Questions tagged [holonomy]
The holonomy tag has no usage guidance.
81 questions
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 ...
- 954
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)...
- 954
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 ...
- 401
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 ...
- 954
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_{\...
- 954
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\...
- 954
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 ...
- 293
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\...
- 213
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 ...
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 ...
- 5,230
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 (...
- 1,173
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 $\mathbb{OP}^2$.
...
- 871
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 ...
- 179
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-...
- 9,237
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 ...
- 131
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 ...
- 272
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 ...
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}(...
- 23
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 ...
- 203
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(\...
- 401
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 ...
- 203
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-...
- 272
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 ...
user117832
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 ...
- 213
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 ...
- 553
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(...
- 13.8k
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 ...
- 277
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 ...
- 463
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 ...
- 2,533
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 ...
- 10.5k
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 ...
- 5,230
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 ...
- 6,023
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 ...
- 1,466
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)$, ...
user117832
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 ...
- 14.4k
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$ ...
- 6,023
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 ...
- 303
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 ...
- 239
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$-...
- 239
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/...
- 643
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)...
- 7,902
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 ...
- 6,023
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 \...
- 31
3
votes
1
answer
550
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 ...
user117832
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 ...
- 437
5
votes
1
answer
568
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 ...
user117832
12
votes
1
answer
406
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-...
- 9,237
5
votes
0
answers
207
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 ...
- 9,330