# 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.

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### Weil homomorphism

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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### 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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### 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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### 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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### 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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### Equivalent notion of Curvature of a connection on a Principle bundle

We know given a Connection 1-form $\omega$ on a Principle bundle $P(M,G)$ we can define a curvature 2-form $\Omega$ of $\omega$. We also know that given a Connection 1-form $\omega$ we can define a ...

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### Nef line bundles over complex analytic spaces

Let $L$ be a line bundle over a compact complex manifold $X$ with a Hermitian metric $\omega$: $L$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth ...

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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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### 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 ...

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### Rolling without slipping interpretation of torsion

This is, in a sense, a follow up to this question.
Hehl and Obukhov try to give an intuitive description of torsion. I am confused about their description. I am looking at the following paragraph on ...

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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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### The exterior derivative of a certain differential form on the space of connections of a surface

Let $Y$ be a closed oriented $2$-dimensional manifold, $G$ a Lie group and $Q \to Y$ a principal $G$-bundle with a given section $q.$ Denote by $\mathcal{A}_Q$ the space of connections on $Q,$ and by $...

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### Flatness as an integrability condition without invoking bundles

Let $\pi:E\rightarrow M$ be a vector bundle, and let $U\subseteq M$ be a trivialization domain for $E$. Assume a linear connection is given on $E$ with local connection form(s) $\omega=(\omega^a_{\ b})...

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### determinant of curvature (notation issue)

This is when studying about Chern classes from Kobayashi and Nomizu.
Let $\pi:E\rightarrow M$ be a complex vector bundle with fibre $\mathbb{C}^r$ and Group $G=GL(r,\mathbb{C})$.
Let $p:P\rightarrow ...

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### Basic questions about crystals and Grothendieck connections

I have a few basic questions about Grothendieck connections and crystals. I know it's bad practice to ask a bunch of different questions at once, however I feel they naturally come bundled together. ...

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### The automorphism group of a symplectic symmetric space

Why is the automorphism group of a sympelctic symmetric space a Lie group?
$\\$
A symplectic symmetric space is a triple $(M, \omega, s)$, where $(M, \omega)$ is a symplectic manifold and $ s \; \...

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### Symplectic connections are (locally) Levi-Civita connections

I was wondering... Is every symplectic connection $\nabla$
on some symplectic manifold $(M,ω)$ the Levi-Civita connection of some metric $g$ on $M$? What about the local statement?

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### Maximally symmetric affine manifold

As a physicist who knows (something) about General Relativity, I'm accustomed to the term "maximally symmetric space" being an $n$-dimensional manifold with $\frac{n(n+1)}{2}$ Killing vectors. A ...

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### holonomy of connection on gerbes

I am reading this notes of Hitchin to understand about gerbes. He defines gerbe by giving a collection of $2$ cocycles $g_{\alpha\beta\gamma}:U_\alpha\cap U_\beta\cap U_\gamma\rightarrow S^1$ with ...

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### Special connection of vector bundle over real manifold

Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...

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### Alternative (easier) Proof of Ambrose Singer Holonomy theorem

Let $P(M,G)$ be a principal bundle. Giving a connection on $P(M,G)$ means two equivalent things. One as an assignment of subspace of $T_pP$ for each $p\in P$ and another as a $\mathfrak{g}$ valued $1$ ...

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### Does the holomorphic curvature determine the connection?

Let $M$ be a simply connected complex manifold (of dimension greater than one), $L$ a line bundle, and $\nabla$ a connection on $L$ with possibly singularities along a divisor $D$. We define the ...

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### Wedge product of Endomorphism-Valued Forms

To define characteristic classes in smooth vector bundles $E\longrightarrow M$ there is a more or less standard procedure: to choose a connection $\nabla$ and to derive the curvature $\Omega$, which ...

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### Applications of Ambrose-Singer theorem on holonomy

I am planning to introduce to a group of Graduate students the notion of connections on Principal bundle, curvature of connection, Holonomy. I want to conclude with the statement of Ambrose-Singer ...

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### Left invariant connections on a Lie group

The exponential map associated with the (-) - connection on a Lie group is generally not surjective. This is because, for this connection, the one-parameter subgroups and geodesics coincide. If we ...

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### Why torsion is only defined for linear connection on TM?

The concept of curvature is defined for any linear connection on any vector bundle $E \to M$, but the concept of torsion is only defined for connection on the tangent bundle $TM$ of a manifold $M^n$, ...

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### Existence of connections in a vector bundle whose parallel transport preserves a function on a total space

Let $p:E \to M$ be a vector bundle over a smooth manifold $M$, $M\times 0$ be the image of its zero section of $p$, $\mathcal{X}(M)$ be the space of vector fields on $M$, and $\Gamma(E)$ be the space ...

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### Is there a connection $\nabla$ for which this particular non geodesible vector field $X$ satisfy $\nabla_X X=0$?

Let $X$ be the following vector field on $\mathbb{R}^2\setminus \{0\}$
\begin{align}
x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\
y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2).
\...

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### When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?

As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion free connection $\nabla_g$, the Levi-Civita connection, that is compatible witht metric.
I was wondering if one can ...

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### Existence of sections of a fibre bundle which are covariantly constant along certain directions

Given a vector bundle $\pi\colon E \rightarrow B$ equipped with a connection $\nabla$, it is well known that a basis of flat sections $s_i$ ($i=1,\dots,\text{rank}(E)$) (i.e. $\nabla_X s_i = 0$ for ...

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### Connection 1-form of the frame bundle associated to a vector bundle with a connection

Let $\lambda = (P,\pi,M;G)$ be a smooth principal $G$-bundle (projection $\pi : P \to M)$, $V$ a finite dimensional vector space, and $\rho : G \to GL(V)$ a smooth representation of $G$ in $V$.
We ...

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### Connections on bundles over algebraic variety

I am asking for a reference for the theory of connections on vector bundles over algebraic varieties. I am particularly interested in the notion of Gauss-Manin connection, the notions of torsion, p-...

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### A question about a paper of Bismut and Lebeau

Let $X$ be a Riemannian manifold, and $Y\hookrightarrow X$ be a closed submanifold of $X$ with normal bundle $N$ with the induced metric.
Then near $Y$, we have $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...

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### Is there a ''simple'' formula for the inverse of the Drinfeld associator?

The Drinfeld associator $\Phi(x_0, x_1)$ encodes the parallel transport of the Knizhnik-Zamolodchikov (KZ) connection $\nabla$ on the bundle $\mathbb{C}\langle\langle x_0, x_1\rangle\rangle$ of formal ...

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### The algebraic connectivity of $P_n$, the path on $n$ vertices does not exceed $\frac{12}{n^2-1}$

The algebraic connectivity of $P_n$, the path on $n$ vertices does not exceed $\frac{12}{n^2-1}$.
Let algebraic connectivity of $P_n$ be denoted by $\mu$. I have proved a result that if $G$ is a ...

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### The bundle of symmetric affine connections as quotient of the second-order frame bundle

This post is not about finding an answer to a certain problem - because the answer already exists - but rather about finding the simplest possible answer.
The problem is: how to define the bundle $C(...

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### Connections in terms of tangent ($\infty$-)categories?

Given a commutative ring $k$ and a commutative $k$-algebra $A$, we know that the Kähler differential $\Omega_{A/k}^1$ could be described through machineries of tangent categories (see, for example, ...

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### When are geodesics straight lines?

Suppose I have a global coordinate system on a manifold, which is affine with respect to an affine connection on that manifold. The connection is flat and torsion free, and the connection coefficients ...

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### Do “associative” connections exist / arise naturally in some context?

Here is a little bit of curiousity that's been itching me, let's hope it doesn't get me killed, meow.
Definition: Let $M$ be a smooth manifold. A connection $\nabla$ on $TM$ is called associative ...

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### Metric connection on $\mathbb{R}^4$ that is locally Kähler but not globally Kähler

in a comment to this question When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?
Robert Bryant mentions that it is possible to construct a metric connection ...

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### Locally Riemannian Connection

Let $\Gamma^a{}_{bc}=\Gamma^a{}_{cb}$ be a symmetric connection whose curvature is $$R^a{}_{bcd}=\partial_c\Gamma^a{}_{bd}-\partial_d\Gamma^a{}_{bc}+\Gamma^a{}_{ec}\Gamma^e{}_{bd}-\Gamma^a{}_{ed}\...

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### Riemannian vs Non-Riemannian curvature

If you neither know the metric nor the holonomy group, how do you recognize a curvature tensor is Riemannian?
I assume a curvature, by definition, satisfies Bianchi identities. I know it is ...

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### Is torsion of a connection always an obstruction to some kind of integrability? [closed]

Let $E$ be a vector bundle over a smooth manifold $M$ equipped with a linear connection $\nabla : \Gamma(E) \to \Omega^1(M;E).$ I say $(M,E,\nabla)$ is flat if it admits trivial local models; i.e. if ...

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### Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...

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### References on principal G bundle and connections

I am trying to understand about principal G bundle given a Lie group $G$. For that I started with action of Lie groups on manifold $M$ and convinced my self that if the action is smooth, proper, free ...

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### Most natural connection on Lie group: comparison of different pictures

Let $G$ be a Lie group (not necessarily compact). One can equip $G$ with the left invariant metric (or
right invariant but in general there is no biinvariant metric in the noncompact case). Once the ...

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### Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?

I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...

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### Riemann-Hilbert correspondence for non-flat connections

First of all, let me warn that my knowledge of the correspondence is rather superficial, and I apologize for any technical inaccuracies below.
Setting
Let $X$ be a smooth complex algebraic variety, ...

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### Kernel of a non-integrable connection

The Riemann-Hilbert correspondence states that the kernel of an integrable (zero curvature) connection is a local system. Here, a connexion on a vector bundle $E$ over a manifold $X$ is a morphism of ...

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### Relation between the geodesics of Finsler norms $F(V)$ and $F(-V)$

I am trying to solve this exercise. Let $(M,F)$ be a Finsler space and define $\tilde{F}(x,y):=F(x,-y)$. Then $(M,\tilde{F})$ is a Finsler space and given a geodesic $t\mapsto \gamma(t)$ of $F$, $t\...