# 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.

67 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 $\overline {𝑓(𝑧)} = g(\bar ... 2answers 75 views ### Parallel displacement on principal bundles Let$\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$. Every ... 0answers 46 views ### Flat$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$-structure is defined be one such ... 0answers 263 views ### Clarification about the definition of Calabi-Yau manifold There are a lot of different definitions of a Calabi-Yau manifold. Roughly, we can divide them in two sets, see Wikipedia https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold . I will refer to ... 1answer 219 views ### Holonomy computation in$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$satisfied some regularity conditions), someone told me that the ... 0answers 51 views ### Why “local holonomy contained in$SU(n)$” is equivalent to “vanishing Ricci curvature”? I found it on Calabi-Yau manifolds' wiki page (https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold#Definitions) and can't figure out why is it true. 1answer 128 views ### Holonomy reduction from constant spinors Let$(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)$lifts to a unique spin ... 2answers 355 views ### basic question about holonomy I'm struggling to understand how conditions on the metric put conditions on the holonomy group and vice-versa. My understanding is that the holonomy principle says that there's a one-to-one ... 1answer 223 views ### For a surface with$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}$as the effect of parallel transport on a ... 1answer 164 views ### On a flat surface, can a holonomy can be nontrivial around certain curves On a flat surface, can a holonomy can be nontrivial around certain curves? How is this possible? 1answer 55 views ### What is a thin loop? I read one definition of a thin loop:$\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$. ... 1answer 277 views ### Holonomy representation: is it actually a class of representations? In D. Joyce's book "Riemannian Holonomy Groups and Calibrated Geometry" (2007) the author writes that if$\nabla$is a connection on a vector bundle$E$(over a connected base) with the fibre$\mathbb ...
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 ...