Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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123
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16answers
18k views

How do I make the conceptual transition from multivariable calculus to differential forms?

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...
103
votes
17answers
14k views

What is torsion in differential geometry intuitively?

Hi, given a connection on the tangent space of a manifold, one can define its torsion: $$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$ What is the geometric picture behind this definition&...
83
votes
17answers
11k 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 ...
80
votes
8answers
8k views

Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function $S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$ Here is a somewhat more conceptual ...
79
votes
16answers
8k 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 ...
77
votes
8answers
9k views

What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.) The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem. The ...
74
votes
11answers
10k views

Is it possible to capture a sphere in a knot?

You and I decide to play a game: To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...
72
votes
13answers
19k views

why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
68
votes
5answers
5k views

Is there a sheaf theoretical characterization of a differentiable manifold?

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
65
votes
5answers
8k views

When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?

As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion free connection $\nabla_g$, the Levi-Civita connection, that is compatible witht metric. I was wondering if one can ...
62
votes
4answers
3k views

Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?
59
votes
2answers
5k views

Complex structure on $S^6$ gets published in Journ. Math. Phys

A paper by Gabor Etesi was published that purports to solve a major outstanding problem: Complex structure on the six dimensional sphere from a spontaneous symmetry breaking Journ. Math. Phys. 56, ...
57
votes
6answers
5k views

Is there an analogue of curvature in algebraic geometry?

I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...
53
votes
8answers
8k views

Is there a complex structure on the 6-sphere?

I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a ...
53
votes
1answer
2k views

Stiefel–Whitney classes in the spirit of Chern-Weil

Chern-Weil theory gives characteristic classes (e.g. Chern class, Euler class, Pontryagin) of a vector bundle in terms of polynomials in the curvature form of an arbitrary connection. There seems to ...
52
votes
12answers
8k views

How much of differential geometry can be developed entirely without atlases? [closed]

We may define a topological manifold to be a second-countable Hausdorff space such that every point has an open neighborhood homeomorphic to an open subset of $\mathbb{R}^n$. We can further define a ...
47
votes
7answers
5k views

Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
47
votes
5answers
3k views

Random manifolds

In the world of real algebraic geometry there are natural probabilistic questions one can ask: you can make sense of a random hypersurface of degree d in some projective space and ask about its ...
46
votes
18answers
15k views

How should one present curl and divergence in an undergraduate multivariable calculus class?

I am a TA for a multivariable calculus class this semester. I have also TA'd this course a few times in the past. Every time I teach this course, I am never quite sure how I should present curl and ...
45
votes
7answers
4k views

Example of a manifold which is not a homogeneous space of any Lie group

Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth ...
45
votes
2answers
4k views

Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...
44
votes
7answers
8k views

What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....
44
votes
5answers
4k views

Kahler differentials and Ordinary Differentials

What's the relationship between Kahler differentials and ordinary differential forms?
44
votes
3answers
2k views

Operations via Morse Theory

I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)homology ...
43
votes
10answers
6k 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 ...
43
votes
1answer
3k views

What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...
43
votes
3answers
3k views

Explicit metrics

Every surface admits metrics of constant curvature, but there is usually a disconnect between these metrics, the shapes of ordinary objects, and typical mathematical drawings of surfaces. Can ...
42
votes
11answers
6k views

Why is the exterior algebra so ubiquitous?

The exterior algebra of a vector space V seems to appear all over the place, such as in the definition of the cross product and determinant, the description of the Grassmannian as a variety, the ...
42
votes
0answers
695 views

Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here. $\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an ...
41
votes
5answers
4k views

Is there a “geometric” intuition underlying the notion of normal varieties?

I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot. One thing that strikes me ...
41
votes
3answers
5k views

$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$

I read (in a paper by Emil Saucan) that the flat torus may be isometrically embedded in $\mathbb{R}^3$ with a $C^1$ map by the Kuiper extension of the Nash Embedding Theorem, a claim repeated in this ...
40
votes
9answers
13k views

What is the exterior derivative intuitively?

Hi, actually I have several related questions, not worth opening different threads: What is the of the exterior derivative intuitively? What is its geometric meaning? A possible answer I know is, ...
39
votes
7answers
8k views

Why are there so many smooth functions?

I do understand that my question might seem a little bit ignorant, but I thought about it a lot and still can't wrap my head around it. Analycity imposes very strong conditions on a map, from ...
38
votes
7answers
6k views

Down-To-Earth Uses of de Rham Cohomology to Convince a Wide Audience of its Usefulness

I'm soon giving an introductory talk on de Rham cohomology to a wide postgraduate audience. I'm hoping to get to arrive at the idea of de Rham cohomology for a smooth manifold, building up from vector ...
38
votes
3answers
4k views

Can every manifold be given an analytic structure?

Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy ...
37
votes
0answers
752 views

Are there periodicity phenomena in manifold topology with odd period?

The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$: $n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...
36
votes
6answers
4k views

Synthetic vs. classical differential geometry

To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...
36
votes
0answers
2k views

Minimal volume of 4-manifolds

This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...
35
votes
1answer
859 views

A dictionary of Characteristic classes and obstructions

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. In an effort to ...
34
votes
5answers
3k views

Beautiful descriptions of exceptional groups

I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
34
votes
3answers
3k views

Connected sum of topological manifolds

A definition of the connected sum of two $n$-manifolds $M$ and $M'$ begins by considering two $n$-balls $B$ in $M$, $B'$ in $M'$, and glueing the varieties $M\setminus \mathring B$ and $M'\setminus \...
34
votes
2answers
2k views

Meaning/Origin of Seiberg-Witten Equations/Invariants

Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from. We take an ...
33
votes
3answers
3k views

What is a foliation and why should I care?

The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential ...
32
votes
4answers
3k views

What are “good” examples of spin manifolds?

I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly: What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin (...
32
votes
2answers
2k views

Drawing of the eight Thurston geometries?

Do you know of a picture, drawing, or other concise visual representation of the eight three-dimensional Thurston geometries? I am imagining something akin to the standard picture (of a sphere, ...
32
votes
1answer
2k views

Many flat totally geodesic surfaces ⇒ flat?

Let $M$ be a 4-dimensional Riemannian manifold. Assume there is a huge number (say 100) of flat totally geodesic 2-dimensional surfaces passing through a point $p\in M$ and assume that their tangent ...
32
votes
3answers
2k views

When is a submanifold of $\mathbf R^n$ given by global equations?

Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...
31
votes
10answers
5k views

de Rham cohomology and flat vector bundles

I was wondering whether there is some notion of "vector bundle de Rham cohomology". To be more precise: the k-th de Rham cohomology group of a manifold $H_{dR}^{k}(M)$ is defined as the set of closed ...
31
votes
9answers
7k 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 ...
31
votes
7answers
6k 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 ...