# Questions tagged [dg.differential-geometry]

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

5,708 questions

**7**

votes

**5**answers

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

**3**answers

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

**1**answer

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

**2**answers

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

**10**answers

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

**4**answers

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

**3**answers

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

**1**answer

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

**3**answers

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

**3**answers

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

**5**answers

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

**6**answers

6k views

### Kahler differentials and Ordinary Differentials

What's the relationship between Kahler differentials and ordinary differential forms?

**20**

votes

**4**answers

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

**3**answers

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

**5**answers

2k views

### Principal bundles, representations, and vector bundles

What is the exact relationship between principal bundles, representations, and vector bundles?

**8**

votes

**1**answer

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

**1**answer

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

**0**answers

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

**5**answers

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

**8**answers

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

**1**answer

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

**4**answers

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

**6**answers

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

**3**answers

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

**1**answer

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

**5**answers

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

**6**answers

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

**3**answers

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

**4**answers

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

**1**answer

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

**2**answers

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

**7**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**3**answers

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

**2**answers

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

**6**answers

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

**4**answers

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

**1**answer

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

**3**answers

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

**2**answers

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

**10**answers

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

**2**answers

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

**10**answers

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

**11**answers

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

**5**answers

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

**1**answer

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

**2**answers

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

**3**answers

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/...