# Questions tagged [connections]

A connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.

226
questions

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### Examples of connection preserving maps in differential geometry

In synthetic differential geometry and tangent categories, linear connections on the tangent bundle are treated as a sort of algebraic gadgets that incorporate the tangent bundle. Like any other ...

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### Simple example of non-integrable holomorphic connection

Let $X$ be a complex manifold with complex dimension $d$ and structure sheaf $\mathcal{O}_X$. Let $E$ be a locally free sheaf on $X$. A $holomorphic$ connection on $E$ is a morphism of sheaves
$$\...

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### Hopf fibration extended to bundle over $\mathbb{C}^2$

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ ...

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137 views

### Characterisation of (integrable) connections on (trivial) principal bundle

Let $M$ be a manifold. Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra.
Let $P(M,G)$ be a principal bundle. Recall that, a connection on $P(M,G)$ is a distribution $\mathcal{H}\subseteq ...

**2**

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108 views

### Canonical connection on $\mathcal{A}\times X$

Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...

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### What is the definition of homotopy flat connections?

What is a definition of a homotopy flat connection - in the context of differential forms with values in a dg algebra

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### Geometric theory for cohomology groups $H^p(M;\mathbb{Z})$

An excerpt from the book Loop Spaces, Characteristic Classes and Geometric Quantization by Jean-Luc Brylinski is mentioned below:
Characteristic classes are certain cohomology classes associated
...

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**1**answer

101 views

### Connections on vector bundles over elliptic curves - concrete computations

This is linked to my question on math.Stackexchange for which I had no answer.
I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me ...

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121 views

### Foliated circle bundles whose Euler class is torsion

Let $X$ be a closed manifold. By a foliated circle bundle $E \rightarrow X$ we mean a circle bundle over $X$ with total space $E$ and structure group $Diff^+(S^1)$, and a codimension one foliation of $...

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76 views

### Does holonomy determine parallel transport? [duplicate]

Let $p: P \longrightarrow M$ be a smooth $G$-principal bundle endowed with a connection that determines the holonomies:
$$\Phi_{\gamma}: P_{x} \overset{\cong}{\longrightarrow} P_{x}$$
for any fiber ...

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76 views

### Pullback connection and diffeomorphism of the base

Let $p \colon E \to B$ be a vector bundle, $\nabla^E \colon E \to E \otimes \Omega^1_B$ a connection on $E$, and $\phi \colon B \xrightarrow{\sim} B$ a diffeomorphism. Further, let there be a natural (...

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213 views

### Katz's paper on $p$-curvature – help with proof understanding

I am studying N. Katz's paper "Nilpotent connections and the monodromy theorem: applications of a result of Turrittin" where I found a fairly good account on $p$-curvatures.
I don't understand the ...

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32 views

### Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates

In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented.
Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...

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161 views

### Map from local systems to holomorphic line bundles on a curve

Let $X$ be a Riemann surface of genus $g > 0$. Let $S$ denote the set of local systems (locally constant sheaves) on $X$ with fiber $\mathbb{C}$. $S$ is in natural bijection with $H^1(X, \underline{...

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276 views

### A struggle with jets and Grothendieck vs Ehresmann connections

Let $X\to Y$ be a $C^\infty$ submersion. Consider the following two sheaves.
The sheaf on $Y$ comprised of jets of sections of $X\to Y$.
The sheaf on $X$ given by the quotient of $\Delta_{X/Y}^{-1}C^\...

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32 views

### One parameter change of a section of $T^*M \otimes End(TM)$ on an affinely flat manifold

Let $\nabla$ be a flat symmetric connection in the tangent bundle of a smooth manifold $M.$ Let $A$ be a global section of $T^*M \otimes End(TM).$ Let $\phi:B(0,1) \to M$ be a local affine chart on $...

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288 views

### Is there an analog of the Levi–Civita connection for schemes?

Is there an analog of the Levi–Civita connection for schemes?
There exists algebraic de Rham theory, $n$-forms on vector bundles (algebraically), and familiar constructions from differential geometry....

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**1**answer

168 views

### A consequence of Ambrose-Singer theorem on holonomy

Consider $\nabla$ a connection in a vector bundle above a smooth manifold $M$.Consider a local frame $\sigma=(\sigma_1, \sigma_2,...,\sigma_m )$ on a contractible open set $U\subset M$ and calculate ...

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56 views

### Poincaré connection encode torsion and curvature

I'm trying to understand something that is written in Baez & Wise paper "Teleparallel Gravity as a Higher Gauge Theory". In section 4, they discuss Poincaré connection and, first of all, split ...

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### Problem arising in metrizability of connections: Simultaneously skewsymmetrizing matrices

Fact: Let $U$ and $V$ be two $ n \times n$ matrices with determinant $ 1.$ Assume that $S_1,S_2,....S_m$ are linearly independent $n \times n$ matrices such that $U^{-1}S_iU$ and $V^{-1}S_iV$ are ...

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222 views

### Why does a principal G-bundle with a discrete structure group G have a unique flat connection?

I'm reading the Dijkgraaf–Witten paper Topological gauge theories and group cohomology (Comm. Math. Phys. 129 (1990) pp 393–429, doi:10.1007/BF02096988) and on page 395, 2nd paragraph they write
...

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198 views

### A question on the nature of the vortex number

In the Yang-Mills-Higgs (also called magnetic Ginzburg-Landau) model in the plane the energy has the form
$$
E(A,\phi)=\int_{\mathbb{R}^2}\left(|(d-iA)\phi|^2+\frac{1}{2}F_{jk}F_{jk}+\frac{1}{4}(1-|\...

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244 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 ...

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119 views

### Introducing connection on principal bundle as lifting of vector field and paths

Let $\pi:P\rightarrow M$ is a principal $G$ bundle. I want to introduce the notion of connection as a way to uniquely lift the structures on $M$ to structures on $P$, namely vector fields and paths.
...

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210 views

### Existence of horizontal lifts in $G$-bundles

I wanted to show that for any smooth principal $G$-bundle $E\xrightarrow\pi B$ any smooth curve $\gamma\colon I\to B$ has a unique horizontal lift from a fixed starting point $u_0\in\pi^{-1}\left(\...

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253 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 ...

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**1**answer

242 views

### Advantages of Atiyah sequence version of connections on a principal bundle

I am reading Lie Groupoids and Lie Algebroids in Differential Geometry
by Kirill Mackenzie.
In appendix (page $291$), before discussing about Atiyah sequence associated to a Principal bundle, the ...

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141 views

### Definition of an equivariant connection and equivariant curvature

Can anyone give me a reference which precisely stated the definition of an equivariant connection and equivariant curvature?
Precisely, If G be a compact lie group acting linearly on a smooth ...

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165 views

### What is the group of symmetries of $\mathbb{R^n}$ with the flat projective structure?

Consider $X = (\mathbb{R^n},c)$, where $c$ is the equivalence class of all torsion free affine connections having straight lines as unparameterized geodesics. What is the group of symmetries of $X$? ...

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### Flatness equivalence

Let $\pi:E\rightarrow M$ be a complex vector bundle and $H$ a hermitian metric over it. If $D$ is a connection over $E$, using the metric $H$, we can decompose it as:
$$
D=D_H+\phi
$$
Where $D_H$ is ...

**4**

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405 views

### Confusion about complex differential forms

I follow Kobayashi "Differential Geometry of Complex Vector Bundles", pages 11-12, prop. 4.9. Given a rank-$r$ Hermitian holomorphic vector bundle $(E,h)$ over a complex manifold $M$, there exists a ...

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### Derivative of expression involving a left-invariant connection

I'm trying to understand a calculation done in this paper. Somewhat simplified, the setup is as follows.
Let $G$ be a Lie group, and $\varrho$ its Lie algebra.
Let $\nabla$ be a left-invariant ...

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### ASD connection for Line bundle over $4$-manifold

Let $(M,g)$ be an oriented closed Riemannian $4$ manifold.
Let $L\to M$ be a complex line bundle.
Q Under what condition, we can find an ASD connection of $L$, i.e. a connection $A$ such that $F^+...

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### Knizhnik-Zamolodchikov equation is a connection on “affine slice”

The question is - what is the precise meaning of the phrase in the title? I heard it from Andrey Okounkov during one of his lectures.
The problem is that he didn't really specified which slice is ...

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161 views

### Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$

I'm reading this paper and at page 67, he states that for any line bundle $L$ over a Rieman surface there is a connection $A$ whose curvature is
$$
F(A)=(\deg L)\omega,
$$
where $\omega$ is a positive ...

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votes

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285 views

### When is the action of the gauge group on the space of connections free?

Let $G$ be a compact Lie group. Let $\mathcal{A}$ be the space of connections on the principal trivial $G$-bundle $G\times \mathbb{R}^4$ possibly with some growth condition (to specify it is a part of ...

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### What exactly is a Cartan radius vector (and its role in Poincaré gauge theories)

I am studying approaches to gravity where the Poincaré group is "gauged". The original motivation of this is to understand what is meant on the statement that "Teleparallel gravity is a gauge theory ...

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### Integrability of connections with partially vanishing curvature

Let $E \rightarrow B$ be a vector bundle with a connection $\nabla$ and a local frame $(e_1, \dots, e_n)$. For any section $V = V^a e_a$ the connection can locally be written as
$$
\nabla V = \left(d ...

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222 views

### Multiplication in Deligne cohomology: explicit formula for $p=q=1$

[This is a double of my question of math.stackexchange https://math.stackexchange.com/questions/3214962/multiplication-in-deligne-cohomology-explicit-formula-for-p-q-1]
In the very beginning of [1] ...

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660 views

### Atiyah Sequence and Connections on a Principal Bundle

Let $G$ be a Lie group and $\pi:E_G\rightarrow M $ be a principal $G$-bundle.
I have seen in many places that a connection on $(E_G,M,G)$ is a splitting of the Atiyah sequence
$$ 0\rightarrow \text{...

**4**

votes

**1**answer

151 views

### Metrics with prescribed Levi-Civita connection

My question involves the symmetries of a (pseudo)-Riemannian metric preserving the Levi-Civita connection (LCC), its unique torsion-free metric connection. For a basic example, one notes that the ...

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### Flattening a connection on a Kähler manifold

Say $M$ is a closed Kähler manifold and $(V, \nabla)$ is a (say) constant Hermitian bundle on $V$ with (say) trivial flat connection. Now $M$ Kähler gives several distinguished classes of closed one-...

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284 views

### Chern-Weil theory and Weil homomorphism of principal bundle

In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $G$ bundle. In this book, they use connection on a principal bundle to ...

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198 views

### Does $\nabla g=\omega(\cdot) g$ imply $\nabla$ is metric w.r.t a conformal rescaling of $g$?

This is a cross-post.
Let $E$ be a smooth vector bundle over a manifold $M$, where $\text{rank}(E) > 1,\dim M > 1$. Suppose that $E$ is equipped with a metric $g$ and an affine connection $\...

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105 views

### Flat covariant derivative

Is it true that for any flat and torsion-free connection $\nabla : \mathfrak{X} (M) \times \mathfrak{X} (M) \rightarrow \mathfrak{X} (M) $ there exist a local systems of coordinates such that the ...

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257 views

### Existence transverse sections in $\mathbb{CP}^1$-bundles over compact Riemann surfaces

We have that every holomorphic $\mathbb{CP}^1$-bundle on a compact Riemann surface admits a holomorphic section, due Tsen and as found in Compact Compact Surfaces of Barth, Peters and Van de Ven, for ...

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369 views

### Explicit Riemann Hilbert correspondence

For simplicity, we assume that $X=\mathbb P_{\mathbb C}^1-\{s_1, s_2, \dots, s_k\}$ and $\infty \in X$.
Consider the trivial bundle $E=\mathcal O_X^r$ with the connection $\nabla$ induced by a ...

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### Compatibility of Kirillov-Kostant-Souriau form and Killing form

Let $\mathfrak{g}$ be a real semisimple Lie algebra. We know that a coadjoint orbit $\mathcal{O} \hookrightarrow \mathfrak{g}^*$ carries a natural symplectic form $\omega$, namely the Kirillov-Kostant-...

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### Inducing linear connections via functors

Let $M$ be a smooth manifold and let $\pi:E\rightarrow M$ be a real vector bundle over it. Let $\nabla$ be a linear (Koszul) connection on $E$ (here in this question I am using covariant derivatives, ...

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212 views

### Is there a notion of a connection for which the horizontal lift of a curve depends on its orientation?

Given a fiber bundle $\pi:E\to M$, a curve $\gamma:[0,1]\to M$, and a point $p \in \pi^{-1}(\gamma(0))$, a connection on the bundle allows us to uniquely lift $\gamma$ to a horizontal curve in E ...