The holonomy tag has no wiki summary.

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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 ...

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### A technical question in Feix's construction of hyperkahler metric on cotangent bundles

I am now reading Feix's paper Hyperkahler metrics on cotangent bundles
and I have a technical question to ask.
In his paper, for an analytic Kähler manifold $(X,J,\omega)$, Feix considered its ...

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**1**answer

187 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 ...

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**1**answer

223 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 ...

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**1**answer

422 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 ...

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**1**answer

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### 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 ...

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votes

**1**answer

180 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 ...

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**1**answer

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### 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, ...

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**1**answer

575 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 $c_1(M)=0$ ...

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votes

**0**answers

266 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 ...

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**1**answer

241 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 ...

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**1**answer

141 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 ...

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**1**answer

335 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 ...

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votes

**1**answer

619 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 ...

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**2**answers

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

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

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**1**answer

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### 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 ...

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### 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 ...