All Questions
Tagged with holonomy dg.differential-geometry
52 questions
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
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 ...
- 12.9k
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 $1$ ...
- 26.3k
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 ...
- 12.8k
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
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\...
- 108k
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\...
- 108k
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
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 ...
- 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-...
- 271
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 ...
- 63.9k
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 ...
- 63.9k
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(\...
- 108k
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}(...
- 3,652
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 ...
- 141
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 \...
CommunityBot
- 1
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 ...
- 188k
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
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 ...
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 ...
- 14.3k
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 ...
- 6,023
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 ...
- 37.8k
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
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?
- 12.3k
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 ...
- 4,713
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 $(M,g_M)$, ...
- 108k
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 ...
- 1,466
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 ...
- 6,023
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 ...
- 13.5k
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 ...
- 26.3k
2
votes
3
answers
613
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 ...
- 26.3k
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-...
- 26.3k
2
votes
2
answers
442
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 ...
- 829
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
3
votes
1
answer
927
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 $Hom(\...
- 8,529
5
votes
0
answers
114
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 ...
- 66.3k
3
votes
0
answers
136
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 ...\...
- 369
4
votes
2
answers
505
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 ...
CommunityBot
- 1
4
votes
2
answers
978
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 ...
- 4,787
8
votes
1
answer
1k
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 ...
- 151k
20
votes
1
answer
1k
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 ...
CommunityBot
- 1
1
vote
1
answer
479
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 ...
CommunityBot
- 1
8
votes
1
answer
400
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, ...
- 108k
12
votes
1
answer
1k
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 $c_1(M)=0$ ...
- 1,804
11
votes
1
answer
540
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 ...
- 108k
14
votes
1
answer
737
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 ...
- 108k