The Volume of Vector Fields on Riemannian Manifolds: Main Results and Open Problems
Gil-Medrano, Olga
- 出版商: Springer
- 出版日期: 2023-08-01
- 售價: $2,460
- 貴賓價: 9.5 折 $2,337
- 語言: 英文
- 頁數: 126
- 裝訂: Quality Paper - also called trade paper
- ISBN: 3031368568
- ISBN-13: 9783031368561
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商品描述
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to inspire further research, it also highlights a selection of intriguing open problems, and exhibits some previously unpublished results.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.
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.
商品描述(中文翻譯)
本書專注於研究黎曼流形上的向量場體積。提供了對定義最小子流形的向量場以及體積最小化者的存在和特徵的研究的全面概述,包括自1986年引入該主題以來獲得的最重要結果的證明。為了激發進一步的研究,本書還突出了一些有趣的開放問題,並展示了一些以前未發表的結果。本書的呈現方式直接,與文獻中常見的方法有很大的偏差,需要對定義、陳述和證明進行重大修訂。
本書涵蓋了廣泛的主題,包括:討論黎曼流形上的向量場在其切空間並帶有Sasaki度量的情況下確定最小子流形的條件;多個最小向量場的例子(包括在穿孔球面上長度恆定的向量場);對奇數維球面及其商空間上的Hopf向量場進行詳細分析;以及描述三維球面空間形式上長度恆定的體積最小化向量場。
每章結束時都有一份最新的調查,提供補充信息並為讀者對材料的理解提供有價值的見解。本書需要對黎曼幾何的基本概念有扎實的理解,對於對幾何分析感興趣的研究人員和博士生非常有用。
作者簡介
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.
作者簡介(中文翻譯)
Olga Gil-Medrano(1956年,西班牙)是西班牙瓦倫西亞大學的退休全職教授。她是研究黎曼流形上向量場體積的領先專家,她的研究還包括幾何分析的其他主題,如葉狀結構的幾何理論、Yamabe問題、度量空間的幾何和張量丛的其他部分,以及這些空間上的變分問題。她於2021年獲得了西班牙皇家數學學會(RSME)的獎章。