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Metrics of Positive Scalar Curvature and Generalised Morse Functions, Part I.

By: Material type: TextTextSeries: Memoirs of the American Mathematical SocietyPublisher: Providence : American Mathematical Society, 2011Copyright date: ©2010Edition: 1st edDescription: 1 online resource (105 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470405977
Subject(s): Genre/Form: Additional physical formats: Print version:: Metrics of Positive Scalar Curvature and Generalised Morse Functions, Part IDDC classification:
  • 516.3/62
LOC classification:
  • QA645 -- .W35 2010eb
Online resources:
Contents:
Intro -- Contents -- Abstract -- Introduction -- 0.1. Background -- 0.2. Main results -- 0.3. The connection with generalised Morse functions and Part II -- 0.4. Acknowledgements -- Chapter 1. Definitions and Preliminary Results -- 1.1. Isotopy and concordance in the space of metrics of positive scalar curvature -- 1.2. Warped product metrics on the sphere -- 1.3. Torpedo metrics on the disk -- 1.4. Doubly warped products and mixed torpedo metrics -- 1.5. Inducing a mixed torpedo metric with an embedding -- Chapter 2. Revisiting the Surgery Theorem -- 2.1. Surgery and cobordism -- 2.2. Surgery and positive scalar curvature -- 2.3. Outline of the proof of Theorem 2.3 -- 2.4. Part 1 of the proof: Curvature formulae for the first deformation -- 2.5. Part 2 of the proof: A continuous bending argument -- 2.6. Part 3 of the proof: Isotoping to a standard product -- 2.7. Applying Theorem 2.3 over a compact family of psc-metrics -- 2.8. The proof of Theorem 2.2 (The Improved Surgery Theorem) -- Chapter 3. Constructing Gromov-Lawson Cobordisms -- 3.1. Morse Theory and admissible Morse functions -- 3.2. A reverse Gromov-Lawson cobordism -- 3.3. Continuous families of Morse functions -- Chapter 4. Constructing Gromov-Lawson Concordances -- 4.1. Applying the Gromov-Lawson technique over a pair of cancelling surgeries -- 4.2. Cancelling Morse critical points: The Weak and Strong Cancellation Theorems -- 4.3. A strengthening of Theorem 4.2 -- 4.4. Standardising the embedding of the second surgery sphere -- Chapter 5. Gromov-Lawson Concordance Implies Isotopy for Cancelling Surgeries -- 5.1. Connected sums of psc-metrics -- 5.2. An analysis of the metric g'', obtained from the second surgery -- 5.3. The proof of Theorem 5.1 -- Chapter 6. Gromov-Lawson Concordance Implies Isotopy in the General Case -- 6.1. A weaker version of Theorem 0.8.
6.2. The proof of the main theorem -- Appendix: Curvature Calculations from the Surgery Theorem -- Bibliography.
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Intro -- Contents -- Abstract -- Introduction -- 0.1. Background -- 0.2. Main results -- 0.3. The connection with generalised Morse functions and Part II -- 0.4. Acknowledgements -- Chapter 1. Definitions and Preliminary Results -- 1.1. Isotopy and concordance in the space of metrics of positive scalar curvature -- 1.2. Warped product metrics on the sphere -- 1.3. Torpedo metrics on the disk -- 1.4. Doubly warped products and mixed torpedo metrics -- 1.5. Inducing a mixed torpedo metric with an embedding -- Chapter 2. Revisiting the Surgery Theorem -- 2.1. Surgery and cobordism -- 2.2. Surgery and positive scalar curvature -- 2.3. Outline of the proof of Theorem 2.3 -- 2.4. Part 1 of the proof: Curvature formulae for the first deformation -- 2.5. Part 2 of the proof: A continuous bending argument -- 2.6. Part 3 of the proof: Isotoping to a standard product -- 2.7. Applying Theorem 2.3 over a compact family of psc-metrics -- 2.8. The proof of Theorem 2.2 (The Improved Surgery Theorem) -- Chapter 3. Constructing Gromov-Lawson Cobordisms -- 3.1. Morse Theory and admissible Morse functions -- 3.2. A reverse Gromov-Lawson cobordism -- 3.3. Continuous families of Morse functions -- Chapter 4. Constructing Gromov-Lawson Concordances -- 4.1. Applying the Gromov-Lawson technique over a pair of cancelling surgeries -- 4.2. Cancelling Morse critical points: The Weak and Strong Cancellation Theorems -- 4.3. A strengthening of Theorem 4.2 -- 4.4. Standardising the embedding of the second surgery sphere -- Chapter 5. Gromov-Lawson Concordance Implies Isotopy for Cancelling Surgeries -- 5.1. Connected sums of psc-metrics -- 5.2. An analysis of the metric g'', obtained from the second surgery -- 5.3. The proof of Theorem 5.1 -- Chapter 6. Gromov-Lawson Concordance Implies Isotopy in the General Case -- 6.1. A weaker version of Theorem 0.8.

6.2. The proof of the main theorem -- Appendix: Curvature Calculations from the Surgery Theorem -- Bibliography.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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