title: An isometric embedding of almost Kähler reduced spaces and contact metric reduced spaces in orbit spaces
Creators: name: Kohli, Ravjot Singh
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2024
Subjects: subject: 510 Mathematics

description: Gegeben eine isometrische und Hamiltonian-Wirkung einer kompakten Lie-Gruppe auf einer almost Kähler-Mannigfaltigkeit, ist der reduzierte Raum im Allgemeinen ein stratifizierter Raum. Entsprechend ergibt sich bei einer isometrischen und contact-Wirkung einer kompakten Lie-Gruppe auf einer contact metric-Mannigfaltigkeit ein contact-reduzierter Raum, der ebenfalls stratifiziert ist. Diese Arbeit untersucht die schichtweise almost Kähler- und contact metric-Strukturen auf dem reduzierten Raum und beweist, dass die natürliche Einbettung des reduzierten Raums in den Orbitsraum schichtweise isometrisch ist. Die auf jeder Schicht des Orbitsraums betrachtete Metrik ist die natürliche Quotientenmetrik, die durch Riemannsche Submersionen induziert wird.
description: Given an isometric and Hamiltonian action of a compact Lie group on an almost Kähler manifold, the reduced space is, in general, a stratified space. Similarly, given an isometric and contact action of a compact Lie group on a contact metric manifold, the contact reduced space is a stratified space. This thesis investigates the stratum-wise almost Kähler and contact metric structures on the reduced space and proves that the natural embedding of the reduced space in the orbit space is stratum-wise isometric. The metric considered on each stratum of the orbit space is the natural quotient metric induced via Riemannian submersions.
description: 1 Introduction ..... 5
2 Stratified Spaces ..... 17
2.1 Smooth Structures on Stratified Spaces ..... 23
2.2 The Whitney Conditions ..... 25
2.3 Topological Local Triviality ..... 27
3 Group Actions and Orbit Spaces ..... 29
3.1 Lie Groups and Proper Actions ..... 29
3.2 Orbit Type Stratification ..... 32
3.3 The Smooth Structure on Orbit Spaces ..... 44
3.4 The Whitney Stratification of Orbit Spaces ..... 46
3.5 Topological Local Triviality of Orbit Spaces ..... 47
4 Riemannian Manifolds and Orbit Spaces ..... 49
4.1 Riemannian Submersions ..... 49
4.2 Riemannian Submersions and Distance Functions ..... 52
4.3 Orbit Spaces and Riemannian Metrics ..... 55
4.4 Orbit Spaces for Linear Group Actions ..... 58
5 Symplectic Manifolds and Reduction ..... 65
5.1 Symplectic Reduction ..... 68
5.2 Orbit Type Stratification of Reduced Spaces ..... 75
5.3 The Smooth Structure and Whitney Stratification of Reduced
Spaces ..... 87
5.4 Topological Local Triviality of Reduced Spaces ..... 90
6 Almost Kähler Manifolds and Reduction ..... 93
6.1 Hermitian Vector Spaces ..... 93
6.2 Almost Kähler Manifolds ..... 97
6.3 Almost Kähler Reduction ..... 98
7 Contact Manifolds and Reduction ..... 103
7.1 Contact Reduction ..... 106
7.2 Orbit Type Stratification of Contact Reduced Spaces ..... 115
7.3 The Smooth Structure and Whitney Stratification of Contact
Reduced Spaces ..... 120
7.4 Topological Local Triviality of Contact Reduced Spaces ..... 123
8 Contact Metric Manifolds and Reduction ..... 125
8.1 Contact Metric Manifolds ..... 125
8.2 Contact Metric Reduction ..... 130
8.3 Reduction of Hermitian Vector Spaces ..... 133
9 Local Quasi-Isometry of Stratum-wise Riemannian Metric ..... 139
9.1 Quasi-Isometries ..... 139
9.2 An Ideal Metric on Slice Bundles ..... 141
9.3 Local Quasi-Isometry on Orbit Spaces ..... 145
9.4 Local Quasi-Isometry on Almost Kähler Reduced Spaces ..... 149
10 Outlook: An Application to L2 Cohomology ..... 153
10.1 Isomorphism of L2 and Intersection Cohomology ..... 153
10.2 An Application to Almost Kähler Reduced Spaces ..... 156