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Topology and Geometry of Intersections of Ellipsoids in R^n

  • Book
  • © 2023

Overview

Authors:
  1. Santiago López de Medrano

    1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

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  • Unified account of the known results on the topology and geometry of intersections of concentric ellipsoids
  • Discusses connections with dynamical systems and complex geometry, among other applications
  • Features an account of the origins of the theory, written by one of its main contributors

Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 361)

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Table of contents (19 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Introduction

    • Santiago López de Medrano
    Pages 1-14

  3. General Intersecions of Quadrics

    1. Front Matter

      Pages 15-15
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    2. General Intersecions of Quadrics

      • Santiago López de Medrano
      Pages 17-37

    3. General Operations on Intersections of Quadrics

      • Santiago López de Medrano
      Pages 39-55

    4. Intersections of Coaxial Quadrics

      • Santiago López de Medrano
      Pages 57-89

    5. Intersections of Coaxial Ellipsoids

      • Santiago López de Medrano
      Pages 91-135

  4. Topological Description of Transverse Intersections of Concentric Ellipsoids

    1. Front Matter

      Pages 137-137
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    2. Characterization of Connected Sums

      • Santiago López de Medrano
      Pages 139-141

    3. Three Coaxial Ellipsoids

      • Santiago López de Medrano
      Pages 143-156

    4. Three Concentric Ellipsoids

      • Santiago López de Medrano
      Pages 157-171

    5. More Than Three Coaxial Ellipsoids

      • Santiago López de Medrano
      Pages 173-183

    6. A Family of Surfaces That Are Intersections of Concentric, Non-Coaxial, Ellipsoids

      • Santiago López de Medrano
      Pages 185-191

  5. Relations With Other Areas of Mathematics

    1. Front Matter

      Pages 193-193
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    2. Dynamical Systems

      • Santiago López de Medrano
      Pages 195-203

    3. Complex Geometry

      • Santiago López de Medrano
      Pages 205-210

    4. Contact and Symplectic Geometry

      • Santiago López de Medrano
      Pages 211-213

    5. Intersections with Dihedral Symmetry

      • Santiago López de Medrano
      Pages 215-230

    6. Polyhedral Products

      • Santiago López de Medrano
      Pages 231-232

  6. Appendices

    1. Front Matter

      Pages 233-233
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Keywords

About this book

This book gives an overview of research in the topology and geometry of intersections of quadrics in $\mathbb{R}^n$, with a focus on intersections of concentric ellipsoids and related spaces. Unifying and organizing material previously spread over many articles, it also contains new results.

The first part provides very detailed foundations of a wide-ranging theory that could be useful for future developments. It includes chapters on general intersections of quadrics, operations on them, and intersections of concentric and coaxial quadrics. Moving from the general to the specific, the second part focuses on a topological description of transverse intersections of concentric ellipsoids, including a complete description of the case of three ellipsoids, and of some large families of more than three of them. The third part looks at relations to other areas of mathematics such as dynamical systems, complex geometry, contact and symplectic geometry, andother applications. An appendix gathers some technical items and also gives an account of the origins, motivations and progression of the subject, including historical recollections of the author, who has been central to its development.

Authors and Affiliations

  • Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Santiago López de Medrano

About the author

​Santiago López de Medrano is a Research Professor in Mathematics at the Universidad Nacional Autónoma de México (UNAM). After completing his PhD at Princeton University in 1968, his thesis was published as the highly influential book Involutions on Manifolds (Springer-Verlag, 1971). After Princeton, he returned to UNAM, where he has been ever since. He was President of the Mexican Mathematical Society (1969–1973) and has been an invited researcher and speaker at numerous international institutes and conferences. His research is primarily in differential topology, singularity theory, dynamical systems, mathematical biology and their interaction. Recently his focus has been on the topology of intersections of ellipsoids in $\mathbb{R}^n$, and its applications to dynamical systems and geometry. He is the author of over 60 publications. In addition to mathematical research, he has also been interested in the improvement of the teaching of mathematics from high schoolto graduate university level.

Bibliographic Information

  • Book Title: Topology and Geometry of Intersections of Ellipsoids in R^n

  • Authors: Santiago López de Medrano

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-3-031-28364-2

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2023

  • Hardcover ISBN: 978-3-031-28363-5Published: 25 May 2023

  • Softcover ISBN: 978-3-031-28366-6Due: 25 June 2023

  • eBook ISBN: 978-3-031-28364-2Published: 24 May 2023

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: XIII, 282

  • Number of Illustrations: 4 b/w illustrations, 68 illustrations in colour

  • Topics: Topology, Geometry

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