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On the Conley-Zehnder index and Sasaki-Einstein manifolds

DC Field Value Language
dc.contributor.advisorKoert, Otto van-
dc.contributor.author홍석민-
dc.date.accessioned2019-05-07T07:00:45Z-
dc.date.available2019-05-07T07:00:45Z-
dc.date.issued2019-02-
dc.identifier.other000000154119-
dc.identifier.urihttps://hdl.handle.net/10371/152904-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2019. 2. Koert, Otto van .-
dc.description.abstract제 2장에서는 저자가 서울대학교 수리과학부에서 학위를 하는 동안 출판한 논문에 대해 소개하였습니다. 구체적으로 Reeb 벡터장을 사교공간상의 경로로 간주하고 그 Conley-Zehnder 지표와 몫공간으로서 생성된 기저 공간의 orbifold 천(Chern) 특성류 사이의 관계를 규명하였습니다. 이렇게 얻어진 관계를 우리에게 매우 익숙한 기본적인 예제들에 적용시켜 구체적인 값을 구하였습니다.



제 3장은 저자가 학위기간 동안 주로 연구한 분야인 사사키-아인슈타인 기하(Sasaki-Einstein geometry)에 대한 조사 보고서입니다. 기본적인 정의, 정리부터 흥미로운 예제, 존재성에 대한 걸림돌 이론(obstruction theory)등에 대해서 살펴보았습니다.		-
dc.description.abstractIn the second chapter, we prove a useful relation between the Conley-Zehnder indices of the Reeb vector flow action along periodic orbits in prequantization bundles and the orbifold Chern class of the base symplectic orbifolds motivated by the well-known case of manifolds. We also apply this method to primary examples.



In the third chapter, we survey interesting properties on Sasaki-Einstein geometry from the elementary definitions and theorems to well-known examples and simple obstructions.		-
dc.description.tableofcontentsAbstract i

1 Introduction 1

2 The Conley-Zehnder indices of the Reeb flow action along S1-fibers over certain orbifolds 4

2.1 The Conley-Zehnder index . . . . . . . . . . . . . . . . . . . . 4

2.1.1 The Maslov index . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 The Conley-Zehnder index . . . . . . . . . . . . . . . . 6

2.1.3 The Robbin-Salamon index . . . . . . . . . . . . . . . 7

2.2 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Classifying spaces . . . . . . . . . . . . . . . . . . . . . 12

2.3 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 The Boothby-Wang fibration . . . . . . . . . . . . . . . 15

2.3.2 The main theorem . . . . . . . . . . . . . . . . . . . . 16

2.3.3 The weighted projective spaces and their complete intersections

. . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.4 Some computations for non-principal orbits . . . . . . 30

2.3.5 Inertia orbifolds . . . . . . . . . . . . . . . . . . . . . . 32

3 A survey on Sasaki-Einstein manifolds 35

3.1 Sasakian structures and Einstein metrics . . . . . . . . . . . . 35

3.1.1 Symplectic manifolds and contact structures . . . . . . 35

3.1.2 Almost contact structures and Sasakian structures . . . 40

3.1.3 General relativity, Einstein manifolds . . . . . . . . . . 45

3.2 Kahler-Einstein metrics . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 Einstein conditions in Kahler metrics . . . . . . . . . . 51

3.2.2 Calabi conjecture and Calabi-Yau manifolds . . . . . . 54

3.2.3 Kahler-Einstein metrics on del Pezzo surfaces . . . . . 57

3.3 Sasaki-Einstein manifolds . . . . . . . . . . . . . . . . . . . . 62

3.3.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . 62

3.3.2 Toric Sasaki-Einstein manifolds . . . . . . . . . . . . . 66

3.3.3 Sasaki-Einstein metrics on Y p		-
dc.description.tableofcontentsq . . . . . . . . . . . . . 75

3.3.4 Simple obstructions . . . . . . . . . . . . . . . . . . . . 80

Abstract (in Korean) 88		-
dc.language.isoeng-
dc.publisher서울대학교 대학원-
dc.subject.ddc510-
dc.titleOn the Conley-Zehnder index and Sasaki-Einstein manifolds-
dc.typeThesis-
dc.typeDissertation-
dc.description.degreeDoctor-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2019-02-
dc.identifier.uciI804:11032-000000154119-
dc.identifier.holdings000000000026▲000000000039▲000000154119▲-
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