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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Factorial term when evaluating the exterior derivative through the connection definition

I found perhaps a hint of what the relation is on page-316 of Penrose's Road to reality. For $ p- $ form $\alpha$ with index epxression, $\alpha_{b...d}$ , and a torsion free connection $\nabla_a$ $$(...
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Pulling back Lie-valued forms to get curvature?

According to Lemma 1 in https://en.wikipedia.org/wiki/Chern%E2%80%93Weil_homomorphism#Definition_of_the_homomorphism , if $\Omega$ is the curvature of a connection on a principal $G$-bundle $P\...
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Are there tensor structures other than a metric which could be defined on a manifold which imply a connection through compatibility criterion?

If we say our connection is torsion free, then the metric compatibility condition completely determines it. While this is geometrically intuitive way to do it, are there other interesting tensor ...
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Weyl algebra and flat connection on the affine line

Given a connection $\nabla_{\partial_x}: \mathbb{C}[x]\rightarrow \mathbb{C}[x]$, we can view it as a $\mathbb{C}[x,\partial_{x}]$ module where $\partial_{x}$ acts on $\mathbb{C}[x]$ by $\partial_x f =...
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Pullback Connection/Riccati-Equation

I'am currently trying to understand the paper: https://epub.uni-regensburg.de/23578/1/MP171.pdf The point where I'am stucked at is in the proof of Theorem 4.6. You dont need to read the whole article, ...
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Levi Civita Connections vs. Ehresmann connection

Forgive me if I mess some of these concepts up or say something incorrect, I am still figuring out all the details of an Ehresmann connection in an associated vector bundle. So, how do we relate these ...
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Covariant and exterior covariant derivative of a bundle-valued $n$-form.

Let $E\to M$ be a vector bundle above a manifold $M$, with a connection $\nabla$ defined on the tangent bundle, and let $\nabla^{E}$ be a linear connection on $E$ and $\omega$ a $n$-form on $M$ with ...
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A coordinate free computation of the acceleration of $ \exp_p(t(v + \frac{t}{2}w))$

$\newcommand{\al}{\alpha}$ Let $M$ be a smooth Riemannian manifold. Fix $p \in M, v,w \in T_pM$, and define $\gamma(t) = \exp_p(t(v + \frac{t}{2}w))$. Is there a coordinate-free proof that $(\nabla_{\...
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Can every tangent vector be realized as an acceleration of a path with a given velocity?

$\newcommand{\al}{\alpha}$ Let $M$ be a smooth Riemannian manifold. Fix $p \in M, v \in T_pM$. Define $$ \mathcal{A}_v:=\{ w \in T_pM\,|\,\exists\alpha:(-\epsilon,\epsilon) \to M, \, \, \alpha(0)=p, \...
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composition of two tangent vector fields and connection

Let $X, Y$ be two tangent vector fields on a manifold. Consider their composition: $(X \circ Y)(f) = X(Y(f)) = X^i\frac{\partial Y(f)}{\partial u^i} = X^i\frac{\partial Y^j}{\partial u^i}\frac{\...
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Solutions of Yang-Mills equation in the case of $G=U_1$

If $P\to M$ is a principal $U_1$-bundle, and $A$ is a connection on $P$, then it's curvature $F_A$ is a $2$-form with coeficient in $P\times_G\mathfrak{u}_1$, where $\mathfrak{u}_1$ is the Lie ...
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$T_{(q, w)}TM$ and $T_q M \times T_q M$ are canonically isomorphic through the Levi-Civita connection

I'm reading an article that states that for any finite dimensional manifold $M$ and any $q \in M, w \in T_q M$, there is a canonical isomorphism between $T_{(q, w)}TM$ and $T_q M \times T_q M$ defined ...
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Concrete example of charting a topological manifold without specifying a connection

I have been trying to understand manifold theory for a long time now, and every single example of charting a manifold I see is after giving the structure of a connection on it. For instance , consider ...
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Why can we use fundamental vector fields for vertical vector fields to prove the form of the curvature of a connection?

There's a standard proof that roughly goes that, to prove the equivalence for a connection form $\omega$ on a $G$-principal bundle $P$ ($u, v \in \Gamma(TP)$): \begin{eqnarray} \Omega(u,v) &=& ...
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Tensor fields defining $G$-structure are parallel

Suppose $G \leq GL_n(\mathbb{R})$ is the stabilizer of some tensors $T^0_1, ..., T^0_k$, let $P$ be a $G$-structure on a manifold, i.e. a principal $G$ subbundle of the frame bundle of $M$ and let $...
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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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covariant derivative of the second form with the induced Levi-Civita connectiom

In a solution to a question of an exam I found the following claim: Let $(M,g)$ a Riemannian manifold and let $\nabla$ be the Levi-Civita connnection induced by the metric $g$. Consider $(\overline{M},...
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Why can't we apply the covariant derivative to normal vector fields?

Assume we are given an embedded Riemannian submanifold $(\mathcal{M},g)\subset (\overline{\mathcal{M}},\overline{g})$, with $\overline{\mathcal{M}}$ having the Levi-Civita connection $\nabla$ ...
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Is a normal connection indeed a connection?

I am reading GTM176. In the first picture, it says that the second fundamental form is bilinear over $C^\infty(M)$, i.e., $(\tilde{\nabla}_XY)^{\bot}$ is a tensor. However, in the second picture, it ...
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Parallel fundamental vector fields

Suppose I have a principal bundle $P$ relative to the group $G$. Suppose I have a torsionless connection on $TP$ for which the fundamental vector fields relative to the $G$ action are parallel. Can I ...
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What is / why the connection one-form from a physics point of view?

Take the Yang-Mills gauge theory for example. Gauge field $A$ is the pullback of the connection one-form to the base manifold. Other concepts of gauge theory also find their definition in fiber ...
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Easy example of a stable pair in the sense of Hitchin

Let $M$ be a compact Riemann surface with metric $g=hdzd{\overline{z}}$ compatible with the conformal structure. Then The Levi-Civita connection is a $U(1)$ connection defined on the canonical bundle ...
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2 votes
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Alternative expression for Riemann curvature tensor

There is the usual expression for the Riemann tensor $$R_{abcd}=\partial_c\Gamma_{adb}-\partial_d\Gamma_{acb}+\Gamma_{ace}{\Gamma^e}_{db}-\Gamma_{ade}{\Gamma^e}_{cb}.$$ However, in the last page of ...
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Equivalence between connections on $\mathcal{E}$ and $\mathcal{P}^1$-linear isomorphisms that induce the identity modulo $\Omega^1_{X|S}$

In Berthelot and Ogus' book "notes on crystalline cohomology", I don't understand the proof of proposition 2.9: Given an $O_X$-module $\mathcal{E}$ on an $S$-scheme $X,$ a connection $\nabla$...
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Exterior covariant derivative on associated bundle

Let $(P,\pi,M;G)$ be a principal fibre bundle over $M$ with connection $1$-form $A:TP\rightarrow \mathfrak{g}$. Let $\rho:G\rightarrow V$ be a representation. The connection $A$ now induces a ...
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What is $dy$ in the 1-form $\alpha |_y := ([\beta,y],dy) $

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. Consider the adjoint action of $ G$ on its Lie algebra, and let $(.,.)$ be a $G$-invariant inner product on $\mathfrak{g}.$ Let $\beta \...
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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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Producing new connections out of given one

Let $E$ be a holomorphic vector bundle over $M$ which is a complex manifold. Let $D$ be a connection on it. So $D$ maps sections of $E$ to sections of $T^*M \otimes E$. Let $D'$ and $D''$ be the $(1,0)...
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4 votes
2 answers
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Reading off connection 1-forms from Cartan's structural equation $de=-\omega\wedge e$

Suppose we have a Lorentzian metric of the form \begin{align} g&=-f(r)^2\,dt^2+ h(r)^2(dr^2+r^2\,d\theta^2+r^2\sin^2\theta\,d\phi^2) \end{align} Where $f,h$ are say strictly positive functions. We ...
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Differential of sections of the vector bundle

I was following Calegari's notes on Differential geometry (https://math.uchicago.edu/~dannyc/courses/riem_geo_2019/differential_geometry.pdf), and I found this part: I got a bit confused when trying ...
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How to use the definition of the extension of $\nabla (X)$ to $T^{0,2}$ to prove a claim about it

Let $(M, g)$ be a Riemannian manifold with Levi Civita connection $\nabla: \Gamma(TM) \rightarrow \Gamma( T^*M \otimes_\mathbb{R} TM)$. I know from here how $\nabla$ induces a map $\Gamma(T^{p,q} M) \...
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3 votes
2 answers
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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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2 votes
1 answer
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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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How to construct isometric tangent bundle embedding

$\newcommand{\D}{\mathop{}\!\mathrm{d}}$ 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 ...
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