Questions tagged [holonomy]

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Holonomy as a right adjoint, monodromy as a left adjoint

This question about the difference between holonomy and monodromy has an interesting answer by Ronnie Brown. An excerpt: So holonomy comes out as a kind of right adjoint, and monodromy as a kind ...
Arrow's user avatar
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7 votes
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199 views

No intermediate denominators growth for holonomic functions?

My question concerns holonomic sequences of rational numbers, meaning assignments $a(n) : \mathbb{N} \to \mathbb{Q}$ fulfilling a linear recurrent relation of the form $$ a(n+k) = \sum_{i=0}^{k-1} p_i(...
Vesselin Dimitrov's user avatar
6 votes
0 answers
345 views

Is central extension of a group equivalent to a bundle with gauge field?

Let $\tilde G$ be a central extension of a group $G$ by $U(1)$. One common elegant definition is that there should be a short exact sequence of groups: $0 \to U(1) \to \tilde G \to G \to 0$ However,...
Ruben Verresen's user avatar
6 votes
0 answers
226 views

On Holonomy in (regular) Riemannian Foliations

Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid: Let $\mathcal{F}\subset M$ be a ...
Llohann's user avatar
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6 votes
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Isometries of hyper-Kähler manifolds

For the purposes of this question, a hyper-Kähler manifold will be a complete connected Riemannian manifold $(\mathcal{M},g)$ whose holonomy representation is isomorphic to the natural representation ...
Paul Reynolds's user avatar
5 votes
0 answers
138 views

Einstein metrics on spheres

We know that a closed oriented manifold $M$ carries a Lorentzian metric iff the Euler characteristic vanishes. My question concerns the existence of those Lorentzian metrics on odd-dimensional spheres ...
Julian Seipel's user avatar
5 votes
0 answers
189 views

Foliations, von Neumann algebras and measurability

In the excellent book Noncommutative Geometry by Alain Connes much of the first chapter is devoted to foliations. At the end of the first chapter the author discusses index theory on measured ...
truebaran's user avatar
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5 votes
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114 views

Holonomy group of a dense, open submanifold

I have a Riemannian manifold $(X,g)$, where $X$ is not necessarily compact or complete, and a dense open submanifold $Y \subseteq X$. In my case $X$ is a smooth quasi-projective variety over the ...
Stefan Kebekus's user avatar
3 votes
0 answers
244 views

Manifolds and Riemannian geometry with a bundle viewpoint

I was wondering if there are any books that builds the theory on manifolds and Riemannian geometry, but at the same time treats these subjects in the general case of bundles (similar to Jeffery Lee's ...
Master.AKA's user avatar
3 votes
0 answers
105 views

Concerning the definition of a 2-crossed module

Question: Is there some generalization of the definition of crossed-module which appropriately fits into the holonomy-considerations I am interested in and has, as an example, the generalization of ...
cheyne's user avatar
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3 votes
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Applications of Ambrose-Singer theorem on holonomy

I am planning to introduce to a group of Graduate students the notion of connections on Principal bundle, curvature of connection, Holonomy. I want to conclude with the statement of Ambrose-Singer ...
3 votes
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Splitting of totally geodesic Riemannian foliations

Let $\mathcal F$ be a non-singular Riemannian foliation on $(M,g)$ whose leaves are totally geodesic. Suppose further that the leaves are Riemannian products of irreducible manifolds $L=L_0\times ...\...
Llohann's user avatar
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A specific question in $G_2$ geometry

Let's take a closed $G_2$ manifold $(M,\Phi)$. $\Phi$ denoting the three-form which defines the $G_2$ structure on $M$. Let's take a closed two form $\theta\in\Omega^2(M).$ Is \begin{align} d(\theta_{\...
Partha's user avatar
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2 votes
1 answer
166 views

Leaf holonomy of Reeb foliation on mobius strip

I am trying to understand the leaf holonomy of the Reeb foliation on the mobius strip, the first problem being visualization. I have been unable to find a visualization of this anywhere. I am ...
Ralphie Chadwick's user avatar
2 votes
0 answers
98 views

Parallelism defect

I have a question that I don't know how to answer. If I have a parallelism defect it is due to the presence of a curvature and therefore we can bring it back to a Riemann tensor. The thing that is not ...
MathDG's user avatar
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222 views

Bryant-Salamon $G_2$ manifold on the spinor bundle over $S^3$

I am trying to understand the spaces constructed in R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy. My first problem is, essentially, about ...
F.T.'s user avatar
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2 votes
0 answers
154 views

Holonomy of hypercomplex manifold

The following is a quote from M. Barberis, I. Dotti, M. Verbitsky, Canonical Boundles of Complex Nilmanifolds with Applications to Hypercomplex Geometry, Math. Res. Lett., 16(2), 331-347, 2009. "Not ...
J.E.M.S's user avatar
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122 views

Calibrated submanifolds in Spin(7) and Calabi-Yau threefold

Suppose I have a Cayley cycle in a $Spin(7)$ holonomy manifold $M$, i.e. a calibrated submanifold. In the special case that $M=CY_3\times T^2$, is it possible that the Cayley cycle reduces to a Kahler ...
sam's user avatar
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1 vote
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Spin(7)-instanton

Let $M$ be a Spin$(7)$-manifold with a spin-bundle $S=S_+\oplus S_-$. There's an obvious connection on $S$ which comes from lifting the Levi-Civita connection. And it induces a connection on the ...
Partha's user avatar
  • 759
1 vote
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Is Stenzel's Ricci-flat metric on $T^*\mathbb{CP}^n$ hyperkahler?

In a well-known paper, Stenzel constructed complete Ricci-flat Kahler metrics on the total spaces of cotangent bundles of $S^n$, $\mathbb{RP}^n,$ $\mathbb{CP}^n$, $\mathbb{HP}^n$, and $\mathbb{OP}^2$. ...
Jesse Madnick's user avatar
1 vote
0 answers
147 views

Fundamental $1$-form for a Riemannian manifold?

Take a Hermitian manifold $(M,I,g)$ where $I$ is the complex structure and $g$ is the Hermitian metric. The associated fundamental $2$-form $ g(\cdot,I(\cdot)) $ captures a lot of the information ...
Dick Johnson's user avatar
1 vote
0 answers
79 views

Incomplete Riemannian-product manifold with group holonomy strict G2

Let $M_1=\mathbb{R}^2 \times K^4$ a Ricci-flat Riemannian-product manifold, where with $K$ I mean $k3$ kummer surface ($1$-complex dimension or $2$ real dimensions), so with $K^4$ I mean real 4-...
MathDG's user avatar
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1 vote
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Is there a class of non-symmetric spaces $G/SO(n)$ such that $Hol(G/SO(n))=SO(n)$?

Is there a certain class (or unique construction of a set) of simply connected non-symmetric spaces $G/SO(n)$ such that the Riemannian holonomy is $Hol(G/SO(n))=SO(n)$? (That is, is there a way to ...
Sergio Charles's user avatar
1 vote
0 answers
160 views

What is the intersection of Spin(7) and U(4)?

I'm just curious from Berger's classification of Riemannian holonomy, how do Spin(7) manifolds intersect the other types of Riemannian manifolds? In particular, what is the intersection of Spin(7) ...
seub's user avatar
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Does every Spin$(7)$-manifold has a unit-length spinor?

Say $M$ be a manifold with a Spin$(7)$-structure. $M$ is spin and hence spin$^c$. Say $S=S_+\oplus S_-$ be a spin$^c$-bundle on $M$. Does $S_+$ has a nowhere vanishing section? The result is true if ...
Partha's user avatar
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Confusion on a term related to connection and holonomy

This question is simply some of my confusions about a specific term. Let $E\to X$ be a trivial complex vector bundle. When one says let $\nabla^E$ be a connection on $E\to X$ with trivial holonomy (...
Ho Man-Ho's user avatar
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The specific connection between the Hecke operator and the t'Hooft Operator

As I was reading some articles concern about the Selberg trace formula and its general form, I have noticed that the Selberg trace formula and its general form can be understand as the energy spectrum ...
loveimissyou123's user avatar
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360 views

hypergeometric representation of Hermite $H_n(x)$

The DLMF has a hypergeometric representation for the Hermite polynomial $H_n(x)$ for real $x$, apparently. $$H_n(x)=(2x)^n{}_2F_0\left(\frac{-\tfrac12n;-\tfrac12n+\tfrac12}{};-\frac{1}{x^2}\right)$$ (...
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