WannabeNewton said:Hi guys. I finished working through D'Inverno's "Introducing Einstein's Relativity" and Schutz's "A First Course in General Relativity" and some of Carroll's "Spacetime and Geometry" but I don't really feel like I learned most of what is out there. I also feel that before I can tackle Wald I need to read up on a proper introductory differential geometry book. If anyone could recommended me one that would be great. Thanks in advance.
George Jones said:As n!kofeyn has stated, contents of differential geometry references vary widely. Another book worth looking at is Differential Geometry and Lie Groups for Physicists by Marian Fecko,
https://www.amazon.com/dp/0521845076/?tag=pfamazon01-20.
This book is not as rigorous as the books by Lee and Tu, but it more rigorous and comprehensive than the book by Schutz. Fecko treats linear connections and associated curvature, and connections and curvature for bundles. Consequently, Fecko can be used for a more in-depth treatment of the math underlying both GR and gauge firld theories than traditionally is presented in physics courses.
Fecko has an unusual format. From its Preface,
A specific feature of this book is its strong emphasis on developing the general theory through a large number of simple exercises (more than a thousand of them), in which the reader analyzes "in a hands-on fashion" various details of a "theory" as well as plenty of concrete examples (the proof of the pudding is in the eating).
The book is reviewed at the Canadian Association of Physicists website,
http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf.
From the review
There are no problems at the end of each chapter, but that's because by the time you reached the end of the chapter, you feel like you've done your homework already, proving or solving every little numbered exercise, of which there can be between one and half a dozen per page. Fortunately, each chapter ends with a summary and a list of relevant equations, with references back to the text. ...
A somewhat idiosyncratic flavour of this text is reflected in the numbering: there are no numbered equations, it's the exercises that are numbered, and referred to later.
Personal observations based on my limited experience with my copy of the book:
1) often very clear, but sometimes a bit unclear;
2) some examples of mathematical imprecision/looseness, but these examples are not more densely distributed than in, say, Nakahara;
3) the simple examples are often effective.
micromass said:If you know some topology, then I recommend "Introduction to smooth manifolds" by John Lee.
If you don't know topology, then I would go for "A comprehensive introduction to differential geometry" by Spivak.
mathwonk said:here (at the bottom of the web page), is a link to a free first course in differential geometry by a student of the great S.S.Chern. Shifrin is an excellent teacher and author and a professional differential geometer as well. I do not know about answers, but most good books do not give answers to exercises. Learning to know whether your answer is right without being told is considered a valuable skill.
http://www.math.uga.edu/~shifrin/
Differential geometry is a branch of mathematics that deals with the study of curves and surfaces in three-dimensional space using the tools of calculus and linear algebra. It has applications in physics, engineering, and computer graphics, among others.
Differential geometry is important because it provides a mathematical framework for understanding the geometric properties and relationships of objects in three-dimensional space. It also has practical applications in fields such as physics, engineering, and computer graphics.
Differential geometry is used by mathematicians, physicists, engineers, and computer scientists, among others. It is also applicable in various fields such as robotics, computer graphics, and general relativity.
Some key concepts in differential geometry include curves, surfaces, manifolds, tensors, and curvature. These concepts are used to describe the shape and properties of objects in three-dimensional space and are essential for understanding more advanced topics in the field.
Some popular books on differential geometry include "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo, "Introduction to Smooth Manifolds" by John M. Lee, and "Riemannian Geometry" by Manfredo P. do Carmo. It is also helpful to consult textbooks on calculus and linear algebra, as these are foundational topics in differential geometry.