Harmonic curvature for real hypersurfaces in complex hyperbolic two plane Grassmannians☆
Introduction
For a Riemannian manifold the Ricci tensor can be regarded as a -form with values in the cotangent bundle . Then a Riemannian manifold is said to have harmonic curvature or harmonic Weyl tensor, if or for the scalar curvature is a tensor of Codazzi type, that is, it satisfies where denotes the exterior differential. Then it is equivalent to the following for the curvature tensor and Weyl curvature tensor respectively. Here the operator denotes an adjoint operator of the exterior derivative on the Riemannian manifold .
For the harmonic tensor, it is seen that in the case of the Riemannian curvature tensor which can be regarded as a -form with values in the bundle is closed and coclosed, that is, and . It means, together with the second Bianchi identity , the Laplacian . So it is said to be harmonic (see Besse [6]).
In the geometry of real hypersurfaces in complex space forms or in quaternionic space forms it can be easily checked that there does not exist any real hypersurface with parallel shape operator by virtue of the equation of Codazzi.
From this point of a view many differential geometers have considered a notion weaker than the parallel Ricci tensor, that is, . In particular, Kwon and Nakagawa [12] have proved that there are no real hypersurfaces in a complex projective space with harmonic curvature for any in . Moreover, Ki, Nakagawa and Suh [9] have also proved that there are no real hypersurface with harmonic Weyl tensor in non-flat complex space forms , , .
Now let us consider the transitive group which denotes the set of -indefinite special unitary matrices and the set of -unitary matrices. Accordingly, the Riemannian symmetric space , , which consists of complex two-dimensional subspaces in indefinite complex Euclidean space , has remarkable geometric properties that it is both a Hermitian symmetric space and a quaternionic Kähler symmetric space. Among all Riemannian symmetric spaces of noncompact type, the symmetric spaces , , are the only ones which are Hermitian symmetric and quaternionic Kähler symmetric. It is said to be complex hyperbolic two-plane Grassmannian whose two structures give us several kinds of interesting geometric problems on .
Moreover, we denote by the Kähler structure and by the quaternionic Kähler structure on , where not containing (see Berndt and Suh [2], [4], [5], Pérez and Suh [14], [15]). Let be a connected hypersurface in and denote by the tangent bundle of . The maximal complex subbundle of is defined by , and the maximal quaternionic subbundle of is defined by .
Now we want to give some mentions about real hypersurfaces in for which both and are invariant under the shape operator of . Now we introduce a few hypersurfaces in with these two properties. We denote by the unique fixed point of the action of the isotropy group on .
First, let us consider the conic (or geodesic) compactification of . The points in the boundary of this compactification correspond to equivalence classes of asymptotic geodesics in . Every geodesic in lies in a maximal flat, that is, a two-dimensional Euclidean space embedded in as a totally geodesic submanifold. A geodesic in is called singular if it lies in more than one maximal flat in . A singular point at infinity is the equivalence class of a singular geodesic in . Up to isometry, there are exactly two singular points at infinity for . The singular points at infinity correspond to the geodesics in which are determined by nonzero tangent vectors with the property and respectively.
As a second, we consider the standard embedding of in . Then the orbit of through is the Riemannian symmetric space embedded in as a totally geodesic submanifold. Every tube around in has the property that both and are invariant under the shape operator.
Finally, let be even, say , and consider the standard embedding of in . Then the orbit of through is the quaternionic hyperbolic space embedded in as a totally geodesic submanifold. Any tube around in has the property that both and are invariant under its shape operator (see Berndt and Suh [5], Suh [16], [19]).
As a converse problem of the results mentioned above, we want to assert that there are no other such real hypersurfaces except only one kind of exceptional case. Related to such a result, we introduce a well known theorem due to Berndt and Suh [4], [5] as follows:
Theorem A Let be a connected hypersurface in , . Then the maximal complex subbundle of and the maximal quaternionic subbundle of are both invariant under the shape operator of if and only if is congruent to an open part of one of the following hypersurfaces: a tube around a totally geodesic in ; a tube around a totally geodesic in , ; a horosphere in whose center at infinity is singular; or the following exceptional case holds: The normal bundle of consists of singular tangent vectors of type . Moreover, has at least four distinct principal curvatures, three of which are given by with corresponding principal curvature spaces If is another (possibly nonconstant) principal curvature function, then we have , and .
On the other hand, Suh [18] has considered a real hypersurface with harmonic curvature in the complex two-plane Grassmannian under the conditions of constant scalar and mean curvatures and gave a classification as follows:
Theorem B Let be a Hopf real hypersurface of harmonic curvature in with constant scalar and mean curvatures. If the shape operator commutes with the structure tensor on the distribution , then is locally congruent to a tube over a totally geodesic in with radius such that .
Now in this paper we consider a real hypersurface in complex hyperbolic two-plane Grassmanians with harmonic curvature. The notion of harmonic curvature in complex hyperbolic two-plane Grassmannian is quite different and discriminating from the result of Theorem B in .
Accordingly, the main result of this paper is to give some classifications of all Hopf real hypersurfaces in with harmonic or Weyl harmonic tensor. First for a real hypersurface in with harmonic curvature tensor, that is, on , we assert a non-existence theorem as follows:
Main Theorem There does not exist a Hopf real hypersurface of harmonic curvature in with constant scalar and mean curvatures, provided that the shape operator commutes with the structure tensor on the distribution .
Remark 0.1 Compared to Theorem B in with our Main Theorem in , the results are remarkable and quite different from each other. Though the conditions assumed in our Main Theorem and Theorem B are being the same, but they are important and valuable to get the corresponding results.
Next let us consider a real hypersurface in with harmonic Weyl tensor. Here is said to be with harmonic Weyl tensor if the Laplacian of Weyl curvature tensor of defined by is identically vanishing, that is, , where and denote the Ricci tensor and the scalar curvature of in respectively. Then by using the second Bianchi identity it can be easily verified that the vanishing Laplacian is equivalent to Then by virtue of our Main Theorem we get the following
Corollary There does not exist a Hopf real hypersurface of Weyl harmonic curvature in with constant scalar and mean curvatures, provided that the shape operator commutes with the structure tensor on the distribution .
In Section 1 we recall Riemannian geometry of complex hyperbolic two-plane Grassmannians and in Section 2 we will show some fundamental properties of real hypersurfaces in . The formula for the Ricci tensor and its covariant derivative will be shown explicitly in this section.
In order to prove our Main Theorem in the introduction, in Section 4 we will give some formulas from the contraction of harmonic curvature tensors related to constant scalar and mean curvatures and prove that the Reeb vector field either belongs to the distribution or . In Sections 5 Real hypersurfaces in, 6 Real hypersurfaces in we will give a complete proof of the Main Theorem according to the Reeb vector field satisfying either or respectively.
Section snippets
Complex hyperbolic two-plane rassmannian
In this section we summarize basic material about complex hyperbolic two-plane Grassmannian manifold , for details we refer to [1], [4], [7], [10], [11], [13], [17], [19], and [20].
The Riemannian symmetric space , which consists of all complex two-dimensional linear subspaces in indefinite complex Euclidean space , becomes a connected, simply connected, irreducible Riemannian symmetric space of noncompact type and with rank two. Let and , and
Real hypersurfaces in complex hyperbolic two-plane rassmannian
Let be a real hypersurface in , that is, a hypersurface in with real codimension one. The induced Riemannian metric on will also be denoted by , and denotes the Levi Civita covariant derivative of . We denote by and the maximal complex and quaternionic subbundle of the tangent bundle of , respectively. Now let us put for any tangent vector field of a real hypersurface in , where denotes the
Proof of the main theorem
Let be a real hypersurfaces in with harmonic curvature, that is, for any vector fields . This condition comes from the 2nd Bianchi identity and , where denotes an adjoint coderivative of the exterior derivative on (see Besse [6]). Then the harmonic curvature of in satisfies
Contraction of codazzi type
In this section, we contract (3.1) with . Then it follows that
Real hypersurfaces in with geodesic eeb flow satisfying
Let us put . Then grad . So this means that the Reeb function is constant on .
In this section we could prove an important part of our main theorem in the introduction for . Then for the case in (3.5) it gives the following Now let us take a symmetric part from (5.1). Then we have
Real hypersurfaces in with geodesic eeb flow satisfying
In this section we consider a Hopf real hypersurface in with harmonic curvature for .
The Reeb vector is said to be a Hopf vector if it is a principal vector for the shape operator of in , that is, the Reeb vector is invariant under the shape operator .
On the other hand, first Suh [19] has shown that the Reeb vector of belongs to the distribution if is locally congruent to a hypersurface of type or a horosphere of type with in
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This work was supported by grant Proj. No. NRF 2018-R1D1A1B-05040381 from National Research Foundation of Korea .