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This semester, I took a course on the geometric theory of curves and sufaces, which are basically now refered to as the classical aspects of (differential) geometry. This is a post to put things in perspective. I want to pick out the essentials of the course, and form a narrative around this subject.
We started by 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 that 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 on 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 Mobius strip was proved! We dabbled a bit into 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.
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. And the map is essentially a derivation of normal, so it's a linear map. We then paused the weingarten map (or the $L_p$ map) study and moved on to studying curves.
We covered $C^k$ curves, reparametrization, regular curves, unit speed ones, their interaction, and then the (signed) curvature of the curves (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 the tangent, normal and also bi-normal, to have a formula that governs their variation, the Serret-Frenet formula. Then came (drumrolls) the geodesics. We pivot around a very simple abstraction from dynamics - 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). This translates to the speed (norm of the tangent vector) being constant. We then proved the existence of geodesics.
That's the end of half the semester :).
It's best if I don't go in the order of lectures from here. The main theme of second part of semester is to study the curvature of surfaces, and more interestingly, in different (equivalent) ways! There are three pieces to this: Weingarten map, curvature of surface a la curvature of curves on surfaces, and then the First and second fundamental forms.
Previously we had already discussed Weingarten map for n-surfaces or level sets. As a map between tangent spaces, giving how the normal changes when moved along a tangent vector -- We continue to study this for locally parametric n-surfaces, and define three kinds of curvature, Principal, Mean, and Gaussian which were formulated in terms of the eigenvalues of this $L_p$ map. We tediously computed mean and gaussian curvatures for a graph of a function and a surface of revolution. We then dabbled briefly into mean curvature equation. The sign of these curvatures dictate some global aspects of surfaces directly!
Let $\lambda_1, \lambda_2$ be the principle curvatures. $K$ be the gaussian curvature. Then we have the following nomenclature:
| Curvature at a point $p$ | Nomenclature for the point $p$ |
|---|---|
| $L_p = 0$ | Planar |
| $\lambda_1(p) = \lambda_2(p)$ | Umblic |
| $K(p) > 0$ | Elliptic |
| $K(p) < 0$ | Hyperbolic |
| $K(p) = 0$ but $L_p \neq 0$ | Parabolic |
With this, we have the following neat results.
| Principle curvature | Global nature of $S$ |
|---|---|
| All $p \in S$ are planar | S is part of a plane |
| All $p \in S$ are umblic | S is part of a plane or sphere |
Interestingly, and very intuitively, it also turns out that for a compact surface $S$, there is a point $p \in S$ which is elliptic, i.e a positive gaussian curvature!
A ground-up discussion of these was done starting from the parametric expansion of the surface around a point and filling it with the curvature information.
The idea is to restrict a curve onto a surface, and probe the curvature of curve which must now encode some information about the surface. When one restricts a unit speed curve $\alpha(t)$ onto a surface with a normal $N$, we can immediately write $\ddot{\alpha}(t)$ as a linear combination of two perpendicular vectors: the normal $N(\alpha(t))$ and $\dot{\alpha}(t) \times N(\alpha(t))$. The coefficients occuring in this are called the normal and geodesic curvatures respectively. This essentially splits the curvature of a curve into two aspects related to the surface!
\[\ddot{\alpha}(t) = K_n N(\alpha(t)) + K_g [\dot{\alpha} \times N(\alpha(t))],~ K_S^2 = K_n^2 + K_g^2\]In the special case of normal sections the normal curvature becomes the key player. A normal section is obtained by intersecting a surface $S$ with a plane $P = \set{p + xv + yN(p) \mid x,y, \in \mathbb{R}^2}$, where $v \in T_pS$ and $N(p)$ is the normal at $p$. The importance of these normal sections comes from the result that there is an open neighborhood $U \subseteq \mathbb{R}^{n+1}$ around $p$ such that $U \cap S \cap P$ is a regular curve!
We then do a unit-speed reparamatrization to find that, the geodesic curvature vanishes! That is, the curvature of this curve at $p$ is completely specified by the normal curvature $K_n$.
Since, the normal curvature $K_n = \ddot{\alpha}\cdot N$, when written as a function on $T_pS$, infact on unit vectors of $T_pS$, $K_n(v) = L_p(v)\cdot v$: we see that it is continous and defined on a compact set. It turns out the maximum and minimum values it attains are precisely the eigenvalues of the Weingarten map $L_p$ and the eigenvectors form an orthogonal basis of $T_pM$! This hints at the connection between the previous way of studying surfaces and through normal curvature. It also happens to be connected to the fundamental forms which would be studied next.
We then studied geodesics on surfaces of revoultion. The content of geodesic curvature became highly relevant when we had later studied the Gauss-Bonnet theorem.
Inorder to compute lengths on the surface, one needs to be able to integrate a quantity like,
\[\int \| \dot{\gamma(t)}\| dt.\]The first fundamental form is precisely the integrand - $\mid\mid\dot{\gamma}\mid\mid$! We start from a parametrization $\sigma(u,v)$ of a surface $S$, with $T_pS$ spanned by \braket{\sigma_u,\sigma_v} the two directional derivatives. And deduce the following (first fundemantal) form,
\[Edu^2 + 2Fdudv + Gdv^2,\]where, $E = \sigma_u\cdot\sigma_u, F = \sigma_u \cdot \sigma_v$ and $G = \sigma_v\cdot \sigma_v.$
Setting, $\gamma(t) = \sigma(u(t),v(t))$, the integrand in particular reads,
\[\int \left(E\dot{u}^2 + 2F\dot{u}\dot{v} + G \dot{v}^2\right)^{\frac{1}{2}}dt.\]Calculating this for few cases like - the plane, the surface of revolution, cylinder, immediately reminds us of the metric (which is indeed a symmetric bi-linear form).
The second fundamental form is then obtained by taylor expanding the parametrization around a point. The change is computed along the normal (away from the tangent plane). This comes out to be in the (second fundamental) form,
\[Ldu^2 + 2Mdudv + Ndv^2,\]where, $L = \sigma_{uu}\cdot \mathbf{N}, M = \sigma_{uv}\cdot \mathbf{N}$ and $N = \sigma_{vv}\cdot \mathbf{N}$.
Although here we appraoched the fundamental forms from a more groud-up procedure, the lectures used the Weingarten map again to extract this information. In terms of this map, the fundamental forms appear from the following defintions,
\[(I)_pv = \braket{v,v}, (II)_pv = \braket{L_pv,v}.\]We then talked about isometries, the fact that the first fundamental form remains same for an isometry (in other words the lengths are preserved). We moved to the Remarkable Theorem due to Gauss on the preserving of gaussian curvature by a local isometry. The proof essentially relies on the fact that the gaussian curvature can be written in terms of the first fundamental form only, which is preserved in isometries.
The down to Earth way of putting this, rather literally, is to say
Any map of any region of the Earth’s surface must distor distances.
This is due to different gaussian curvatures of sphere and plane!
The course ended with the other two major theorems on surfaces, the Gauss-Bonnet theorem which relates the (gaussian) curvature to the topology of the surface and the Stokes’ theorem which relates
There are many fundamentals which are clear here and but many of these aspects go deeper than what we could cover in the course like the more geometric significance of gaussian curvature (area, measure e.t.c) or any curvature (of curves and surfaces) for that matter.
This subject almost marks the end of the study of geometry of objects in one and two dimensions, mostly due to Gauss. Any efforts from here on to study higher geometry would take us to the realm of Riemannian Geometry.The story goes that Riemann gave a one hour lecture on his thesis work in this area when Gauss was more interested in higher dimensional geometry than the other two Riemann had planned better for.