《黎曼:芬斯勒几何导论》

《黎曼:芬斯勒几何导论(英文版)》讲述了：The subject matter of this book had its genesis in Riemann's 1854 "habil-itation" address: "Uber die Hypothesen, welche der Geometrie zu Grundeliegen" （On the Hypotheses, which lie at the Foundations of Geometry）.Volume II of Spivak's Differential Geometry contains an English translationof this influential lecture, with a commentary by Spivak himself. Riemann, undoubtedly the greatest mathematician of the 19th century,aimed at introducing the notion of a manifold and its structures. The prob-lem involved great difficulties. But, with hypotheses on the smoothness ofthe functions in question, the issues can be settled satisfactorily and thereis now a complete treatment. Traditionally, the structure being focused on is the Riemannian metric,which is a quadratic differential form. Put another way, it is a smoothlyvarying family of inner products, one on each tangent space. The resultinggeometry —— Riemannian geometry —— has undergone tremendous develop-ment in this century. Areas in which it has had significant impact includeEinstein's theory of general relativity, and global differential geometry. In the context of Riemann's lecture, this restriction to a quadratic dif-ferential form