# Tagged Questions

**6**

votes

**0**answers

102 views

### Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids

When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie ...

**24**

votes

**8**answers

3k views

### Why differential forms are important?

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated ...

**4**

votes

**5**answers

393 views

### What is “Data” involved in a mathematical construction?

What exactly do mathematicians mean when they refer to "the data" involved in a construction?
I've encountered this many times and I can usually figure out what's going on, but I am curious about the ...

**5**

votes

**3**answers

846 views

### group of diffeomorphisms of a manifold

How much has been the group of diffeomorphisms of a manifold " been studied.
I got this information from wiki.
" Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra ...

**11**

votes

**4**answers

1k views

### How Many 4-Manifolds are Symplectic?

As an honest question (probably with some subjectivity), how many smooth oriented 4-manifolds are actually symplectic? Can I say half (perhaps under some mild assumptions)? I ask this question because ...

**5**

votes

**1**answer

563 views

### Convenient definition of “category of Riemannian manifolds”?

Has a notion of "category of Riemannian manifolds" been defined and used in the literature?
For which reasons is it or would it (not) be a useful notion?
I think the objects should be all (perhaps ...

**25**

votes

**2**answers

2k views

### What else is Seiberg-Witten Theory equal to?

In low-dimensional topology there have been a bunch of invariants defined, and Seiberg-Witten Theory seems to make its appearance in [a lot of] them:
1) Heegaard Floer homology = SW Floer homology ...

**35**

votes

**8**answers

4k views

### Possibility of an Elementary Differential Geometry Course

I have to admit I'm not sure if this is an appropriate question. It's related to research in math education, but not directly to math.
I've found that in talking to professional physicists and ...

**15**

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**9**answers

4k views

### Open Questions in Riemannian Geometry

What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.

**49**

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**13**answers

4k views

### What makes four dimensions special?

Do you know properties which distinguish four-dimensional spaces among the others?
What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds ...

**4**

votes

**5**answers

1k views

### When do we study maps into an object or from the object to another object?

In many Mathematical theories, to study an object, we usually consider the set of all maps from that object to some other object. For example, in differential geometry, we study the smooth maps from a ...

**51**

votes

**15**answers

7k views

### Are there examples of non-orientable manifolds in nature?

Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...

**5**

votes

**2**answers

564 views

### Why did the word “exterior” get chosen for the idea of “exterior derivative”?

What are the intuitive and historical reasons for choosing the word "exterior" for the concept of an exterior derivative of a form?
The reasoning I've heard about it is the following: let p(t) be a ...

**12**

votes

**3**answers

2k views

### Why are they called isothermal coordinates?

On a Riemannian manifold, a coordinate system is called "isothermal" if the Riemannian metric in those coordinates is conformal to the Euclidean metric:
$$g_{ij} = e^{f} \delta_{ij}$$
My question ...

**8**

votes

**3**answers

772 views

### What's the “correct” smooth structure on the category of manifolds?

As will become clear, this is in some sense a follow up on my earlier question Why should I prefer bundles to (surjective) submersions?. As with that one, I hope that it's not too open-ended or ...

**12**

votes

**7**answers

2k views

### Why should I prefer bundles to (surjective) submersions?

I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...

**7**

votes

**1**answer

346 views

### Can minimal surfaces be characterized by some universal property?

As objects which are minimal (in some respect), this seems entirely plausible, but I'm not sure what category we should be working in, and what restrictions we would need, to actually have a situation ...

**9**

votes

**3**answers

2k views

### Number Theory and Geometry/Several Complex Variables

This is a question for all you number theorists out there...based on my skimming of number theory textbooks and survey articles, it seems like most of the applications of geometry and complex ...

**3**

votes

**2**answers

459 views

### Broken Symmetry

I have a tangled web of ideas about natural transformations, vector spaces, equivalence classes, local coordinates, etc. in my head that I'm trying to unravel. So here are some of the questions I ...

**29**

votes

**7**answers

4k views

### Why is it useful to study vector bundles?

I have this question coming from an earlier Qiaochu's post. Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just ...