# Questions tagged [dg.differential-geometry]

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

5,708 questions
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### (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 ...
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### 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 ...
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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...