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This Open Access textbook presents Carnot-Carathéodory spaces from the perspective of Lie groups. Its main objective is to illustrate how these non-smooth geometries manifest in various mathematical domains, including metric geometry and geometric group theory. In contrast to other sources, this book utilizes the formalism of Lie groups to showcase how this theory facilitates the development of geometry and analysis on the non-smooth structure of Carnot-Carathéodory spaces. Major results are presented with rigorous mathematical proofs, and references for further exploration are provided. Open problems in these areas are discussed, offering insights into recent developments and avenues for future research.
Prerequisite topics such as differential geometry, measure theory, and group theory are incorporated in the main flow of the chapters, ensuring a comprehensive understanding. Junior researchers seeking an introduction to the field of sub-Riemannian geometry will find this an invaluable introductory companion. The book is also suitable for those entering research subjects on the interplay between geometry, analysis, and group theory.
List of contents
- 1. Introduction.- 2. The Main Example: The Heisenberg Group.- 3. A Review of Metric and Differential Geometry.- 4. General Theory of Carnot-Carathéodory Spaces.- 5. A Review of Lie Groups.- 6. Metric Groups and Homogeneous Spaces.- 7. Sub-Finsler Lie Groups.- 8. Riemannian Lie Groups.- 9. Nilpotent Lie Groups.- 10. Metrics on Nilpotent Groups.- 11. Carnot Groups.- 12. Limits of CC Spaces.- 13. Rank-One Symmetric Spaces.- 14. Heintze Groups and their Visual Boundaries.
About the author
Enrico Le Donne is a mathematician and researcher who has made significant contributions to the study of Carnot groups, sub-Riemannian geometry, and other non-smooth geometries. His research focuses on the regularity and classification of distinguished sets and maps within Carnot groups, as well as the development of tools and techniques for analyzing their geometric and analytic properties. Le Donne received his first degree at Scuola Normale Superiore in Pisa (Italy) and his PhD from Yale University (USA). Following research stays in Zurich, Berkeley, and Paris, he obtained his first permanent position at the University of Jyväskylä in Finland, just three years after graduating. He later served as a professor at the University of Pisa and in 2020, joined the Department of Mathematics at the University of Fribourg in Switzerland. Throughout his career, Le Donne has fostered extensive international collaborations with researchers from America, Australia, Asia, and various European locations. His contributions have been recognized through notable achievements such as becoming a research fellow of the Academy of Finland in 2015 and receiving a prestigious "ERC Starting Grant" from the European Research Council in 2017, amounting to 1.25 million Euros. Presently, he is funded by the Swiss National Science Foundation for his project titled "Regularity of sub-Riemannian geodesics and applications."
Summary
This Open Access textbook presents Carnot-Carathéodory spaces from the perspective of Lie groups. Its main objective is to illustrate how these non-smooth geometries manifest in various mathematical domains, including metric geometry and geometric group theory. In contrast to other sources, this book utilizes the formalism of Lie groups to showcase how this theory facilitates the development of geometry and analysis on the non-smooth structure of Carnot-Carathéodory spaces. Major results are presented with rigorous mathematical proofs, and references for further exploration are provided. Open problems in these areas are discussed, offering insights into recent developments and avenues for future research.
Prerequisite topics such as differential geometry, measure theory, and group theory are incorporated in the main flow of the chapters, ensuring a comprehensive understanding. Junior researchers seeking an introduction to the field of sub-Riemannian geometry will find this an invaluable introductory companion. The book is also suitable for those entering research subjects on the interplay between geometry, analysis, and group theory.
