# Questions tagged [connections]

In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. (Def: https://en.wikipedia.org/wiki/Connection_(vector_bundle))

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### Gauge equivalence of Lie-valued forms on the base space of a principal bundle

Given a principal $G$-bundle $P\xrightarrow{\pi} M$: Assuming the bundle is globally trivial, we define two Lie$G$-valued 1-forms $A_1,A_2$ on $M$ to be gauge-equivalent if there is a principal bundle ...
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### Obtaining an explicit formula for a connection applied to an almost complex structure

Let $(M,J)$ be an almost complex manifold and $\nabla$ be a connection on $TM$. I am trying to see how we can obtain an explicit formula for $\nabla_X J$. I know that the way to extend $\nabla$ to a ...
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### Do unique horizontal lifts of path homotopic paths through a point are always path homotopic?

Let $\pi:E \rightarrow M$ be a principal $G$-bundle for a Lie group $G$. Let $\omega$ be a connection on the principal bundle. It is a well known fact, that if a path $\gamma:[0,1] \rightarrow M$ is ...
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### What is the "gauge field" on the base space in a gauge field theory?

Suppose we have a principal $G$-bundle $P\xrightarrow{\pi} M$, and we want to consider a classical gauge field theory (with a field Lagrangian) on $M$ for this bundle, in the physics sense. In the ...
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### What is the most generic way to write a Lagrangian quadratic in velocities?

I'm trying to generalize the expression for a Lagrangian to a manifold that doesn't posses a metric. To be more clear, when the configurations space has a metric, we write the lagrangian quadratic in ...
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### In what coordinates do the Christoffel symbols of this flat connection vanish?

Let $M$ be a submanifold of a pseudo-Riemannian manifold $N$ such that the Levi-Civita connection of $N$ is flat. For simplicity, since this seems to already include the general case, let us restrict ...
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### Inconsistency in the definition of the connection coefficients

I am new to general relativity and I am currently facing an apparent inconsistency in the definition of the connection coefficients. Some references I've been consulting (e.g. the lecture notes by S. ...
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### Parallel fundamenal vector fields on circle bundle

Since this question received no answer, let me go through a simpler case first. Let $P$ be a $U(1)$-bundle and suppose I have a metric on it that makes the fundamental vector field of the action ...
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### Moduli Space of Flat connection over Homology 3-Shpere

I'm trying to understand the space of flat connections over the trivial $SU(2)$ -bundle of a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer homology of it)...
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### How does a connection act on a one form?

It is well discussed how a connection acts on a vector with parallel transport. For example, have a look at 17:29 of this video by Eigen -Chris. However, I have never seen a visualization of the ...
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### Relation between G-connection and second fundamental form when embedding is in principal G-bundle

I'm pretty familiar with intrinsic geometry utilized in say General relativity for instance, and I understand the intrinsic curvature $\Omega$ 2-form of a connection $A$ on a manifold $M$ of dimension ...
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### Calculate the induced covariant derivative on the pullback bundle $\pi^*\mathcal{E}$

Let $\pi: \mathcal{E}= M \times E \rightarrow M$ be a trivial vector bundle (where $M$ is smooth and $E$ is a finite dimensional real vector space). Let $\nabla = d + \omega$ be a covariant ...
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### Connections on Principal bundles & Covariant derivatives on Vector bundles

Nowadays I'm reading "Differential geometry" written by Taubes. I have some problems and I guess that there may be some typos or I must get something wrong. Suppose vector bundle $E$ is ...
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### What exactly is the relationship between an Ehresmann connection and splitting of the jet sequence.

An Ehresmann connection on a vector bundle $\pi : E \to X$ is a splitting of the sequence, $$0 \to V \to TE \to \pi^* TX \to 0$$ which respects the linear structure on $E$ (meaning the section is ...
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### Tangent bundle of a tensor product bundle

Let $E\to M$ and $F\to M$ be vector bundles. The structure of their tangents $TE$ and $TF$ is well known. In particular, connectors map $K_E: TE \to E\times_M E$ and $K_F: TF \to F\times_M F$ induce ...
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### Corollary of Lefschetz decomposition theorem

Let me develop the setup. Let $M$ be compact Kahler manifold and $E$ be a smooth complex vector bundle on it. Then I'm told that the smooth $E$-valued forms have a Lefschetz decomposition. If $E$ is ...
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### How do we go from a covariant derivative on a principal bundle to a covariant derivative on an associated bundle

Let $M$ be a smooth manifold and $\pi:P\to M$ a principal $G$ bundle over $M$. Suppose that $P$ is equipped with a connection one form $\omega$. We can define an exterior covariant derivative on $P$ ...
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### Confusion regarding definition of gauge transformation

Let $E \to M$ be a principal $G$-bundle. The gauge group is the group of $G$-bundle automorphisms of $E$. A connection on $E$ can be thought of as a global $g$-valued 1-form on $E$ where $g$ = Lie$(G)$...
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### Confusion regarding connection form

I have the following two notions of connection. For a vector bundle we have a covariant derivative from sections of $E$ to sections of $E \otimes T^{*}M$ which is a $\mathbb C$-linear map and ...
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### Connection on a principal $S^1$ bundle

Let $\pi:M\to B$ be a principal $S^1$-bundle over a symplectic manifold $(B,\omega)$. Is it always possible to construct a vector field $R\in \mathfrak{X}(M)$ such that the $S^1$ action on $M$ is ...
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$\newcommand{\D}{\mathop{}\!\mathrm{d}}$ Given an isometrically embedded manifold $ι : M ↪ ℝ^N$ I want to to embed the tangent bundle $TM$ isometrically. This question provides an idea for a smooth ...