Tensor Analysis on Manifolds

connection (i.e. covariant derivative) that is geometrically clear; we have to axiomatize this notion when there's no natural space to embed in).

Though the book may not be suitable as a first text, it can be used in conjunction with a more elementary text. Alternatively, it could be used for a graduate course. Though there are now a plethora of other good treatments around, this book remains one of the classics,and furthermore its price makes it particularly appealing.


good introductory text   July 22, 1998
17 out of 19 found this review helpful

Of all of these dover books, this is the best one to learn differential geometry from (you can safely ignore just about any other dover text with the word "tensor" in the title). The authors keep the exposition very clear with good examples.


A bit difficult for the non-professional but overall a fascinating book   May 8, 2008
magellan (Santa Clara, CA)
3 out of 3 found this review helpful

I came to this book with the minimum background--calculus and advanced calculus, differential equations, and some linear algebra, and found it a bit tough going, but still enjoyable. In fact, for me, not being a mathematician but a math hobbyist, really, whose education is mostly in biology and art history, I found it pretty difficult but also quite fascinating and even mind-blowing. I only had the vaguest ideas about tensors, fields, and manifolds before this, although I knew that the theory of manifolds underlies differential geometry and Einstein's famous General Relativity theory.

I understand that the notation in this book is considered old-fashioned and may contribute to the difficulty of reading it. Not having had anything different I don't know if it was harder for me or not, but overall I didn't find the notation too bad. The authors make the interesting point in the introduction that notational developments have occupied much of the work in manifolds, which I found funny. This implies that you can be good at math notation but not that good at the math. So maybe there's hope for me yet. :-)

That issue aside, I found this a very complete and well presented discussion on the subject. Some of it seemed pretty abstract and even counter-intuitive; for example, the concept of distance between two points isn't necessary to have a manifold, and yet having a coordinate neighborhood, or a manifold consisting of differentiable functions is, or other similar properties. It is a little strange to consider that one can perform differentiation on a manifold without the concept of spatial distance, when to my mind taking delta y over delta x at the limit is just shrinking the distance down to nothing in order to obtain the derivative of a function, not to mention that this seems problematic given the requirement of either uniform or non-uniform convergence. How do you know the function converges without some concept of distance? If you're better at this stuff than I am perhaps you could leave me a brief comment if I'm getting something wrong here.

But I still learned a lot, and much of it is pretty amazing and even mind-blowing stuff. People wouldn't need psychedelics if they knew enough to be learning about tensors, manifolds, and topology. They could blow their minds just on this stuff. :-)

So go out and get yourself a book on tensor manifolds and blow your mind the natural way. Higher mathematics is just awesome stuff even if I'm not quite smart enough to really understand it, but I can at least appreciate it, and I probably got a lot further with it than most biology and art history majors. :-)