Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [dg.differential-geometry]

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

7
votes
5answers
2k views

Indexing the line bundles over a Grassmannian.

As is well known, the line bundles over *CP*$^1$ are indexed by the integers. My question is how are the line bundles over *CP*$^n$, $n > 1$, and *Gr*$(n,k)$ indexed? Moreover, do there exist any ...
12
votes
3answers
2k views

What are CR manifolds like?

The complex structure on a complex manifold pulls back to what's called a CR structure on any real codimension 1 submanifold. The structure induced on a submanifold of higher codimension is a CR ...
20
votes
1answer
956 views

Is Dependent Choice all we really need?

http://en.wikipedia.org/wiki/Axiom_of_dependent_choice Is DC sufficient for the understanding of objects that are countable in some suitable sense? For example, is DC sufficient for the full ...
5
votes
2answers
693 views

Historical question Cauchy-Crofton theorem vs. Radon transform

The Radon transform apparently was discovered around 1917 if Wikipedia is to be believed. The Cauchy-Crofton theorem is a much older theorem (mid 19th-century). But both ideas are more or less the ...
45
votes
10answers
7k 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 ...
21
votes
4answers
1k views

What is the relationship between various things called holonomic?

The following things are all called holonomic or holonomy: A holonomic constraint on a physical system is one where the constraint gives a relationship between coordinates that doesn't involve the ...
8
votes
3answers
882 views

Integration in equivariant K-theory

Let F be a smooth classifying space for K-theory (ordinary or equivariant). If X is a smooth compact manifold and W is a real vector space of dimension n, there is an integration map from the ...
6
votes
1answer
506 views

The 2-sphere and $\mathbb{CP}^1$

As is very well known, the algebraic variety $S^2$ is isomorphic to projective variety $\mathbb{CP}^1$ as a complex manifold. As is also well known, the coordinate ring of $S^2$ is given by $< x,y,...
1
vote
3answers
200 views

Transformations induced by geodesics of boundary

I have a general question in Riemannian geometry: Let M be a compact manifold and $\partial M \neq \emptyset$. Then shoot a geodesic from any boundary point perpendicularly into the interior of M. How ...
10
votes
3answers
2k views

Circle bundles over $RP^2$

Does anybody know if orientable, closed $3$-manifolds that are circle bundles over $RP^2$ have been classified? One can determine the isomorphism classes of bundles using obstruction theory, but I am ...
9
votes
5answers
1k views

Global Proof of Serre Duality

Does anyone know of a global proof (involving no local argument) of Serre Duality at the level of varieties or manifolds (as opposed to schemes).
51
votes
6answers
6k views

Kahler differentials and Ordinary Differentials

What's the relationship between Kahler differentials and ordinary differential forms?
20
votes
4answers
5k views

Formal Geometry

[edit: I posted an answer to this which summarizes one that I received verbally a few weeks after posting this question. I hope it is useful to someone.] I am presently seeking references which ...
6
votes
3answers
853 views

Where does the generic triangle live?

This is a reformulation of my question Characterizing triangles unembeddedly. Motivation 1: There is such a thing as a generic group. In category theory this is done by constructing "theory" of ...
19
votes
5answers
2k views

Principal bundles, representations, and vector bundles

What is the exact relationship between principal bundles, representations, and vector bundles?
8
votes
1answer
844 views

Hodge Index Theorem for Gr(n,k)

I'm trying to understand the Hodge-Index Theorem at the moment. What does it say explicitly for the case of the complex Grassmannian Gr($n,k$), and can this be established without recourse to the ...
3
votes
1answer
278 views

Hodge-Index Theorem for $\mathbb{C}P^2$

I'm trying to understand the Hodge-Index Theorem at the moment. What does it say explicitly for the case of $\mathbb{CP}^2$?
21
votes
0answers
1k views

Milnor's cartography problem

Let $\Omega$ be a round disc of radius $\alpha<\pi/2$ on the unit sphere $\mathbb{S}^2$. It is easy to construct a $(1,\tfrac{\alpha}{\sin\alpha})$-bi-Lipschitz map from $\Omega$ to the plane. Is ...
22
votes
5answers
2k views

Algebraic description of compact smooth manifolds?

Given a compact smooth manifold $M$, it's relatively well known that $C^\infty(M)$ determines $M$ up to diffeomorphism. That is, if $M$ and $N$ are two smooth manifolds and there is an $\mathbb{R}$-...
12
votes
8answers
5k views

Lie Groups and Manifolds

I'm trying to get a better handle on the relation between Lie groups and the Manifolds they correspond to. Firstly, is the relationship injective? that is, does each Lie group correspond to a unique ...
5
votes
1answer
1k views

Are smooth functions on an uncountable sum continuous?

Consider the linear space $\sum_{\mathbb{R}} \mathbb{R}$. As in the Frolicher-Kriegl-Michor view, we make this into a Frolicher space as follows. Equip it with the locally convex topology of the ...
9
votes
4answers
940 views

How are invariants represented in category theory?

I'm trying to better understand how to think about invariance in the setting of category theory. In some cases it seems there's an obvious interpretation: for instance, the fundamental group $\pi_1$ ...
3
votes
6answers
1k views

Dolbeault cohomology

Hello I am trying to get a good book that explains the Dolbeault Cohomology, does anyone know of a good one?
9
votes
3answers
1k views

Minimize Perimeter(S)/Area(S) for S inside the unit square.

This is a very silly question. For all regions S contained inside the unit square, what is the infimum of the quantity Perimeter(S)/Area(S)? This ratio being considered is not scale invariant, so it ...
7
votes
1answer
409 views

Is the space of nondegenerate classical paths connected?

I have a fairly specific question. My intuition says the answer is "yes", but there is a natural generalizations in which I take out all the "physics", and then I think the answer is "no". Edit ...
5
votes
5answers
947 views

A walk on a compact 2D surface embedded in 3-space that never returns home

At the risk of asking an uninformed question... Imagine an ant on a compact two-dimensional surface embedded in 3-space. The ant is placed at a point on the surface with random orientation. Once ...
38
votes
6answers
4k views

Universal definition of tangent spaces (for schemes and manifolds)

Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the ...
11
votes
3answers
3k views

Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?

Let $X$ be a complex analytic space. It is a 'well known fact' that the categories of local systems on $X$ (i.e. locally constant sheaves with stalk $C^n$), and of (holomorphic) vector bundles on $X$ ...
21
votes
4answers
2k views

Minimal surface in a ball

Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space and $r$ is the distance from $\Sigma$ to the center of the ball. Is it true that $$\mathop{\rm area} \Sigma\ge ...
6
votes
1answer
623 views

Infinite dimensional Newlander-Nirenberg theorem

The Newlander-Nirenberg theorem states that an almost complex structure is integrable if and only if the Nijenhuis tensor vanishes. I heard that this statement is not true in infinite dimensions, ...
11
votes
2answers
2k views

What is the geometric significance of Cartan's structure equations?

The Cartan structure equations for a connection and various associated 1-forms can be checked in a straightforward algebraic manner. But is there a geometric or global significance to the equations- ...
58
votes
7answers
11k 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....
3
votes
1answer
229 views

limits of algebraic varieties

I'm looking for a reference which deals with limits of families of algebraic varieties as the degree increases (or at least keywords from this subject). For the kind of example I have in mind, ...
11
votes
3answers
2k views

Non-Lie Subgroups

A result of Borel and Lichnerowicz states that the holonomy group of a connection on a principal $G$-bundle is a Lie subgroup of $G$ (Cartan had earlier asserted this, but apparently without proof). ...
6
votes
1answer
700 views

simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex. Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...
6
votes
3answers
696 views

Intrinsic characterization of a star shaped domain

Let A be a closed (compact no boundary), embedded (no self intersections), smooth surface of R^3. We say that the interior of A is star shaped if there exists a point p in A, such that for any point q ...
4
votes
2answers
1k views

(how) are vector bundles and homotopy groups related?

Hello, homotopic maps induce isomorphic pullback bundles, and so isomorphism classes of vector bundles over X correspond to homotopy classes of maps from the grassmannian to X. But I think, in the ...
23
votes
6answers
3k views

Least number of charts to describe a given manifold

Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it. E.g. a circle requires at least two charts, and ...
10
votes
4answers
845 views

Abel's equation for the dilog

Abel's identity for the dilogarithm (see the wikipedia page about polylogarithms) plays a role in web geometry as it is one of the abelian relations of the first example of exceptional web (Bol's 5-...
4
votes
1answer
468 views

Lower bound on volume of minimal hypersurface contained in a unit ball with curvature bounds

I was just wondering, if I have a geodesic ball of radius one in a manifold M whose sectional curvature lies between -epsilon and epsilon for epsilon small, and the injectivity radius of my manifold ...
12
votes
3answers
1k views

In how many ways can an iterated tangent bundle (T^k)M be viewed as a fibre bundle over (T^(k-1))M?

Let M be a smooth manifold. The double tangent bundle, TTM,can be viewed as a fibre bundle over TM in two ways, with the projection maps given by T_πM (i.e. the derivative of the projection from TM ...
12
votes
2answers
629 views

Is a smooth closed surface in Euclidean 3-space rigid?

Classical theorem of Cohn-Vossen: A closed convex surface in Euclidean 3-space cannot be deformed isometrically. Robert Connelly found an example of a polyhedral surface that can be deformed ...
67
votes
10answers
22k 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 ...
1
vote
2answers
2k views

Gaussian curvature of a z=f(x,y) function [closed]

This is not a homework question. Just wondering if there is a general formula for the gaussian curvature at point (x,y,f(x,y)) in terms of x, y, and f(x,y). I didn't see any thing like that on the ...
37
votes
10answers
6k views

Looking for an introduction to orbifolds

Is there any source where the basic facts about orbifolds are written and proved in full detail? I found the article by Satake "The Gauss-Bonnet Theorem for V-manifolds", but I'd like to have a more ...
51
votes
11answers
7k 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 ...
18
votes
5answers
2k views

Point singularity of a Riemannian manifold with bounded curvature

Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold ...
7
votes
1answer
469 views

Universal covers of domains in complex projective space

The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
58
votes
2answers
6k 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 ...
8
votes
3answers
683 views

Geometric calculations using Grassmann variables

Physicists seem to get huge computational value by introducing Grassmann variables and Grassmann integration into differential geometric calculations. See: http://en.wikipedia.org/wiki/...