Aspects of Differential Geometry II

Peter Gilkey, JeongHyeong Park, Ramon Vazquez-Lorenzo

  • 出版商: Morgan & Claypool
  • 出版日期: 2015-04-01
  • 售價: $1,540
  • 貴賓價: 9.5$1,463
  • 語言: 英文
  • 頁數: 158
  • 裝訂: Paperback
  • ISBN: 1627057838
  • ISBN-13: 9781627057837
  • 相關分類: 微積分 Calculus物理學 Physics
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商品描述

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment.

Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Kähler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition Theorem, Poincaré duality, and the Künneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. The exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. The de Rham cohomology of compact Lie groups and the Peter--Weyl Theorem are treated. In Chapter 7, material concerning homogeneous spaces and symmetric spaces is presented. Book II concludes in Chapter 8 where the relationship between simplicial cohomology, singular cohomology, sheaf cohomology, and de Rham cohomology is established.

We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the total curvature and length of curves given by a single ODE is new as is the discussion of the total Gaussian curvature of a surface defined by a pair of ODEs.

Table of Contents: Preface / Acknowledgments / Additional Topics in Riemannian Geometry / de Rham Cohomology / Lie Groups / Homogeneous Spaces and Symmetric Spaces / Other Cohomology Theories / Bibliography / Authors' Biographies / Index

商品描述(中文翻譯)

微分幾何是一個廣泛的領域。我們選擇專注於某些適合入門的方面,並沒有試圖進行百科全書式的處理。

第二冊涉及比第一冊更高級的材料,面向研究生水平。第4章涉及黎曼幾何的其他主題。研究了由單個常微分方程給出的實解析曲線和由一對常微分方程給出的曲面的性質,並處理了测地球的體積。介紹了全纯幾何和Kähler幾何。在第5章中,討論了de Rham上同調的基本性質,證明了Hodge分解定理、Poincaré對偶和Künneth公式,並簡要介紹了特徵類理論。在第6章中,處理了李群和李代數。討論了指數映射、經典群和在雙不變度量的背景下的测地球。處理了緊致李群的de Rham上同調和Peter-Weyl定理。在第7章中,介紹了關於齊次空間和對稱空間的材料。第二冊在第8章結束,建立了單形上同調、奇異上同調、層上同調和de Rham上同調之間的關係。

我們提供了一些與傳統證明不同的證明,這些卷中還有一些新材料。例如,對由單個常微分方程給出的曲線的總曲率和長度的處理是新的,同樣,對由一對常微分方程定義的曲面的總高斯曲率的討論也是新的。

目錄:前言/致謝/黎曼幾何的其他主題/de Rham上同調/李群/齊次空間和對稱空間/其他上同調理論/參考文獻/作者簡介/索引