by ( ¯ ∇XA)Y = q(X)JAY for any vector fields X and Y on M and q denotes a certain 1-form defined on M. Recall that a nonzero tangent vector W ∈ T[z]Qm is called singular if it is tangent to more than one maximal flat in Qm . There are two types of singu- lar tangent vectors for the complex quadric Qm (see Berndt and Suh [2] and Reckziegel [20]): • If there exists a conjugation A ∈ A such that W ∈ V (A)(the (+1)-eigenspace of a conjugation A in T[z]Qm ), then W is singular. Such a singular tangent vector is called A-principal. • If there exists a conjugation A ∈ A and orthonormal vectors X, Y ∈ V (A) such that W/||W|| = (X + JY )/ √ 2, then W is singular. Such a singular tangent vector is called A-isotropic. When we consider a real hypersurface M in the complex quadric Qm , under the assumption of some geometric properties the unit normal vector field N of M in Qm can be either A-isotropic or A-principal (see [27] and [28]). In the first case, where N is A-isotropic, Suh has shown in [27] that M is locally congruent to a tube over a totally geodesic CPk in Q2k . In the second case, when the unit normal N is A-principal, he proved that a contact hypersurface M in Qm is locally congruent to a tube over a totally geodesic and totally real submanifold Sm in Qm (see [28]). Jacobi fields along geodesics of a given Riemannian manifold M̄ satisfy a well known differential equation. Naturally the classical differential equation inspires the so-called Jacobi operator. That is, if R̄ is the curvature operator of M̄, the Jacobi operator with respect to X at z ∈ M̄, is defined by (R̄XY )(z) = (R̄(Y, X)X)(z) for any Y ∈ TzM̄. Then R̄X becomes a symmetric endomorphism of the tangent bundle TM̄ of M̄, that is, R̄X ∈ End(TzM̄). Clearly, each tangent vector field X to M̄ provides a Jacobi operator with respect to X (see Pérez and Santos [14], and Pérez, Santos and Suh [15], [16]). From such a view point, for a real hyper- surface M in Qm with unit normal vector field N the normal Jacobi operator R̄N is defined by R̄N = R̄( ·, N)N ∈ End (TzM), z ∈ M, where R̄ denotes the curvature tensor of Qm . Of course, the normal Jacobi opeartor R̄N is a symmetric endomorphism of TzM (see Jeong, Kim and Suh [6], Jeong, Machado, Pérez and Suh [7] and [8]). We introduce the notion of paral- lelism with respect to the normal Jacobi operator R̄N of M in M̄. It is defined by (∇XR̄N )Y = 0 for all tangent vector fields X and Y on M. This has a geometric meaning that the eigenspaces of R̄N are parallel, that is, invariant under any parallel displacements along any curves on M in M̄. Using this notion, specially, in [6] they gave a non-existence theorem for Hopf hypersurfaces with parallel normal Jacobi operator in G2(Cm+2 ). Moreover, Suh [30] gave a non-existence theorem for the case of a Hopf hypersurface with parallel normal Jacobi operator in Qm . Here M is called Hopf if the Reeb vector field ξ defined by ξ = −JN is principal, that is, Sξ = αξ, where S is the shape operator of M associated with the unit normal N. As a generalized notion of parallel normal Jacobi operator, in this paper we want to introduce the definition of the normal Jacobi operator of Codazzi type from the view point of exterior derivative (see Besse [2], and Derdzinski and Shen [3]). Let E be a vector bundle over a manifold M̄. For any section ω of Vp M̄ N E the exterior derivative d∇ ω is the section of Vp+1 M̄ N E such that for X0, · · ·, Xp in TzM̄, z ∈ M̄, extended to vector fields X̃0, · · ·, X̃p in a neighborhood as follows: (d∇ ω)(X0, · · ·, Xp) = X i (−1)i ∇Xi (ω(X̃0, · · ·, ˆ X̃i, · · ·, X̃p)) + X i6=j (−1)i+j ω([X̃i, X̃j], X̃0, · · ·, ˆ X̃i, · · ·, ˆ X̃j, · · ·, X̃p). In case of the normal Jacobi operator R̄N , then the exterior derivative of the normal Jacobi operator gives (d∇ R̄N )(X, Y ) = ∇X(R̄N Y ) − ∇Y (R̄N X) − R̄N ([X, Y ]) = (∇XR̄N )Y − (∇Y R̄N )X. Now we apply above equation to R̄N , which is a tensor field of type (1, 1) on a real hypersurface M in M̄. The normal Jacobi operator R̄N of M in M̄ is said to be of Codazzi type if the normal Jacobi operator R̄N satisfies d∇ R̄N = 0, that is, the normal Jacobi operator is closed with respect to the exterior derivative d∇ related to the induced connection ∇. Then by the above formula we have the following (∇XR̄N )Y = (∇Y R̄N )X for any X, Y ∈ TzM, z ∈ M. Related to this definition, Machado, Pérez, Jeong, and Suh considered the case of a real hypersurface in G2(Cm+2 ) and gave a non-existence theorem as follows. Remark 1. There does not exist any connected Hopf real hypersurface in com- plex two-plane Grassmannians G2(Cm+2 ), m ≥ 3, whose normal Jacobi operator is of Codazzi type if the distribution D or the D⊥ -component of the Reeb vector field is invariant under the shape operator (see [13]). Remark 2. Many geometers have studied the various tensor of Codazzi type (1,1), for example the shape operator, the structure Jacobi operator Rξ := R( ·, ξ)ξ ∈ End(TM), Ricci tensor etc., on a real hypersurface M in G2(Cm+2 ) (see [4], [22], [25] and so on). In particular, in [22] the third author deal with the parallelism for the shape operator on M. But we know that the proofs contain for the case of Codazzi type with respect to the shape operator. On the other hand, the Ricci operator Ric of M in Qm is said to be of harmonic curvature if the Ricci operator Ric satisfies (∇XRic)Y = (∇Y Ric)X for any X, Y ∈TzM, z∈M. In the study of real hypersurfaces in the complex quadric Qm we considered some notions of parallel Ricci tensor or more gen- erally, harmonic curvature, that is, ∇Ric = 0 or (∇XRic)Y = (∇Y Ric)X re- spectively(see Suh [28] and [29]). But from the assumptions of Ricci parallel or harmonic curvature, it was difficult for us to derive the fact that either the unit normal N is A-isotropic or A-principal. So in [28] and [29] we gave a classification with the further assumption of A-isotropic. But fortunately, when we consider the normal Jacobi operator R̄N of Codazzi type, first we can assert that the unit normal vector field N becomes either A-isotropic or A-principal. Theorem 1. Let M be a Hopf real hypersurface in Qm , m ≥ 3, with normal Jacobi operator of Codazzi type. Then the unit normal vector field N is singular, that is, N is A-isotropic or A-principal. u t Then motivated by such a result, next we give a non-existence theorem for Hopf hypersurfaces in the complex quadric Qm with normal Jacobi operator of Codazzi type as follows: Theorem 2. There does not exist any Hopf real hypersurface in Qm , m ≥ 3 with normal Jacobi operator of Codazzi type. u t 2 Outline of Proofs It is well known that complex quadric Qm = SOm+2/SOmSO2 is a compact Kählelr manifold with the metric and complex structure induced from CPm+1 with constant holomorphic sectional curvature 4. Moreover it becomes a Hermi- tian symmetric space of rank 2 being equipped with both a complex structure J and a real structure (or complex conjugation) A satisfying AJ = −JA and TrA