Curvature in Mathematics and Physics (Paperback)

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This original text for courses in differential geometry is geared toward advanced undergraduate and graduate majors in math and physics. Based on an advanced class taught by a world-renowned mathematician for more than fifty years, the treatment introduces semi-Riemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool.
Starting with an introduction to the various curvatures associated to a hypersurface embedded in Euclidean space, the text advances to a brief review of the differential and integral calculus on manifolds. A discussion of the fundamental notions of linear connections and their curvatures follows, along with considerations of Levi-Civita's theorem, bi-invariant metrics on a Lie group, Cartan calculations, Gauss's lemma, and variational formulas. Additional topics include the Hopf-Rinow, Myer's, and Frobenius theorems; special and general relativity; connections on principal and associated bundles; the star operator; superconnections; semi-Riemannian submersions; and Petrov types. Prerequisites include linear algebra and advanced calculus, preferably in the language of differential forms.

這本針對微分幾何課程的原始文本適用於數學和物理的高年級本科生和研究生。該教材基於一位世界知名數學家五十多年來所教授的高級課程，介紹了半黎曼幾何學及其主要的物理應用，即愛因斯坦的廣義相對論，並以Cartan外微分作為主要工具。

從介紹嵌入歐幾里得空間的超曲面相關曲率開始，本書進一步回顧了流形上的微分和積分計算。接著討論了線性聯繫及其曲率的基本概念，以及Levi-Civita定理、李群上的雙不變度量、Cartan計算、Gauss引理和變分公式等。其他主題包括Hopf-Rinow定理、Myer定理、Frobenius定理、特殊相對論和廣義相對論、主並聯和相關並聯上的聯繫、星算子、超聯繫、半黎曼子流形和Petrov類型。先備知識包括線性代數和高級微積分，最好使用微分形式的語言進行學習。