Questions tagged [holonomy]

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

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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 ...
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56 views

Proof of the holonomy principle for $1$-forms

Let $M^n$ be a Riemannian manifold with a connection $\nabla$. For a fixed $x\in M$, we define $H:=\text{Hol}_x(\nabla)$ and identify $T_xM\equiv \mathbb{R}^n$. The Holonomy Principle states that: ...
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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\...
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Holonomy vector of saddle connection

I have a translation surface of genus 2 and I know that it has a single zero of order 2 given by the differential form $\omega = 3z^2dz$ and a total angle is $6\pi$ by turning around this point. So I ...
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Conditions for a Connection to be a Metric(or Chern) Connection

Given a Hermitian metric on a holomorphic vector bundle we can easily define its Chern connection. But if we are given a connection $\mathcal{A}$, $$[De=\mathcal{A}e,]$$ where $e$ is a holomorphic ...
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19 views

Example on Local Systems: Flat Connections

I am reading about the local system, and a local system $\mathcal{L}$ on $X$ with values in $\mathcal{C}$ is defined as a functor $$ \mathcal{L}:\Pi(X)\to \mathcal{C} $$ where $\Pi(X)$ is the ...
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34 views

Flat connection: holonomy is invariant under homotopy of loops

I am looking for an elementary proof of the fact (and probably not any broader results) that when the connection is flat, holonomy does not change under homotopy of the loop. Although this is a first ...
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Flat tori as a riemannian product

A flat torus is defined as the torus with the metric inherited from its representation as the quotient $\mathbb{R}^2/\Lambda$ where $\Lambda$ is a discrete subgroup of $\mathbb{R}^2$ which is ...
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128 views

Riemannian holonomy of a covering

Suppose I have a connected Riemannian manifold $X$ and a covering $\pi:Y\to X$ with the pulled back metric on $Y$, making $\pi$ into a local isometry between Riemannian manifolds. Suppose we have a ...
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28 views

What is the Riemannian holonomy of a quotient of Lie groups?

Is there an expression for the Riemannian holonomy of a quotient of Lie groups (not necessarily compact or simply connected) such that $G/H$ is simply connected? In particular, is there a general way ...
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Correspondence between flat connections and fundamental group representations

Let $M$ be a manifold. Two stackexchange posts state that there is a correspondence $$ \{ (P,A): P \text{ a $G$-bundle}, A \text{ flat connection} \} \leftrightarrow \{ \text{morphisms } f:\pi_1(M) \...
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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 ...
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Combinatorial analog of holonomy on a planar graph with quadrilateral faces

The concept of holonomy comes from differential geometry. It describes the behaviour of a vector on a surface when it is moved via parallel transport along a closed curve on the surface. A similar ...
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42 views

Translation Holonomy

I'm trying to get a mental image of translation holonomy. I start with rotational holonomy, which corresponds to intrinsic curvature. This is the quantitative failure of a continuous process of ...
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42 views

Is the comparator $U(y,x)$ in Gauge Theory the same as a holonomy?

For Gauge theories you have a comparator that transforms as $$U(y,x) = e^{i\alpha(y)}U(y,x) e^{-i\alpha(x)}$$ Is this the same thing as the holonomy?
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Holonomy and curvature on a surface

I have read in a few places that the change in direction a vector experiences when parallel transported along a closed circuit $\partial D$ (holonomy) is equal to the surface integral of the gaussian ...
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Differential geometry of line bundles

I am trying to understand concepts of Gerbes and their differential geometry as generalisation of line bundles and their differential geometry using Hitchin’s notes. I am familiar with concepts of ...
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Recovering a principal connection from its monodromy

Given a principal bundle $P \to M$ with structure group $G$ ($M$ and $G$ are connected), it is well known that one can recover the data of $P$ and a flat connection $\Gamma \in \Omega^{1}(P, \mathfrak{...
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Is the determinant of the holonomy gauge invariant / significant?

I am currently reading the book Baez & Muniain - "Gauge Fields, Knots, and Gravity". Chapter 2 of part II defines the holonomy on a bundle $E\to M$ with a gauge group $G$ and connection $D$. They ...
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In what way can the Riemannian curvature be regarded as a two-form with values in $\mathfrak{hol}$?

Denote $G:=Hol(g) \subset SO(n)$ and $\mathfrak{g}:=\mathfrak{hol}{g} \subset \mathfrak{so}(n)$. The Levi-Civita connection $A \in \Omega^1(GL(M), \mathfrak{gl}(n))$ reduces to a connection $\hat{A} \...
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105 views

Holonomy group and irreducible $\mathrm{SU}(2)$-connections

Let $P\to M$ be a $\mathrm{SU}(2)$-principal bundle over a closed connected manifold $M$ and let $A$ be a flat connexion form on $P$. Fixing $x\in M$ we have a homomorphism $$ \mathrm{Hol}_x(A) : \...
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What is the holonomy of homogenous spaces?

Let $M$ be a Riemannian symmetric space. $M$ is then of the form $G/H$ for a semi-simple Lie group $G$ and a closed Lie subgroup $H \subset G$. In this classification, the metric $\eta$ on $M$ is ...
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Recovering Curvature Endomorphism out of Holonomy

In $\S 3$ of this document the following is stated: Let $M$ be a $2$-dimensional Riemannian manifold and $R$ denote the curvature tensor. Let $p$ be a point in $M$ and $X_0, Y_0\in T_p M$ be linearly ...
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What is meant by an Orthogonal $1$-form

In this article, titled Holonomy Groups in Riemannian Geometry, on pg 37, the following line appears (before equation 2.5.3) Let $\theta$ be a horizontal, orthogonal $1$-form on $\mathcal F_{GL}$ ...
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Showing that a 7-manifold has $G_{2}$ holonomy

I have to show that the direct product of the multi-center Taub-NUT metric with $\mathbb{R}^{3}$ corresponds to a 7-manifold with G2 holonomy. The metric of the Taub-NUT is: $ds_{TN}^{2}=V(r)(dr^{2}+...
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When total space of canonical bundle of Kaehler-Einstein manifold admits a hyperkaehler structure?

Let $M$ be a Kaehler-Einstein manifold of positive scalar curvature and real dimension $4n-2$ (e.g. $\mathbb{C}P^{2n-1}$). Then the total space of canonical bundle $K(M)$ has an explicit Ricci-flat ...
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Realize $SU(3)$ as a subgroup of $SO(6)$ (or $SO(7)$) in a surprising way

Consider coordinates $z_k=x_k+iy_k (k=1,2,3)$ on $\mathbb C^3$. Consider a 2-form $$ \omega = \sum_{k=1}^3 dx_k \wedge d y_k $$ and a 3-form $$ \rho= \mathrm{Im}(dz_1\wedge dz_2 \wedge dz_3) $$ A ...
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Integrability of the holonomy invariant distribution

Assume that $(M,g)$ is a Riemmanian manifold with holonomy group $G$. For a point $p\in M$ we decompose the tangent space $T_p M$ to irreducible $G$ invariant subspaces. This gives us natural ...
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Why does orientability imply SO(n) holonomy?

In each reference I check I am told, without proof, that orientable Riemannian manifolds have holonomy contained in SO$(n)$ by obviousness. I am sure I will feel bad about myself when I hear the ...
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199 views

Torsionless connection and parallel transport over loops

Let suppose I have a Riemannian manifold and I define a connection on it. Is it true and safe to say that if a connection is torsionless and I parallel transport a vector along a loop made by ...
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34 views

Convexity of holonomic extension

I'm currently reading Introduction to h-Principle by Eliashberg. There the author makes a comment that the space of holonomic extensions is a convex space. To elaborate, consider $\pi:X\to V$ to be a ...
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A question about the holonomy group at different basepoints

if $\gamma :I \rightarrow M$ where I is the unit interval, M is a manifold with a connection $\nabla$, $\gamma(0)=x,\gamma(1)=y$ and $P_\gamma$ is the parallel transport map from x to y, why is it ...
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196 views

The holonomy group is a Lie group

I'm reading about holonomy groups and the book says: Theorem. Let $P(M,G)$ be a principal fibre bundle whose base manifold $M$ is connected and paracompact. Let $\Phi(u)$ and $\Phi^0(u)$, $u \in P$,...
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Differential Geometry - Holonomy Angle

Let $M$ = $S^2$ = ${\{(x, y, z)\in\mathbb R}$; $x^2+y^2+z^2 = 1\}$ $\gamma = \gamma_1 \cup \gamma_2 \cup \gamma_3$ where $\gamma_j$, $j = 1, 2, 3$ is given below: $\gamma_1 = \{(cost, sint, 0)\ |\ 0\...
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1answer
113 views

The restricted holonomy group is a normal subgroup

I've been studying theory of connections, and in order to prove that the restricted holonomy group is a normal subgroup of the holonomy group, the book says: 'If $\tau$ and $\mu$ are two loops at $x$ ...
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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 $...
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814 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 ...
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210 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)} = g(\bar ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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128 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 ...
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355 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 ...
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223 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 ...
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1answer
164 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?
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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 $\gamma$. ...
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1answer
277 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 ...
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1answer
25 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 ...