LECTURES ON DIFFERENTIAL GEOMETRY Wulf Rossmann (Updated  Aug 2004) (Picture from section 3.4)

 Contents Chapter 1. Manifolds 1.1 Review of calculus 1.2 Manifolds:definitions and examples 1.3 Vectors and differentials 1.4 Submanifolds 1.5 Riemann metrics Chapter 2. Tensor Calculus 2.1 Tensors:definitions 2.2 Differential forms 2.3 Differential calculus 2.4 Integral calculus 2.5 Lie derivatives Chapter 3. Connections and curvature 3.1 Connections 3.2 Geodesics 3.3 Riemann curvature 3.4 Gauss curvature 3.5 Cartan’s mobile frame 3.6 Levi-Civita’s connection 3.7 Curvature identities Chapter 4. Special topics 4.1 General Relativity 4.2 Schwarzschild metric 4.3 The group SO(3) 4.4 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 of it.

A comment 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 formal matters.

W.R.