Jong Taek Cho

On slant curves in Sasakian 3-manifolds

A classical theorem by Lancret says that a curve in Euclidean 3-space is of constant slope if and only if its ratio of curvature and torsion is constant. In this paper we study Lancret type problems for curves in Sasakian 3-manifolds.

PSEUDO-SYMMETRIC CONTACT 3-MANIFOLDS

Contact Homogeneous 3-manifolds are pseudo-symmetric spaces of constant type. All Sasakian 3-manifolds are pseudo-symmetric spaces of constant type.

SLANT CURVES IN CONTACT PSEUDO-HERMITIAN 3-MANIFOLDS

By using the pseudo-Hermitian connection (or Tanaka‐Webster connection) b, we construct the parametric equations of Legendre pseudo-Hermitian circles (whose b-geodesic curvatureb is constant andb r-geodesic torsionb is zero) in S 3 . In fact, it is realized as a Legendre curve satisfying theb-Jacobi equation for theb-geodesic vector field along it.

Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Structure Jacobi Operator

Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type (A) in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.

Parabolic Geodesics in Sasakian

We give explicit parametrizations for all parabolic geodesics in 3-dimensional Sasakian space forms. Department of Mathematics, Chonnam National University, Gwangju, 500–757, Korea e-mail: jtcho@chonnam.ac.kr Department of Mathematical Sciences, Faculty of Science, Yamagata University, Yamagata 990-8560, Japan e-mail: inoguchi@sci.kj.yamagata-u.ac.jp Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea e-mail: jieunlee12@gmail.com Received by the editors August 18, 2008. Published electronically March 10, 2011. The third author was partially supported by the National Research Foundation of the Korean Government (NRF-2009-351-C00008). AMS subject classification: 58E20.

3

Abstract Ricci soliton contact metric manifolds with certain nullity conditions have recently been studied by Ghosh and Sharma. Whereas the gradient case is well-understood, they provided a list of candidates for the nongradient case. These candidates can be realized as Lie groups, but one only knows the structures of the underlying Lie algebras, which are hard to be analyzed apart from the three-dimensional case. In this paper, we study these Lie groups with dimension greater than three, and prove that the connected, simply-connected, and complete ones can be realized as homogeneous real hypersurfaces in noncompact real two-plane Grassmannians. These realizations enable us to prove, in a Lie-theoretic way, that all of them are actually Ricci soliton.

-Manifolds

Every Riemannian symmetric space of noncompact type is isometric to some solvable Lie group equipped with a left-invariant Riemannian metric. The corresponding metric solvable Lie algebra is called the solvable model of the symmetric space. In this paper, we give explicit descriptions of the solvable models of noncompact real two-plane Grassmannians, and mention some applications to submanifold geometry, contact geometry, and geometry of left-invariant metrics.