The holonomy tag has no usage guidance.

**3**

votes

**0**answers

122 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: $0 \to U(1) \to \tilde G \to G \to 0$
...

**5**

votes

**0**answers

78 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 ...

**4**

votes

**2**answers

143 views

### Are Wolf spaces flag manifolds?

It's all in the title: Are Wolf spaces flag manifolds? Both are group quotients of semi-simple Lie groups. In the Grassmannian case this is so, and I always tacitly assumed it extended to the general ...

**7**

votes

**1**answer

326 views

### How algebraic is the holonomy map?

Let $G$ be either a compact, simple, simply-connected Lie group, or a simply-connected complex reductive group (so either $SU(n)$ or $SL(n,\mathbb{C})$, for instance), or even complex affine. I don't ...

**2**

votes

**2**answers

198 views

### Connections having the same holonomy along loops at a point

I find myself stuck with the following question, which seems very classical but for which I have not been able to find a reference.
Consider a smooth vector bundle $E$ of rank $r$ over a compact ...

**1**

vote

**0**answers

95 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) ...

**1**

vote

**1**answer

160 views

### Isomorphisms of Positive and Negative Spinor Bundles

Here is an extract of the doctoral thesis of C. Lewis under the supervision of D. Joyce (https://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf, 1998):
2.6 Spin Bundles and the Dirac Operator
...

**4**

votes

**0**answers

315 views

### 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 ...

**8**

votes

**1**answer

323 views

### Dubins car shortest paths: Decidable?

A Dubins car follows a
Dubins path
in $\mathbb{R}^2$, with constant wheel speed and
limited turning radius.
It is known that the shortest Dubins path in the absence
of obstacles follows circular
arcs ...

**1**

vote

**1**answer

247 views

### Possible homotopy-theoretical approach to Gauss-Bonnet

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...

**17**

votes

**1**answer

491 views

### Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...

**8**

votes

**1**answer

297 views

### Are there principal $G$-bundles whose holonomy group is $G$?

While studying universal constructions on principal bundles, I've stuck on a quite a basic question, namely:
Given a Lie group $G$, does there exist a principal $G$-bundle $\pi \colon P \to B$, for ...

**-1**

votes

**1**answer

219 views

### Holonomy group of a fiber bundle

Let $M$ and $N$ be a Riemannian manifold. Assume $N$ is flat. Let $G$ be a finite group acting on $M$ and $N$ as an isometry group. Assume that the $G$-action on $N$ is free. Then we have new ...

**6**

votes

**1**answer

205 views

### Holonomy group of Enriques surface

I expect that the holonomy group of an Enriques surface $S$ is $SU(2)\times C_2$. I think this can be proven by the fact that its double cover, which is a K3 surface, has the full $SU(2)$ holonomy, ...

**9**

votes

**1**answer

753 views

### Why Yau's theorem implies the existence of hyperkähler metric on complex symplectic manifolds?

Every expository article on hyperkähler manifolds that I have read states without detailed proof the following fact:
It follows from Yau's theorem (i.e. a compact Kähler manifold $M$ with $c_1(M)=0$ ...

**3**

votes

**0**answers

334 views

### Representation variety vs. space of flat connections

The holonomy provides a bijection from
the space of flat G-connections (modulo gauge equivalence) on a trivial G-bundle over M
to
a connected component of the representation variety ...

**9**

votes

**1**answer

271 views

### Minimum requirements for a Kähler manifold to be hyperkähler

In 'panoramic view of Riemmannian geometry' when introducing hyperkähler manifolds, Berger states, informally, that a hyperkähler manifold is a Riemmannian manifold which is Kähler for more than one ...

**1**

vote

**1**answer

166 views

### Holonomy groups of quotient Riemannian manifolds?

Let $(X,g)$ be a Riemannian manifold with holonomy group $Hol(X,g)$. Suppose that a finite group $G$ acts on $X$ freely and the metric $g$ is invariant under $G$. What can one say about the the ...

**13**

votes

**1**answer

385 views

### Algebraic characterization of the curvature operator of symmetric spaces

My question is the following :
Given an algebraic curvature operator $R\in S^2_B(\Lambda^2\mathbb{R}^n)$, is there an a simple criterion to know if this curvature operator can occur as the ...

**26**

votes

**3**answers

4k views

### What is the difference between holonomy and monodromy?

What is the difference between holonomy and monodromy?
And what is the simplest example in which one is trivial and the other is not?

**2**

votes

**1**answer

701 views

### Isometry groups of Riemannian submersions with totally geodesic fibers

Suppose $F\to M\stackrel{\pi}{\to} B$ is a Riemannian submersion with totally geodesic fibers, all manifolds compact. In general, unless $M=B\times F$ is a Riemannian product, the isometry groups of ...

**1**

vote

**2**answers

356 views

### Holonomy group of $\mathbb{O}P^1$

What is the holonomy group of the 1-dimensional octonionic projective space ?

**3**

votes

**1**answer

606 views

### Holonomy Groups and the Hopf Fibration

I am trying to understand holonomy groups at the moment and am focusing on the example of the Hopf fibration $SU_2 \to S^2$. Since $S^2$ is path connected we can talk about the holonomy group of a ...

**16**

votes

**2**answers

2k views

### Are there Ricci-flat riemannian manifolds with generic holonomy?

This may well be an open problem, I'm not sure.
In Berger's classification (refined by Simons, Alekseevsky, Bryant,...) of the holonomy representations of irreducible non-symmetric complete ...