The Volume of Vector Fields on Riemannian Manifolds: Main Results and Open Problems

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Olga Gil-Medrano

The Volume of Vector Fields on Riemannian Manifolds

Main Results and Open Problems

Olga Gil-Medrano

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English

Springer International Publishing AG
01 August 2023

Calculus & mathematical analysis; Differential & Riemannian geometry

Series: Lecture Notes in Mathematics

This book focuses on the study of the volume of vector fields on Riemannian manifolds. Providing a thorough overview of research on vector fields defining minimal submanifolds, and on the existence and characterization of volume minimizers, it includes proofs of the most significant results obtained since the subject’s introduction in 1986. Aiming to inspire further research, it also highlights a selection of intriguing open problems, and exhibits some previously unpublished results. The presentation is direct and deviates substantially from the usual approaches found in the literature, requiring a significant revision of definitions, statements, and proofs.

A wide range of topics is covered, including: a discussion on the conditions for a vector field on a Riemannian manifold to determine a minimal submanifold within its tangent bundle with the Sasaki metric; numerous examples of minimal vector fields (including those of constant length on punctured spheres); athorough analysis of Hopf vector fields on odd-dimensional spheres and their quotients; and a description of volume-minimizing vector fields of constant length on spherical space forms of dimension three.

Each chapter concludes with an up-to-date survey which offers supplementary information and provides valuable insights into the material, enhancing the reader's understanding of the subject. Requiring a solid understanding of the fundamental concepts of Riemannian geometry, the book will be useful for researchers and PhD students with an interest in geometric analysis.

By: Olga Gil-Medrano
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 1st ed. 2023
Volume: 2336
Dimensions: Height: 235mm, Width: 155mm,
Weight: 221g
ISBN: 9783031368561
ISBN 10: 3031368568
Series: Lecture Notes in Mathematics
Pages: 126
Publication Date: 01 August 2023
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

- 1. Introduction. - 2. Minimal Sections of Tensor Bundles. - 3. Minimal Vector Fields of Constant Length on the Odd-Dimensional Spheres. - 4. Vector Fields of Constant Length of Minimum Volume on the Odd-Dimensional Spherical Space Forms. - 5. Vector Fields of Constant Length on Punctured Spheres.

Olga Gil-Medrano (1956, Spain) is a retired Full Professor at the University of Valencia, Spain. A leading specialist in the study of vector field volumes on Riemannian manifolds, her research also includes other topics of geometric analysis, such as the geometrical theory of foliations, the Yamabe problem, the geometry of spaces of metrics and other sections of tensor bundles, and variational problems on these spaces. She received the Medal of the Royal Spanish Mathematical Society (RSME) in 2021.

Reviews for The Volume of Vector Fields on Riemannian Manifolds: Main Results and Open Problems

“The monograph under review provides a quite complete survey of up-to-date results on the minimality of vector fields on odd-dimensional spheres and more general Riemannian manifolds. Besides, it can be recommended as a self-consistent textbook introducing the reader to modern techniques for studying the extrinsic geometry of vector fields. the monograph includes a lot of intriguing open questions and conjectures which may serve as a novel source of inspiration for further researches … .” (Vasyl Gorkaviy, zbMATH 1536.53003, 2024)