intro tensors book

gvk

Being educated as a physicist, I understand many people who complain about "bourbaki" style of writing math textbooks, and I would not recommend to read the books by F. Warner and M. Spivak as a first introductory reading in modern geometry. (Spivak is only good to understand the historical line of development, but you have to have some background and being familiar with modern terminology for that.) In my opinion more or less suitable book, written by mathematicians for physicists and engineers, is
Dubrovin, Novikov, Fomenko, Modern Geometry v. 1,2,3.
http://www.amazon.com/exec/obidos/tg...glance&s=books

This is three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics.
Topics of 1st volume starts from curves and surfaces and include tensors and their differential calculus, vector fields, differential forms, the calculus of variations in one and several dimensions, and even the foundations of Lie algebra. So, the first volume would be enough for start. I looked in 2 and 3 v. and think its close to the front of modern geometry and definitly prepares for the reading more special books...
The material of books is explained in simple and concrete language that is in terminology acceptable to physicists. There are some exercises, but should be more to get practical skills. If I will find the special problem book on modern geometry to accompanying this textbook, it would be excellent pair for any beginner.

mathwonk

gvk, i suggest you post your review of fomenko et al, at amazon.com for possibly wider distribution.

I have not seen this book but am very favorably impressed with writing of most russian texts.