Okay, so I did classical differential geometry this semester. The teaching style was not so rigorous. It was not that abstract either. This is a post to put things in perspective. I want to pick out the essentials of the course, so that I don't regret not learning an interesting subject.
After doing a course on Rings and Modules, I was afraid of algebra. Mostly because it has been clearly proved that with courses in SPS going full fledged I cannot do justice to the beautiful abstraction of math. Since I had courses like General Theory of relativtiy and Quantum Field theory II which are intensive, I thought I shouldn't stress myself (especially after the recent health events). I thought I should take a course like Complex Analysis or Differential Equations, something that I know can be less abstract than algebra yet rigorous. I love abstraction, it's just that I can give it only so much time while I do physics courses, especially the 9 hr labs every week. Both complex analysis and Differential Equations clashed with my other physics courses. One with QFT and other with Non-Linear optics. Also, I will have to work on my algebra soon...
Let me summarise how the course progressed. We first started out with defining the basic tools that we were going to use - vectors, vector fields, integral curves (also maximal), Tangent space. All defined on any open set in $\mathbb{R}^n$. We next worked on n-surfaces, in the framework of level sets. Tangent spaces on such surfaces (relation to the gradient of the function(s) in discussion), and then the important theorem of Lagrange Multiplier's to extremize a given function with constraint written in the form of level set(s). This theorem has got a good intution which Abha and I managed to figure out during a study session. A simple application of this goes into proving a symmetric matrix is diagonalizable; just define appropriate function (which looks like a mixed polynnomial, and can be written in inner product form) and restrict it to a level set. Smoothness of functions on surface, Tangent vector fields surfaces and how there exist (maximal) integral curves on them, orientation of surface, the two-oritentation proof for a connected surface; the highly intuitive Gauss map (basically capturing all the unit normals onto a sphere; yes turns out Gauss map is onto for a compact surface!).
Remember we only defined an n-surface in the framework of a level set. We now extend the concept of n-surfaces to be either parametric or locally parametric by defining what we mean by a parametrization (and also reparametrization). We basically look for a function that takes us from a low dimenstion euclidean space (locally atleast) to a higher one, where our surface recides. Yea, the concept of charts and atlasses sneek in, but we didn't go there, although we did use the terminology of chart, as a three tuple ($D\subset\mathbb{R}^2,\phi:D\mapsto S, S\subset\mathbb{R}^3 $)An n-surface usually means a level set, unless the (local) parametrization is specified.
We come back to studying the unit normal, basically the gauss map. We define the orientability of a locally parametric n-surface. It is dependent on sign of the determinant of the derivate of the transitions functions. Recall that a compact n-surface had two orientations. Guess what, the non-orientability of M\"obius strip was proved! We dabbled a bit on how a compact, orientable and locally parametrizable n-surface could be seen as an n-surface, by questioning the existence of the so called tubular neighborhood (through Inverse function theorem), where there is a unique normal line passing through each point. I'm not clear on how this leads to an n-surface. This is good direction to probe, there are as the instructor briefly alluded about a few conditions to consider and check around.
Now for the legendary study on the unit normals/gauss map. We define Weingarten map or the Shape operator which tell you how much the normal changes along the tangent. It maps one tangent vector to another by varying the normal at the initial point, of which the tangenet vector belongs to. This turns out to be a very strong concept, that gives out information on the curvature of surfaces. It's a self adjoint operator, meaning it has real eigen values. Oh yeah, the map is essentially a derivation of normal, so it's a linaer map. We then paused the weingarten map (or the $L_p$ map) study and moved on to studying curves.
$C^k$ curves, reparametrization, regular curves, unit speed ones, their interaction, and then the curvature (signed) of the curves; yeah, on how much the curve curves... Relating this to variation of the angle between any vector and the tangent. Then about the space curves, one could define tangent, normal and also bi-normal, and have a formula that governs their variation - the Serret-Frenet formula. Then came the drumrolls, geodesics. Simple abstraction - the double derivative of the curve at every point has to be normal to the tangent space at that point, geodesics got no acceleration along the direction they are going to move (tanget space)- translates to speed (norm of the tangent vector) being constant. We then proved the existence of geodesics.
That's the end of half the semester :).
Back to the Weingarten map, probably the most important concept in the entire course. We discussed it in the context of locally parametric n-surfaces, and defined three kinds of curvature—Principal, Mean, and Gaussian—in terms of the eigenvalues. We tediously computed mean and Gaussian curvature for a graph of a function and a surface of revolution. After the section where we discuss minimal surfaces (those with vanishing mean curvature everywhere), I found a great book called Gauge Fields, Knots and Gravity and got completely absorbed in mathematical physics, thinking about future moves: QFT II/GR reading projects, thesis, master's after master's, PhD, and all that. I couldn't resist diving into it,I must admit I was distracted from the class a little. But I didn't lose track of the concepts. I kept notes on the main ideas, and also it was easy to refer to Abha's clear notes later and understand the material properly. Moving on, we defined a normal section: a plane passing through a point on the surface, its unit tangent, and its unit normal. Its intersection with the surface turned out to be a regular curve! (to be continued...)