Showing papers by "Young Jin Suh published in 2005"

Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ξ-parallel

Abstract: We classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Lie parallel in the direction of the structure vector field.

Compact hypersurfaces in a unit sphere

Abstract: We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

Kinematic formulas for integral invariants of degree two in real space forms

Abstract: In this paper we investigate the kinematic formulas for submanifolds in real space forms, and we give explicit expressions of the kinematic formulas for integral invariants defined by invariant homogeneous polynomials of degree two on second fundamental forms in the sense of Howard, completely. They are actually certain extensions of the Chern-Federer kinematic formula and of the one by Chen.

Real hypersurfaces in complex two-plane Grassmannians related to the Ricci curvature

Abstract: In this paper we introduce a new notion of the Ricci tensor derived from the curvature tensor of real hypersurfaces in complex two-plane Grassmannians G2(C). Moreover, we give a characterization of real hypersurfaces of type A in G2(C), that is, a tube over a totally geodesic G2(C) in G2(C) in terms of integral formulas related to the Ricci curvature Ric(ξ, ξ) along the direction of the structure vector field ξ for real hypersurfaces in G2(C). 0. Introduction In the geometry of real hypersurfaces in complex space forms or in quaternionic space forms there have been many characterizations of model hypersurfaces of type A1, A2, B,C, D and E in complex projective space CP , of type A0, A1, A2 and B in complex hyperbolic space CH or A1, A2, B in quaternionic projective space QP, which are completely classified by Cecil and Ryan [4], Kimura [5], Berndt [1], Martinez and Pérez [6] respectively. Among them there were some characterizations of homogeneous real hypersurfaces of type A1, A2 in complex projective space CP and of type A0, A1, A2 in complex hyperbolic space CH . As an example, we say 2000 Mathematics Subject Classification. Primary 53C40; Secondary 53C15. The first author was supported by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2005. 12 Young Jin Suh and Yoshiyuki Watanabe that the shape operator A and the structure tensor φ commute with each other, that is Aφ − φA = 0, is a model characterization of hypersurfaces, which are tubes over a totally geodesic CP k in CP (See Okumura [8]), a tube over a totally geodesic CH in CH or a horosphere in CH (See Montiel and Romero [7]). Now let us denote by G2(C) the set of all two-dimensional linear subspaces in C. This Riemannian symmetric space G2(C) has a remarkable geometrical 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, G2(C) is the unique compact, irreducible, Kähler, quaternionic Kähler manifold which is not a hyperkähler manifold. So, in G2(C) we have the two natural geometrical conditions for real hypersurfaces M that [ξ] = Span {ξ} or D⊥ = Span {ξ1, ξ2, ξ3}, which are spanned by almost contact 3-structure vector fileds {ξ1, ξ2, ξ3} such that TxM = D⊕D⊥, are invariant under the shape operator A of M (See [2] and [3]). The almost contact structure vector field ξ mentioned above is defined by ξ = −JN , where N denotes a local unit normal vector field of M in G2(C) and the almost contact 3-structure vector fields {ξ1, ξ2, ξ3} are defined by ξν = −JνN , ν = 1, 2, 3, where {Jν} denotes a canonical local basis of a quaternionic Kähler structure J. The first result in this direction is the classification of real hypersurfaces in G2(C) satisfying both conditions. Namely, Berndt and the first author [2] have proved the following Theorem A. Let M be a connected real hypersurface in G2(C), m ≥ 3. Then both [ξ] and D⊥ are invariant under the shape operator of M if and only if (A) M is an open part of a tube around a totally geodesic G2(C) in G2(C), or (B) m is even, say m = 2n, and M is an open part of a tube around a totally geodesic QP in G2(C). In the paper [3] due to Berndt and the first author we have given a characterization of real hypersurfaces of type (A) in Theorem A when the shape operator A of M in G2(C) commutes with the structure tensor Real hypersurfaces with the Ricci curvature 13 φ. This is equivalent to the condition that the Reeb flow on M is isometric, that is Lξg = 0, where L(resp. g) denotes the Lie derivative(resp. the induced Riemannian metric) of M in the direction of the Reeb vector field ξ as follows: Theorem B. Let M be a connected orientable real hypersurface in G2(C), m ≥ 3. Then the Reeb flow on M is isometric if and only if M is an open part of a tube around some totally geodesic G2(C) in G2(C). Now the purpose of this paper is to show non-existence properties related to the Ricci curvature along the direction of the structure vector ξ of a compact real hypersurface in G2(C). In order to do this we recall some integral formulas due to Watanabe [14] (See also Yano [15]) on a compact Riemannian manifold and give some relations between the Ricci curvature and the covariant derivative for the structure vector field ξ of a real hypersurface in G2(C) as follows: ∫ M {Ric(ξ, ξ) + ‖∇ξ‖2} ∗ 1 = 0 and ∫ M { Ric(ξ, ξ) + 1 2 ‖Lξg‖ − ‖∇ξ‖2 − (div ξ) } ∗ 1 = 0. By virtue of these formulas we are able to assert the following theorems respectively: Theorem 1. There does not exist any compact real hypersurface in G2(C), m≥3, satisfying Ric (ξ, ξ)≥0 and TrA2≤4 ∑3 ν=1 ην(ξ) + 2‖Aξ‖2 − TrA g(Aξ, ξ)− 4(m + 1). Theorem 2. There does not exist any compact real hypersurface in G2(C), m≥3, satisfying Ric(ξ, ξ)≤0 and TrA2≤4(m + 1)− 4 ∑3 ν=1 ην(ξ) + TrA g(Aξ, ξ). In this paper we also give a characterization of real hypersurfces of type A in G2(C) by the second integral formula mentioned above. Then if we use the expression of the shape operator A of a compact real hypersurface M in G2(C), we assert the following: 14 Young Jin Suh and Yoshiyuki Watanabe Theorem 3. Let M be a compact real hypersurface in G2(C), m≥3. If it satisfies ∫