0
votes
1answer
29 views

Tangent bundle of the 2-sphere

I'm reading through Tu's Introduction to manifolds and today I learned about tangent bundles and vector bundles. I was surprised to learn that $TS^2$, tangent bundle of the 2-sphere, isn't trivial ...
0
votes
0answers
61 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
10
votes
2answers
160 views

Applications of Geometry to Computer Science

How is differential geometry (or any type of theoretical math) related to computer graphics and/or computer programming? A friend of mine has only a bachelors degree in pure math and got hired by ...
4
votes
1answer
168 views

Russian Texts on Geometry

I recently saw a question today pertaining to Russian mathematics and I have a similar question but of a slightly different flavor. I've always heard that the Soviet Union had a history of producing ...
1
vote
1answer
107 views

Path to Differential Geometry

What do I need to learn to start on the rigorous study of differential geometry? I'm about to start my 3rd undergrad year at school, and have taken Cal 1-3, Linear Algebra, Elementary Number Theory, ...
9
votes
1answer
80 views

Relationship between radii in Riemannian manifolds

S.T. Yau raised an (somewhat) interesting question in a lecture today, which I thought I would ask for thoughts on from the good people here. The context is as follows: Consider a 3-manifold $M$ with ...
38
votes
5answers
1k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
10
votes
0answers
136 views

Soft question: How does basic differential geometry “fit together”?

I'm self-studying diff geom from Lee's Introduction to Smooth Manifolds. He warns the reader that there's a lot of machinery to construct, which is fine, and he explains things with wonderful clarity. ...
2
votes
1answer
119 views

What should a student (with algebraic-geometry minded) study in differential geometry?

One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a ...
1
vote
3answers
114 views

Are truth tables a valid method to prove an iff statement?

I recently had a homework assignment returned to me (for a Differential Geometry course, undergrad level) in which my instructor wrote "You cannot use truth tables to prove an if and only if ...
5
votes
3answers
208 views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
1
vote
1answer
37 views

Why use Gauss and mean curvature to characterize a surface's deviation from being “flat” at one point?

We know for a 2-dimensional surface there are two orthogonal principal directions at every point, where the principal curvatures $\kappa_1$ and $\kappa_2$ are the two ends of the curvature spectrum ...
3
votes
4answers
93 views

Is there a shorter path to these results?

I'm a student of Physics, however I usually study mathematics on texts aimed at mathematicians to gain a deeper understanding. Currently I'm studying differential geometry on Spivak's book and one of ...
1
vote
2answers
136 views

Why do people stick with Riemann-Integration when dealing with differential geometry?

I asked a question yesterday that is, "Is there an introductory differential geometry text using Lebesgue integration?" Then, i got an answer that "since we are dealing with differential geometry we ...
2
votes
0answers
60 views

Is it generally difficult to memorize 'multivariable calculus' theorems?

There are many weak forms of "Fubini's theorem" with strong hypotheses in elementary calculus texts. However, these strong hypotheses are very unnatural and are thus hard to memorize. Compared to ...
3
votes
1answer
96 views

Recommendation on studying differential geometry

Below are what i studied so far: Rudin - Principles of Anlysis (only except one chapter, namely differential forms) Munkres - Topology (only point-set topology) Rudin - RCA (Only first 4 ...
11
votes
4answers
230 views

Topics in Differential geometry $\cap$ Algebraic geometry

I find (both: differential and algebraic) geometry fascinating. I'm just beginning my graduate studies, but I'd like to know some topics/theorems in the intersection of these two (since I don't really ...
2
votes
1answer
83 views

Literature on Chern-Weil Theory and the Chern-Gauß-Bonnet Theorem

At my university there are plans for a graduate seminar on Chern-Weil Theory and Chern's generalisation of the Gauß-Bonnet Theorem. Unfortunately I am having a though time in finding adequate and ...
22
votes
1answer
418 views

Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. ...
3
votes
0answers
70 views

Curve on Riemannian Manifold

A curve on Riemannian Manifold is $c:I\rightarrow M$. We study many properties about it, like parallel $\bigtriangledown_\dot{c}X=0$ and geodesic $\bigtriangledown_\dot{c}\dot{c}=0$. And we apply the ...
1
vote
1answer
46 views

What are some helpful pre-requisites/hints/encouragement for going through Theodore Frankel's “Geometry of Physics” in a self-study?

I plan on working through "Geometry of Physics" by Frankel. I keep on running into little snags here and there, and I am wondering if that's just part of the process, or whether I am ill-prepared. ...
1
vote
2answers
107 views

Recommendation on studying Smooth Manifold & Differential Geometry and related subjects.

My major was physics, but i'm changing my major to mathematics this year. I took 2 years off to study mathematics myself and now i'm going back in this year. Here's the list of books i have studied ...
9
votes
0answers
189 views

Big geometry grad schools - for an average applicant

What are some schools that have a lot of geometry going on, but that might accept some middle-of-the-range applicants? Let me add some context... I left grad school (UC Davis) with an MS in 2012 ...
1
vote
0answers
135 views

Some Advice on My Undergraduate Paper

My teacher wants me to read something about "Differential Geometry in $R^3$" and choose a topic as a paper. Now I have finished these books. And I am interested in some topics below: $(1)$ ...
4
votes
2answers
143 views

Why do differential geometry textbooks bother with equivalence classes of smooth structures?

In contemporary textbooks on differential geometry, the definition of smooth manifolds is given in a (IMHO) awkwardly obfuscated way, by saying that a smooth manifold is a topological space endowed ...
5
votes
1answer
101 views

Is there any way to define differentiablity without any reference to the Euclidean space?

We define metric spaces based on the properties of the real numbers $\Bbb{R}$. In the same spirit we define smooth manifolds. But there is a more general and elegant way to formulate our intuition of ...
0
votes
1answer
73 views

Meaning of differentiability

Could anyone give an intuitive idea of the meaning of differentiability in general in any dimension and any space?
3
votes
3answers
125 views

Ideas for a Project on Differential Geometry

Currently trying to find a topic for a roughly fifteen page paper on Differential Geometry with presentation, with the rough target being a second year graduate student audience. I was looking in ...
3
votes
1answer
170 views

What is the exact motivation for the Minkowski metric?

In introductory texts about Lorentz Geometry, one always learns about the Minkowski space, i.e. $R^4$ with the Minkowski metric $$ m(x, y) := -x_0 y_0 + x_1y_1 + x_2y_2+ x_3 y_3 $$ Using this ...
3
votes
1answer
84 views

motivating the conservation of symplectic area by way of general (coordinate) covariance

I'm trying to motivate why a symplectic structure captures exactly the right structure one needs to do classical mechanics. The easiest part of this story goes like this: we need a procedure for ...
2
votes
3answers
58 views

What separates rotations from other co-ordinate transformations?

I am confused about some seemingly elementary ideas. From what I have understood, a rotation is just a specific class of co-ordinate transformations. If this is true, what exactly separates a rotation ...
10
votes
1answer
506 views

Research in differential geometry

I am an 3rd year undergrad interested in mathematics and theoretical physics. I have been reading some classical differential geometry books and I want to pursue this subject further. I have three ...
0
votes
3answers
169 views

A little confusion about compactness and connectedness

This question may be a bit simple or even naive for some people but it indeed confuses me for a long time. Thank you all if you provide any explanation. I know concepts: compactness means any open ...
5
votes
2answers
191 views

Coordinate independence of geometrical objects.

I am still trying to get a good grasp on the motivations behind various concepts in Differential Geometry. But I am struggling to come to terms with how certain concepts have this added attribute of ...
4
votes
3answers
207 views

Geometry and Physics

I have to do a presentation on Geometry and Physics. I am asking it here (rather than physics.se) because I have to focus on Geometry More than Physics. The intended audience is Undergraduate Seniors ...
11
votes
3answers
521 views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
2
votes
3answers
306 views

What's the motivation to add inner product and wedge product together in geometric product

I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining $$ ab = a\cdot b + ...
7
votes
2answers
274 views

Applications of information geometry to the natural sciences

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
3
votes
1answer
954 views

Differential Geometry Video Lectures

I know there's a similar question here, however since what I found there wasn't what I was looking for I thought on creating a new question. I'm studying Differential Geometry through Spivak's book "A ...
5
votes
3answers
262 views

Does Differential Topology or Differential Geometry play a larger role in Chaos Theory?

I'm an undergraduate on somewhat of a time constraint in school. I have room in my remaining schedule for a semester of either Differential Geometry or Differential Topology. I understand the ...
8
votes
1answer
236 views

what's the role of fiber bundles play in understanding the base space?

Suppose $\pi: E\to B$ be a fiber bundle, and assume $B$ is connected, I want to know: 1, What do the fiber bundles (e.g., the covering spaces) on $B$ help understand the topology, or the structure on ...
5
votes
1answer
270 views

Differential Geometry without General Topology

I want to ask if there is some book that treats Differential Geometry without assuming that the reader knows General Topology. Well, many would say: "oh, but what's the problem ? First learn General ...
2
votes
1answer
198 views

Selecting Differential Geometry Exercises

I'm self-studying differential geometry with Do Carmo's books "Differential Geometry of Curves and Surfaces" and "Riemannian Geometry" and I find those books very good, however I feel a little ...
7
votes
3answers
253 views

Are the Ricci and Scalar curvatures the only “interesting” tensors coming from the Riemannian curvature tensor?

In studying Riemannian geometry, one learns about the Riemann curvature tensor $$Rm(X,Y,Z,W) = \langle \nabla_X\nabla_YZ -\nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z, W\rangle$$ and its various symmetries. ...
3
votes
1answer
105 views

What are all isometry classes of the 2-sphere?

In topology, one learns how to classify the compact surfaces up to homeomorphism. And in fact, since "homeomorphic" and "diffeomorphic" coincide in dimension 2, we can classify the compact (smooth) ...
24
votes
3answers
1k views

Roadmap to study Atiyah-Singer index theorem

I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
8
votes
5answers
670 views

Differential Geometry past an introductory course?

I'll be doing an independent study with one of my profs in differential geometry next semester (my university did not happen to offer an intro diff. geometry course next semester like it usually ...
2
votes
1answer
155 views

Getting Practice on Finding Charts for a Manifold

I just want to ask for a suggestion on the study of differential geometry. When I study it I understand the theorems, their proofs, I understand perfeclty the concepts and so forth, but I'm having ...
7
votes
3answers
319 views

Why do people care about principal bundles?

I've started to learn a little about principal bundles (in the smooth category) and while I see how notions like connections and curvature are abstracted from the setting of vector bundles and brought ...
3
votes
2answers
130 views

Explanation about frames as distinct from a co-ordinate system

I am quite confused as to what is the difference between a frame and a co-ordinate system. The wikipedia page was not very helpful for me. I would be very happy if someone could give me a non-rigorous ...