ON DIFFERENTIAL GEOMETRY
(Picture from section 3.4)
Chapter 1. Manifolds
Manifolds:definitions and examples
Chapter 2. Tensor Calculus
Chapter 3. Connections and curvature
Cartan’s mobile frame
Chapter 4. Special topics
Weyl’s gauge theory paper of 1929
TO THE STUDENT
This is a
collection of lecture notes which I put together while teaching courses on manifolds,
tensor analysis, and differential geometry.
I offer them to you in the hope that they may help you, and to
complement the lectures. The style is
uneven, sometimes pedantic, sometimes sloppy, sometimes telegram style,
sometimes long--winded, etc., depending on my mood when I was writing those
particular lines. At least this set of notes is visibly finite. There are a
great many meticulous and voluminous books written on the subject of these
notes and there is no point of writing another one of that kind. After all, we are talking about some fairly old mathematics, still
useful, even essential, as a tool and still fun, I think, at least some parts
about the nature of the subject (elementary differential geometry and tensor calculus)
as presented in these notes. I see it as a natural continuation of analytic
geometry and calculus. It provides some
basic equipment, which is indispensable in many areas of mathematics (e.g.
analysis, topology, differential equations, Lie groups) and physics (e.g.
classical mechanics, general relativity, all kinds of field theories). If you
want to have another view of the subject you should by all means look around,
but I suggest that you don't attempt to use other sources to straighten out problems
you might have with the material here.
It would probably take you much longer to familiarize yourself
sufficiently with another book to get your question answered than to work out
your problem on your own. Even though
these notes are brief, they should be understandable to anybody who knows
calculus and linear algebra to the extent usually seen in second-year courses.
There are no difficult theorems here; it is rather a matter of providing a
framework for various known concepts and theorems in a more general and more
natural setting. Unfortunately, this
requires a large number of definitions and constructions which may be hard to
swallow and even harder to digest. (In this subject the definitions are much
harder than the theorems.) In any case, just by randomly leafing through the
notes you will see many complicated looking expressions. Don't be intimidated: this stuff is easy.
When you looked at a calculus text for the first time in your life it probably
looked complicated as well.
Let me quote
a piece of advice by Hermann Weyl from his classic
Raum–Zeit–Materie of 1923 (my translation). Many
will be horrified by the flood of formulas and indices which here drown the
main idea of differential geometry (in spite of the author's honest effort for conceptual
clarity). It is certainly regrettable
that we have to enter into purely formal matters in such detail and give them
so much space; but this cannot be avoided.
Just as we have to spend laborious hours learning language and writing
to freely express our thoughts, so the only way that we can lessen the burden
of formulas here is to master the tool of tensor analysis to such a degree that
we can turn to the real problems that concern us without being bothered by