# Questions tagged [holonomy]

The holonomy tag has no usage guidance.

66
questions

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### What is the relationship between $\mathrm{SO}(2)$ and $\mathrm{PSL}(2,\mathbb{R})$?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$The holonomy of a hyperbolic surface $S$ in terms of differential geometry is either $\SO(2)$ or $\mathrm{O}(...

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115 views

### Quaternion-Sasakian manifolds and special holonomy Sasakian manifolds

Two well-known slogans are
A Sasakian manifold is the odd dimensional analogue of a Kähler manifold
and
A $3$-Sasakian manifold is the odd dimensional analogue of a hyper-Kähler manifold
Does this ...

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99 views

### Quaternonic Kähler Chern connection

For a Riemannian manifold, the natural connection is of course the Levi-Civita connection. For a complex manifold, the natural connection is the Chern connection, which coincides with the Levi-Civita ...

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70 views

### Holonomy of a triangle in an affine symmetric space

Let $G/H$ be an affine symmetric space with involution $\sigma$, and $\mathfrak{g}=\mathfrak{m}\oplus \mathfrak{h}$ the Cartan decomposition of its Lie algebra. We can identify $G/H$ and $\exp(\...

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95 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 ...

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66 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-...

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659 views

### Are there mistakes in Kovalev's “Twisted connected sums and special Riemannian holonomy”?

This is kind of a strange and vague question... sorry about that.
I am really interested in $G_2$ Twisted Connected sums as described in this paper: https://arxiv.org/abs/math/0012189 "Twisted ...

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153 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 ...

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194 views

### Integrability condition for flat connections

I am reading Kobayashi's book "Differential geometry of complex vector bundles". More precisely, I am on section 2 of chapter 1, page 5.
Kobayashi is trying to prove that if $E$ is a vector ...

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108 views

### Holonomy groups of Hermitian, and hyper-Hermitian, manifolds

An $n$-dimensional complex manifold $M$, endowed with an Hermitian metric $g$, is Kähler if and only if the holonomy group of $g$ is contained in $U(n)$. If $g$ is Hermitian but not Kähler do we ...

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78 views

### Does holonomy determine parallel transport? [duplicate]

Let $p: P \longrightarrow M$ be a smooth $G$-principal bundle endowed with a connection that determines the holonomies:
$$\Phi_{\gamma}: P_{x} \overset{\cong}{\longrightarrow} P_{x}$$
for any fiber ...

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168 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(...

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226 views

### Examples of Calabi-Yau manifolds with $\mathbb{T}^2$ symmetry

I want to know if there exists examples of Calabi-Yau manifolds with $\mathbb{T}^2$-invariant $SU(n)$-structure. In particular these actions are both Killing and holomorphic. I am especially ...

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201 views

### A consequence of Ambrose-Singer theorem on holonomy

Consider $\nabla$ a connection in a vector bundle above a smooth manifold $M$.Consider a local frame $\sigma=(\sigma_1, \sigma_2,...,\sigma_m )$ on a contractible open set $U\subset M$ and calculate ...

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493 views

### Flat connections, curvature and holonomy

Let $A$ be a flat connection on a principal $G$-bundle $G\hookrightarrow P\to M$.
Consider an homotopically trivial loop $\gamma \subset M$. For simplicity, suppose $\gamma = \partial D$ is the ...

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195 views

### 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 ...

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285 views

### Nowhere vanishing section implies reduction of structure group

Description
I noticed a repeating theme in vector bundle theory, and wonder if there's a theorem that describes this kind of phenomenon.
Given a vector bundle $E$ over a manifold $X$. If there is a ...

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295 views

### Holonomy map on a connected manifold determines the connection and the bundle

I am reading Parallel transport on principal bundles over stacks. I quote from their paper :
Let $G$ be a Lie group and $M$ a $C^{\infty}$ manifold. Recall that a
choice of a connection $1$-form ...

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84 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 ...

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179 views

### Holonomy of a Warped Product Metric

A warped product metric on the "cone" $\tilde{M} = \mathbb{R}^{+} \times M$ is $\tilde{g} =dr^2 + r^2g_M$ where $g_M$ is the metric on $M$.
If we know the holonomy group of the manifold $(M,g_M)$, ...

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30 views

### $C^1$ partially hyperbolic diffeomorphism have Hölder stable holonomies (reference request)

I have spent an insane amount of time searching for a preprint I have printed a few months ago but misplaced. I cannot find it anymore and this drives me crazy.
It might not have been meant for ...

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### Alternative (easier) Proof of Ambrose Singer Holonomy theorem

Let $P(M,G)$ be a principal bundle. Giving a connection on $P(M,G)$ means two equivalent things. One as an assignment of subspace of $T_pP$ for each $p\in P$ and another as a $\mathfrak{g}$ valued $1$ ...

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366 views

### 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 ...

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119 views

### Flatness in a neighborhood of a point condition

Suppose that we have a Riemannian Manifold $(M,g)$ whose
curvature vanishes in an open neighborhood U of a point p.
When does this imply that the metric is Flat ?
In particular, does it happen ...

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132 views

### 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 ...

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453 views

### How does one complexify a real $n$-dimensional Riemannian manifold $(M,g)$?

If $V$ is a real vector space, then the complexification of $V$ is formally defined as $V^{\mathbb{C}}=V\otimes_{\mathbb{R}}\mathbb{C}$. Is there an analogous complexification operation for a real $n$-...

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269 views

### Holonomy groups of compact Riemannian symmetric spaces

Let $M$ be a compact Riemannian symmetric space. By the classification of Cartan, it belongs to the table of homogeneous spaces given in the Wikipedia page:
https://en.wikipedia.org/wiki/...

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### Riemannian holonomy of generic manifolds

It is well known, as well as absolutely intuitive, that the Riemannian holonomy of a generic Riemannian manifold is $O(n)$, the Riemannian holonomy of a generic orientable Riemannian manifold is $SO(n)...

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273 views

### holonomy of connection on gerbes

I am reading this notes of Hitchin to understand about gerbes. He defines gerbe by giving a collection of $2$ cocycles $g_{\alpha\beta\gamma}:U_\alpha\cap U_\beta\cap U_\gamma\rightarrow S^1$ with ...

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245 views

### $\DeclareMathOperator\SU{SU}$$\SU(2)\times \SU(2)$ invariant $\SU(3)$-structure on $\{t\} \times M^6$

$\DeclareMathOperator\SU{SU}$I am reading Jason Lotay and Goncalo Oliveira's paper $\SU(2)^2$ invariant $G_2$-instantons, and have a few questions from the same.
If we consider the space $M = S^3 \...

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391 views

### English translation of paper: Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)

I am interested in the history of $G_2$ manifolds and want to read this paper in english:
Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)
Does anyone know where I can find a ...

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122 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 ...

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358 views

### Killing vector fields on a compact $G_2$ manifold

I am trying to show that there can not be any nonvanishing Killing vector fields on a compact $G_2$ manifold.
For the definition of a $G_2$ manifold just see the Wikipedia page.
I know that since ...

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290 views

### Compact quaternionic Kahler manifolds of negative curvature: examples

There is a well known problem of LeBrun-Salamon:
are there any non-symmetric compact quaternionic-Kahler
manifolds of positive scalar (and Ricci) curvature?
It is hard and still unsolved:
Quaternionic-...

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134 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 ...

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### 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 ...

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489 views

### Manifolds with special holonomy especially $G_2$

I am interested in learning about $G_2$ manifolds and am aware that one of the canonical references is Joyce's Compact Manifolds with Special Holonomy. I am certain that my background is, at this ...

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108 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 ...

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116 views

### 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 ...\...

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### Analytic conjugacy of vanishing holonomy groups implies analytic conjugacy of foliations

i am reading and trying to do some exercises and problems of the book Lectures on Analytic Differential Equations- Y. Ilyashenko, S. Yakovenko.
I can not solve the problem 11.6 that says
Consider ...

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230 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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373 views

### Holonomy of a Ricci-flat affine connection

There is some link between Ricci-flatness and reduction of holonomy. For example a Kahler manifold is Ricci-flat if and only if it has at most $SU(n)$ holonomy rather than $U(n)$, and it's apparently ...

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280 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,...

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171 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 ...

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232 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 ...

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### 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 ...

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631 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 ...

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150 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) ...

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### 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
To ...

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671 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 ...