The failure of "parallel transport around a closed loop" to be the identity map. Studied in differential geometry, it is intimately tied with curvature.
Questions tagged [holonomy]
117 questions
39
votes
2 answers
Does the Levi-Civita connection determine the metric?
Can I reconstruct a Riemannian metric out of its Levi-Civita connection?
In other words: Given two Riemannian metrics $g$ and $h$ on a manifold $M$ with the same Levi-Civita connection, can I conclude that $g=h$ up to scalars?
If not, what can I say…
archipelago
- 4,448
13
votes
2 answers
What does holonomy measure?
I have difficulty understanding conceptually what holonomy measures.
it can return a phase shift of the vector transported parallel along the connection. If there is no phase shift, it means that the connection is flat, and if there is phase shift,…
user333046
12
votes
1 answer
Flat connection with non-trivial holonomy? I cannot get it
maybe this is a dumb question, but I cannot understand how a principal $G$-bundle can have non-trivial holonomy with a flat connection. Maybe I'm missing something, but doesn't Ambrose-Singer theorem say that the holonomy is generated by the…
user40276
- 5,543
12
votes
3 answers
Flat non-trivial $U(1)$-bundle? Is it possible?
maybe this is a very stupid question and I'm missing something very trivial.
It's well known that $U(1)$-bundles are classified by the Euler class or the first Chern class. More precisely, the isomorphism $$c: \check{H}^1 (X, \mathscr{C}^{\infty}…
user40276
- 5,543
10
votes
1 answer
Calibrations vs. Riemannian holonomy
I've began to study the relationship between calibrations and holonomy, mainly through D.D. Joyce's Riemannian Holonomy Groups and Calibrated Geometry and partly through internet material.
Pretty much everyone explains this relationship by the…
rmdmc89
- 10,709
- 3
- 35
- 94
9
votes
1 answer
Holonomy of the sphere
I saw an example in which the holonomy of $\mathbb{S}^n$ with the standard metric is calculated. I'm just starting to get familiar with holonomy groups and I wanted to know what could one do by knowing that the holonomy of the sphere is $SO(n)$.…
Sak
- 4,027
9
votes
1 answer
What does $\text{Sp}(n)\cdot\text{Sp}(1)$ mean in Berger's holonomy list?
This is probably a silly question.
I was looking at Berger's classification for holonomy groups, and the fourth element is "Quaternion-Kähler manifolds, $\,\dim M=4n, \,\text{Hol}=\text{Sp}(n)\cdot\text{Sp}(1)$".
First I tought…
rmdmc89
- 10,709
- 3
- 35
- 94
8
votes
2 answers
About two notions of holonomy
I have found something called "holonomy" in two apparently different contexts:
Let $M$ be a smooth manifold, $E\to M$ a vector bundle and $\nabla $ a connection on $E$. Then you have a notion of parallel transport, and thus a notion of holonomy as…
Lor
- 5,738
8
votes
1 answer
Can one exchange fibre and base space in a fibre bundle?
The first trivial example of a fibre bundle $E$ is a product bundle $E=F \times B$, with fibre $F$ and base space $B$. Of course in this trivial example, one can exchange base space and fibre and think of the product bundle $E$ as a fibre bundle…
Earthliŋ
- 2,592
8
votes
1 answer
Holonomy of Lie groups
Simple compact Lie groups have unique bi-invariant metrics. Hence, they are Riemannian manifolds in a unique way, so we can ask what is their holonomy group. Is there a relationship between the group $G$ and its holonomy group? For example, is the…
Simon Parker
- 4,421
8
votes
1 answer
Holonomy computation in $S^2$
If $\gamma$ is a closed Loop in $S^2$ and $p\in S^2$, where $\gamma$ is the boundary curve of some region $X$ in $S^2$ (and $\gamma$ satisfied some regularity conditions), someone told me that the holonomy map $H_\gamma:TS^2_p\rightarrow TS^2_p$ is…
Urs
- 81
7
votes
1 answer
holonomic D-modules
I am trying to develop an intuition about holonomic D-modules and find the literature formidable (I study physics). My question is, given a linear differential operator in n-variables, $x=(x_1,...,x_n)$ (using multi-index notation), $…
alphanzo
- 193
7
votes
1 answer
Holonomy and Differential Characters
This question is going to be rather vague, but I'm just trying to see if there are obvious connections between these two concepts.
So the holonomy of a vector bundle with Lie group $G$ is
$$h(A)=\mathcal{P}\exp\left(\int_\gamma A\right)$$
where…
levitopher
- 2,685
7
votes
1 answer
Berger's theorem on holonomy
Can someone clarify to me what the correct hypothesis of Berger's theorem are (if at all what I write is correct)?
Theorem: assume $M$ is a Riemannian manifold, with irreducible reduced holonomy group; moreover that $M$ is not homogeneous, and it is…
jj_p
- 2,500
- 1
- 15
- 31
7
votes
1 answer
Holonomy group of quotient manifold
Let $(M,g_M)$ be a compact Riemannian manifold with holonomy group $Hol(M,g_M)$. Suppose that a finite group $G$ acts on $M$ freely and preserves the metric $g$.
What can one say about the holonomy group $Hol(M/G,g_{M/G})$ of $M/G$ equipped with…
M. K.
- 5,191
- 2
- 33
- 55