Skip to main content

Questions tagged [holonomy]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
7 votes
2 answers
834 views

Holonomy as integration of curvature for principal $G$-bundles?

Holonomy and curvature may seem to be slightly advanced topics in geometry. However, their origins are easily imaginable. Namely, picture the surface of earth $S$, and pick an arbitrary contractible ...
3 votes
0 answers
118 views

Decomposition of forms in $\operatorname{SU}(4)$-manifold

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let $(X,\Omega,\omega,J)$ be a manifold with an $\SU(4)$ structure. Since $\SU(4)\subset\Spin(7)$, $X$ also has a $\Spin(7)$-structure. I ...
3 votes
4 answers
3k views

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 $1$ ...
2 votes
1 answer
241 views

Leaf holonomy of Reeb foliation on Möbius strip

I am trying to understand the leaf holonomy of the Reeb foliation on the Möbius strip, the first problem being visualization. I have been unable to find a visualization of this anywhere. I am ...
3 votes
0 answers
281 views

A question in $\operatorname{Spin}(7)$ geometry

$\DeclareMathOperator\Spin{Spin}$I am looking for a proof of a fact (I think it's true intuitively due to representation theory) in $\Spin(7)$ geometry. Let's take a closed $\Spin(7)$-manifold $(M^8,g)...
2 votes
0 answers
45 views

De Rham product decomposition theorem in a particular setting

Let $G$ be a Lie group, and $H$ a Lie subgroup of $G$ such that $G/H\sim \mathcal M$ is a homogeneous space diffeomorphic to $\mathbb{R}^n$, equipped with an invariant Riemannian metric $g$. If this ...
1 vote
0 answers
72 views

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 ...
2 votes
0 answers
109 views

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_{\...
2 votes
1 answer
199 views

Decomposition of forms on a Spin$(7)$ manifold

Let's take a $G_2$ manifold $(M,\Phi)$, then we get a Spin$(7)$ manifold by taking $(M\times\mathbb{R},\Psi:=\Phi\wedge dt+*_M\Phi),$ where $t$ is the coordinate in the $\mathbb{R}$-direction. $\Phi\...
2 votes
1 answer
306 views

Learning roadmap for holonomy theory

During my Master's thesis I encountered the theory of holonomy for the first time. Unluckily it was only tangentially related to the topic of my thesis, so I couldn't dive into it. The book I was ...
9 votes
1 answer
653 views

Does the curvature locally determine the connection?

Let $E$ be a Euclidean vector bundle over the unit ball centered at the origin $B^n(0)$. Let $\nabla$ and $\nabla'$ be two metric connections such that the curvatures coincide globally, i.e. $F_\nabla\...
0 votes
0 answers
87 views

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 (...
1 vote
0 answers
129 views

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 $\mathbb{OP}^2$. ...
3 votes
1 answer
551 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 ...
5 votes
0 answers
142 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 ...
6 votes
1 answer
462 views

Holonomy bounded in terms of area and the curvature

I suppose the following result follows from Ambrose-Singer theorem, but I cannot find a reference, and the arguments I found in the literature are usually weaker. The idea is that holonomy over a null-...
3 votes
0 answers
283 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 ...
2 votes
0 answers
101 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 ...
0 votes
0 answers
97 views

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 ...
5 votes
1 answer
162 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(\...
1 vote
1 answer
372 views

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}(...
1 vote
1 answer
138 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 ...
3 votes
1 answer
277 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 \...
1 vote
0 answers
159 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 ...
1 vote
0 answers
80 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-...
8 votes
1 answer
856 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 ...
2 votes
0 answers
230 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 ...
4 votes
2 answers
992 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 ...
7 votes
1 answer
735 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 ...
1 vote
1 answer
230 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 ...
7 votes
0 answers
203 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(...
4 votes
1 answer
332 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 ...
4 votes
1 answer
339 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 ...
9 votes
0 answers
263 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 ...
3 votes
1 answer
2k 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 ...
5 votes
1 answer
560 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 ...
3 votes
0 answers
124 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 ...
53 votes
3 answers
11k 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?
6 votes
0 answers
830 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 ...
3 votes
1 answer
1k 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 ...
2 votes
1 answer
248 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 $(M,g_M)$, ...
2 votes
1 answer
72 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 ...
3 votes
2 answers
374 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 ...
3 votes
0 answers
476 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 ...
3 votes
1 answer
174 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 ...
1 vote
0 answers
137 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 ...
8 votes
1 answer
750 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 $n$-...
6 votes
1 answer
465 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/...
7 votes
1 answer
963 views

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)...
2 votes
0 answers
167 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 ...