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1answer
113 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 ...
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227 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 ...
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0answers
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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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1answer
217 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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1answer
231 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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1answer
443 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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1answer
224 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 ...
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1answer
192 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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1answer
178 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, ...
8
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1answer
591 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
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0answers
281 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
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1answer
248 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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1answer
147 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
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1answer
343 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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1answer
634 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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2answers
345 views

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

What is the holonomy group of the 1-dimensional octonionic projective space ?
3
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1answer
558 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 ...
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2answers
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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 ...