2
votes
1answer
214 views

undergraduate handle decomposition. Reference

As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.
13
votes
2answers
425 views

How useful/pervasive are differential forms in surface theory?

Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...
0
votes
0answers
179 views

About the parallel transport and choice of connection

Thought Experiment Consider a 2-sphere, $S^2$, and let $p$ be a point at the equator. Case 1 Let us parallel transport a vector, $V$ from $p$ using the recipe: Move one unit of length East. Move ...
6
votes
1answer
544 views

Teaching stacks to differential geometry students

Does anyone have any experience teaching stacks over the category of manifolds to students whose background is, say, a semester-long course on manifolds? Does anyone know of any publicly available ...
11
votes
2answers
869 views

There are two points on the Earth's surface that … ?

At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...? What is the strongest, most impressive statement one can make here? The ...
4
votes
2answers
355 views

Surface Laplace-Beltrami without coordinates, exterior calculus?

Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator ...
9
votes
3answers
919 views

History surrounding Gauss Theorema Egregium and differential geometry

I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian ...
8
votes
3answers
597 views

Reference for working with the implicit function theorem

I just had a student come to my office hours and ask me a ton of questions, the answer to all of which was "that's a slight variant to the implicit function theorem, which is proved by formal ...
15
votes
4answers
2k views

What is the best way explain to undergraduates that all 1-dimensional manifolds are orientable?

Let's suppose that $M$ is a connected $1$-dimensional smooth manifold (Haussdorf and paracompact). We know that there are exactly two types, up to diffeomorphism (even up to homeomorphism), namely ...
22
votes
7answers
2k views

[STILL OPEN] Why are two notions of Gaussian curvature are the same - what is the simplest & most didactic proof?

This question is still wide open - all of the answers so far rely on magical calculations. I've only accepted an answer because, by bounty rules, otherwise one would be accepted automatically. I can't ...
21
votes
5answers
2k views

References for “modern” proof of Newlander-Nirenberg Theorem

Hi, I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
10
votes
17answers
9k views

Undergraduate Differential Geometry Texts

Can anyone suggest any basic undergraduate differential geometry texts on the same level as Manfredo do Carmo's Differential Geometry of Curves and Surfaces other than that particular one? (I know a ...