Hypersurfaces in the nearly Kähler S6 and S3 × S3 = 유사 켈러다양체 S6와 S3×S3의 초곡면

다국어 초록 (Multilingual Abstract)

In this thesis, we study hypersurfaces in the nearly Kähler S6 and S3 × S3. First, we show that neither the nearly Kähler S6 nor S3 × S3 have a hypersurface that satisfies the condition of contact hypersurface, which is the notion defined in the Kähler manifold. Moreover, we prove that if M is a hypersurface of the nearly Kähler S6, then M satisfies the condition Aϕ+ϕA = 2kϕ if and only if M is locally congruent to a geodesic sphere, and, if M is a hypersurface of the nearly Kähler S3 × S3, there exist no hypersurface satisfying the condition Aϕ+ ϕA = 2kϕ. We characterize Hopf hypersurfaces in the nearly Kähler S6 with 1, 2, or 3 distinct principal curvatures using some geometric operators in the nearly Kähler S6 and its hypersurfaces.

국문 초록 (Abstract)

이 논문에서 우리는 유사 켈러다양체 S6와 S3×S3의 초곡면에 대해 연구한다. 먼저,
우리는 유사 켈러다양체 S6와 S3×S3가 켈러다양체에서 정의된 접촉초곡면의 조건을
만족하는초곡면을갖지않음을보인다.또한S6의초곡면M이조건Aϕ+ϕA=2kϕ을
만족할 필요충분조건은 M이국소적으로측지구와동형이나S3×S3의 초곡면중에는
조건 Aϕ+ϕA =2kϕ을 만족하는 초곡면이 존재하지 않음을 증명한다. 그리고 S6와 그
초곡면에서정의된기하적연산자들을이용하여1개,2개또는3개의서로다른주곡률을
갖는S6의Hopf 초곡면을특성화한다.

목차 (Table of Contents)

• 1. Introduction 1
• 2. Preliminaries 5
• 2.1. Vector cross product in R7 5
• 2.2. Riemannian manifolds 6
• 2.3. Nearly Kahler manifolds 8
• 2.4. The homogeneous NK structure on S6 9
• 2.5. The homogeneous NK structure on S3 S3 10
• 2.6. Hypersurfaces of nearly Kahler manifolds 12
• 2.7. Hypersurfaces of NK S6 and NK S3 S3 15
• 3. Nearly Kahler manifolds and their hypersurfaces 19
• 3.1. Proof of Theorem 3.3 20
• 3.2. Proof of Theorem 3.4 21
• 3.3. Proof of Theorem 3.5 22
• 4. Characterizations of Hopf hypersurfaces in S6 37
• References 47
• Abstract(Korean) 50