Seoung Dal Jung

The first eigenvalue of the transversal Dirac operator

Abstract On a foliated Riemannian manifold with a transverse spin structure, we give a lower bound for the square of the eigenvalues of the transversal Dirac operator. We prove, in the limiting case, that the foliation is a minimal, transversally Einsteinian with constant transversal scalar curvature.

Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation

Abstract On a foliated Riemannian manifold with a Kahler spin foliation, we give a lower bound for the square of the eigenvalues of the transversal Dirac operator. We prove, in the limiting case, that the foliation is a minimal, transversally Einsteinian of odd complex dimension with nonnegative constant transversal scalar curvature.

Structure and characterization of ruled surfaces in Euclidean 3-space

In this paper, using the elementary method we study ruled surfaces, the simplest foliated submanifolds, in Euclidean 3-space. We define structure functions of the ruled surfaces, the invariants of non-developable ruled surfaces and discuss geometric properties and kinematical characterizations of non-developable ruled surface in Euclidean 3-space.

Structures and properties of null scroll in Minkowski 3-space

In this paper, we study structures and properties of Null scrolls. We define the (relative) invariants for Null scrolls by using a kind of standard equation. Using these (relative) invariants of Null scrolls, we give some new characterizations and classifications of Null scrolls and B-scrolls.

Eigenvalues of Type r of the Basic Dirac Operator on Kahler Foliations

Abstract. In this paper, we prove that on a Kahler spin foliatoin of codimension q = 2n,any eigenvalue λ of type r (r ∈ {1,··· ,[ n+12 ]}) of the basic Dirac operator D b satisfies theinequality λ 2 ≥ r4r−2 inf M σ ∇ , where σ ∇ is the transversal scalar curvature of F. 1. IntroductionOn a K¨ahler spin foliation (M,F) of codimension q = 2n, any eigenvalue λ ofthe basic Dirac operator D b satisfies(1.1) λ 2 ≥n+14ninf M K σ if n is odd [6,7],n4(n−1)inf M K σ if n is even [4],where K σ = σ ∇ + |κ| 2 with the transversal scalar curvature σ ∇ and the meancurvature form κ of F. In the limiting cases, F is minimal. For the point foliation,see [9,10]. Since the limiting cases of (1.1) are minimal, the inequalities (1.1) yieldthe following:(1.2) λ 2 ≥n+14ninf M σ ∇ if n is odd,n4(n−1)inf M σ ∇ if n is even.In this paper, we give an estimate of the eigenvalues λ of type r of the basic Diracoperator D b on a K¨ahler spin foliation. Recently, G. Habib and K. Richardson [5]proved that the spectrum of the basic Dirac operator does not change with respectto a change of bundle-like metric. And the existence of a bundle-like metric suchthat δ