Riemannian Geometry in an Orthogonal Frame by Vladislav V Goldberg (PDF)

Description

Foreword by S S Chern In 1926-27, Cartan gave a series of lectures in which he introduced exterior forms at the very beginning and used extensively orthogonal frames throughout to investigate the geometry of Riemannian manifolds. In this course he solved a series of problems in Euclidean and non-Euclidean spaces, as well as a series of variational problems on geodesics. In 1960, Sergei P Finikov translated from French into Russian his notes of these Cartan’s lectures and published them as a book entitled Riemannian Geometry in an Orthogonal Frame. This book has many innovations, such as the notion of intrinsic normal differentiation and the Gaussian torsion of a submanifold in a Euclidean multidimensional space or in a space of constant curvature, an affine connection defined in a normal fiber bundle of a submanifold, etc. It has now been translated into English by Vladislav V Goldberg, currently Distinguished Professor of Mathematics at the New Jersey Institute of Technology, USA, who also edited the Russian edition.

User’s Reviews

Editorial Reviews: Review The reviewer expresses his hearty appreciation to the authors for their utmost endeavor of translating E Cartan’s ever valuable lectures. — Zentralblatt fur Mathematik

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Before you read this book, please note that you have a sound foundation of differential geometry on surface ( Do Carmo has a very good text book on it ).I have heard a lot of complaints like – they don’t understand Master’s work of their time or just simply said it is over-simplied – This book is a truly master work: it is the direct extension of the geometry on surface – an orthogonal frame is used and imply more – how to generate results to Cantan’s moving frame.A good book not only teach knowledges to student, but reveal the intellectual pathway to the problem. From this criteria, I give the highest mark to this book.

⭐As with all great masterpieces of math, I am not a critic for a good critique assumes expertise. I am an expert self-study beginner in pure and applied mathematics. My comments is for people in that disposition. If you get a first look at the inside, then you quickly learn that the notation is very hard to understand for an introductory textbook. In addition, it is very dense and concise. Not much of an explanation to help first time beginners. It isn’t meant to be introductory! I am sure that it is a standard classic for experts. Elie Cartan was a truly great differential geometer! That is the reason I have the book; it’s from a master. In terms of approachability of the book, it is not that aproachable. If you are studying differential geometry, then it maybe right for you. I can say that I wanted an introductory self-study book and was disappointed. It remains on the bookself and even got lost among other books for later review. When I have some more differential geometry courses or time I will try to decipher the notation. I can read math at the undergraduate level. This book is for sure for graduate students! I just hope that in the future I can still make use of the book. I plan to go study differential geometry as a course with a professor. Then it would be nice to have something like this for reference work. All in all, a masterpiece not easy to understand. It is how works of genius is. I better stick to reading math history books for introductions to math and self-studies. A better elementary book on Vector and Tensor Analysis is by Harry Lass which covers the fundamentals for elementary beginners like me. If you like Harry Lass’s book then look forward to my other reviews. I am a struggling beginner that is not yet in graduate school for pure mathematics. Preparation is key and this book is an indication of what is ahead.

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