Aspects of Differential Geometry IV

Calvino-Louzao, Esteban, Garcia-Rio, Eduardo, Gilkey, Peter

  • 出版商: Morgan & Claypool
  • 出版日期: 2019-04-18
  • 售價: $2,780
  • 貴賓價: 9.5$2,641
  • 語言: 英文
  • 頁數: 167
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 1681735636
  • ISBN-13: 9781681735634
  • 相關分類: 微積分 Calculus線性代數 Linear-algebra
  • 海外代購書籍(需單獨結帳)

商品描述

Book IV continues the discussion begun in the first three volumes.

Although it is aimed at first-year graduate students, it is also intended to serve as a basic reference for people working in affine differential geometry. It also should be accessible to undergraduates interested in affine differential geometry. We are primarily concerned with the study of affine surfaces which are locally homogeneous. We discuss affine gradient Ricci solitons, affine Killing vector fields, and geodesic completeness. Opozda has classified the affine surface geometries which are locally homogeneous; we follow her classification. Up to isomorphism, there are two simply connected Lie groups of dimension 2. The translation group ℝ is Abelian and the ���� + �� group is non-Abelian. The first chapter presents foundational material. The second chapter deals with Type �� surfaces. These are the left-invariant affine geometries on ℝ . Associating to each Type �� surface the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue �� = -1 turns out to be a very powerful technique and plays a central role in our study as it links an analytic invariant with the underlying geometry of the surface. The third chapter deals with Type �� surfaces; these are the left-invariant affine geometries on the ���� + �� group. These geometries form a very rich family which is only partially understood. The only remaining homogeneous geometry is that of the sphere �� . The fourth chapter presents relations between the geometry of an affine surface and the geometry of the cotangent bundle equipped with the neutral signature metric of the modified Riemannian extension.

商品描述(中文翻譯)

第四冊延續了前三冊的討論內容。雖然它主要針對一年級研究生,但也旨在為從事仿射微分幾何的人提供基本參考。同時,本書也適合對仿射微分幾何感興趣的本科生閱讀。我們主要關注的是局部均質的仿射曲面的研究。我們討論了仿射梯度Ricci擬準穩態、仿射Killing向量場和測地完備性。Opozda對局部均質的仿射曲面幾何進行了分類,我們遵循了她的分類。在同構的情況下,有兩個維度為2的單連通Lie群。平移群ℝ是阿貝爾群,而𝕊² + ℝ²群則是非阿貝爾群。第一章介紹了基礎材料。第二章討論了Type 𝕊²曲面。這些是ℝ上的左不變仿射幾何。將每個Type 𝕊²曲面與對應特徵值𝑙 = -1的拟爱因斯坦方程解空間相關聯,這被證明是一種非常強大的技術,並在我們的研究中起著核心作用,因為它將一個分析不變量與曲面的底層幾何聯繫起來。第三章討論了Type 𝕊³曲面;這些是𝕊² + ℝ²群上的左不變仿射幾何。這些幾何形成了一個非常豐富的家族,但我們對其只有部分理解。唯一剩下的均質幾何是球面𝕊³。第四章介紹了仿射曲面的幾何與配備修改的黎曼擴展的中性簽名度量的餘切空間幾何之間的關係。