The failure of "parallel transport around a closed loop" to be the identity map. Studied in differential geometry, it is intimately tied with curvature.

learn more… | top users | synonyms

2
votes
3answers
76 views

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 ...
0
votes
0answers
32 views

Derive the formula: $f(z)=2u(\frac{1}{2}z,\frac{1}{2i}z)-2u(0,0)$

Let $𝑢(𝑥, 𝑦)$ be a harmonic function which is the real part of a holomorphic function $𝑓 (𝑧)$, so that $$𝑢(𝑥, 𝑦)=\frac{1}{2}(f(z)+\overline {f(z)})$$ Argue that $\overline {𝑓(𝑧)} = ...
1
vote
2answers
31 views

Parallel displacement on principal bundles

Let $\pi : P \to M$ be a principal bundle with structure group $G$ and connection $\Gamma$. For a fixed $x \in M$, denote by $\Omega(x)$ the space of piecewise differentiable loops based at $x$. Every ...
1
vote
0answers
24 views

Flat $G$-structures have Hol=Id

Exercise I've been given the task to show, given a flat $G$-structure, we have that $\text{Hol}=\text{Id}$ (here "Hol" is the holonomy group; furthermore a flat $G$-structure is defined be one such ...
4
votes
0answers
115 views

Clarification about the definition of Calabi-Yau manifold

There are a lot of different definitions of a Calabi-Yau manifold. Roughly, we can divide them in two sets, see Wikipedia https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold . I will refer to ...
7
votes
1answer
106 views

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 ...
2
votes
0answers
36 views

Why “local holonomy contained in $SU(n)$” is equivalent to “vanishing Ricci curvature”?

I found it on Calabi-Yau manifolds' wiki page (https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold#Definitions) and can't figure out why is it true.
2
votes
1answer
73 views

Holonomy reduction from constant spinors

Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin, manifold, and let us denote by $S$ the corresponding spinor bundle. The Levi-Civita connection $\nabla$ on $(M,g)$ lifts to a unique spin ...
4
votes
2answers
112 views

basic question about holonomy

I'm struggling to understand how conditions on the metric put conditions on the holonomy group and vice-versa. My understanding is that the holonomy principle says that there's a one-to-one ...
0
votes
1answer
95 views

For a surface with $K=0$ everywhere, show that the holonomy group reduces to the identity element.

Consider a connected surface $S$ embedded in $\Bbb R^3$ and let $\alpha$ be a closed path in $S$ connecting $p\in S$ back to itself. Now we define $P_{\alpha}$ as the effect of parallel transport on a ...
2
votes
1answer
84 views

On a flat surface, can a holonomy can be nontrivial around certain curves

On a flat surface, can a holonomy can be nontrivial around certain curves? How is this possible?
0
votes
1answer
41 views

What is a thin loop?

I read one definition of a thin loop: $\gamma$ is a thin loop if there exists a homotopy of $\gamma$ to the trivial loop with the image of the homotopy lying entirely within the image of $\gamma$. ...
2
votes
1answer
71 views

Holonomy representation: is it actually a class of representations?

In D. Joyce's book "Riemannian Holonomy Groups and Calibrated Geometry" (2007) the author writes that if $\nabla$ is a connection on a vector bundle $E$ (over a connected base) with the fibre $\mathbb ...
0
votes
1answer
22 views

Multiple points in the parallel transport equation

Let $E \to M$ be a vector bundle with connection $\nabla$, $\gamma \colon [0,1] \to M$ a smooth curve. A section $s = \gamma^* \tilde s$, $\tilde s \in \Gamma(E)$ of $\gamma^* E$ is called parallel if ...
2
votes
1answer
126 views

Homology and topological propeties

i have this theorem with it's proof but i don't understand the last part They use this proposition: My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where ...
7
votes
1answer
382 views

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 ...
4
votes
0answers
62 views

Holonomy group as a quotient group of $\text{GL}(k,\mathbb{R})$

Currently, I'm reading the script of a Global Analysis lecture at my university. There, we look at a real vector-bundle $(E,\pi,M)$ of rank $k$ with a given connection $\nabla$ and define the ...
3
votes
2answers
273 views

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 ...
6
votes
1answer
150 views

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, ...
3
votes
2answers
115 views

If a connection on a principal $G$-bundle restricts to an $H$-subbundle, must its holonomy lie in $H$?

Let $P \to M$ be a principal $G$-bundle, equipped with a principal connection $D$. Let $Q \subset P$ be a principal subbundle with fiber $H$, where $H \leq G$ is a (let's say closed and connected) ...
6
votes
0answers
70 views

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 ...
4
votes
1answer
67 views

Why is the restricted holonomy the identity component of the holonomy group?

Let $M$ be a connected smooth paracompact manifold, $E$ a vector bundle over $M$ with fibre $\mathbb R^k$, and $\nabla$ a connection on $E$. It is known that Hol$^0(\nabla)$ is a connected Lie ...
5
votes
1answer
315 views

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 ...
1
vote
0answers
310 views

Ambrose Singer Theorem

I wish to learn about holonomy groups of Riemannian manifolds and the Ambrose- Singer theorem. Please advise some references other than the original paper of Ambrose and Singer.
0
votes
1answer
24 views

Absolutely rough planes

What is an absolutely rough plane? Can you give an analytical expression of one? Why cannot a rigid body slide on a rough plane? Why does the point of contact not move in an infinitesimal movement of ...
3
votes
1answer
258 views

What is curvature, in terms of holonomy functors?

It is well known and understood that linear connections, as holonomy functors, are composition-preserving mappings from the path groupoid to a structure group $G$. This extends the idea of a 1-form ...
4
votes
1answer
474 views

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 ...
2
votes
1answer
560 views

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 ...
1
vote
1answer
157 views

How does the holonomy act on the tangent space at a point?

Suppose $(X,h)$ is a compact $n$-dimensional Hermitian manifold, with holonomy group $H$. Now we know,since $X$ is a complex manifold, that $H\subset U(n)$, and that there is a representation of $H$ ...
11
votes
2answers
498 views

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 ...
6
votes
1answer
129 views

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 ...