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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Doubts on the proof of the definition of the local connection form in Nakahara's book

I've been reading Nakahara's book on topology and geometry for physicists and got stuck on the proof of definition of the local connection form $\omega $. So, given a Lie-algebra-valued one form $\...
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The Grassmann connection is a connection

Let $\mathcal{A}$ be a *-algebra and $p\in M_N(\mathcal{A})$ an orthogonal projection. I need to show that $\nabla=p\circ d$ defines a connection on $\mathcal{E}=p\mathcal{A}^N$, where $d$ is acting ...
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Levi-Civita conncetion after contraction with a vector field

I'm reading "Elements of Noncommutative Geometry" by Garcia-Bondía. I have a questions regarding the the contraction of the Levi-Civita connection $\nabla^g$ with a vector field $Z$. First I ...
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Local description of the Levi-Civita connection and the Christoffel symvols

I'm reading "Elements of Noncommutative Geometry" by Garcia-Bondía. I have some questions regarding the local description of the Levi-Civita connection $\nabla^g$. First I will give you some ...
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Ehresmann connection and (Koszul) connections on base manifold

Once we define an Ehresmann connection over the double tangent space, how does this explicitly determine canonically a (Koszul) connection on the base manifold? Given a smooth manifold $ M $, let $ TM ...
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Affine connections: alternatives to the Levi-Civita connection

When reading about the notion of affine connection, the Levi-Civita connection appears naturally as the unique affine connection that preserve the metric and is torsion free. In this case, it is ...
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Local expression of covariant derivative on associated vector bundle

I'm using Instantons and four manifolds by Freed and Uhlenbeck to write a seminar on Yang-Mills instantons. I used Kobayashi and Nomizu's Foundations of differential geometry to fill the gap I had on ...
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How to determine the notion of parallel transport within the conformal or projective and general connections?

I'd like to know how the parallel transportation behaves in non-Levi-Civita connections and how does one realize it formally. I know that parallel transportation along some piece-smooth curve is ...
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How does $\partial_\mu$ act on $A_\nu$ in the local structure equation for a curvature 2-form written in coordinates?

Let $\pi: P \rightarrow M$ be a principal bundle and $s: U \rightarrow P$ be a local section where $U \subset M$ is open. Let $A$ be a connection 1-form and $F$ a curvature 2-form. Define $A_s = s^* A$...
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Coordinatefree Cartan-Maurer

If $E$ is a real bundle and $\nabla$ a connection, each $e \in E$ gives us $\omega_e := \nabla_e \in Hom(T,E)$. The cartan-maurer equation is that using the connection twice we get $$\theta_e = d\...
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A connection along a curve is $C^{\infty}(I)$-linear.

Let $I \subset \mathbb{R}$ and $c:I \to M$ smooth. We define $$\mathfrak{X}(M)_c := \Big\{X \in C^{\infty}(I,TM): X(t) \in T_{c(t)}M, \quad \forall t \in I\Big\}.$$ Then $X \in \mathfrak{X}(M)_c$ is ...
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Connections restricted to an open set

Currently trying to understand the restriction of a connection $\nabla: \Gamma(E) \to \Gamma(T^*M \otimes E)$ to an open set $U$. Is the definition for this $$\nabla^U: \Gamma(U,E) \to \Gamma(U, T^*M\...
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Connection Leibniz rule

This is my definition for a connection on a vector bundle. Let $E \to M$ be a vector bundle. A connection on $E \to M$ is a bilinear map $$\nabla : \mathfrak{X}(M) \times \Gamma(E) \to \Gamma(E), \ \ ...
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Intuition behind connection 1-forms and Ehresmann connections

I am learning about mathematical gauge theory and so far I have been able to develop an intuition behind all the objects I've read about such as principal bundles, associated bundles, and vertical/...
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Tensor change of coordinates - why one way is inverse matrix of the other?

I am working on tutorial on connections from Gravity and Light winter school. I have a problem understanding solution from walkthrough on youtube. To calculate connection coefficients in different ...
Marek G.'s user avatar
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Motivation for Connections over smooth manifolds

I was reading about connections on Wikipedia and I found this section about "Motivation": My question is the following one. If I look at the vector bundle $E$ like a manifold, then locally ...
wood's user avatar
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Why does the adjoint representation appear in the definition of a connection 1-form?

When defining the a connection 1-form $A$ on a principal $G$-bundle we require that $$r^*_gA = Ad_{g^{-1}} \circ A, \quad \forall g \in G.$$ Here $r_g^*$ is the pullback of right multiplication by $g$,...
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Understanding the Cartan connection in the context of General Relativistic spacetime.

I'm trying to understand the Cartan formalism in the context of General Relativity. As I understand it given a pseudo-Riemannian spacetime manifold $M$ we can consider the group of spactime ...
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Invariant differential forms?

In the first page of the paper "On the spaces of maps inducing isomorphic connections" by T.R. Ramadas, one can read that the automorphisms of a connection $\nabla$ on a principal $G$-bundle ...
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Affine connection on non smooth vector fields

The usual definition of an affine connection $\nabla_X Y$ requires $X,Y$ to be smooth vector fields on the ambiant differential manifold $M$, i.e. $C^\infty$. However at any point $p\in M$, $(\nabla_X ...
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Connections and curvature on line bundles

I'm trying to understand some basic definitions in relation to line bundles (following Woodhouse's 'Geometric Quantization'). Let $V\rightarrow M$ be a vector bundle (in particular, a complex line ...
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Vector Bundle Connection and Agreement a on Curve

Problem 10.3 (Tu - Differential Geometry) Let $\nabla$ be a connection on vector bundle $E \to M$, $p \in M$, and $X_p \in T_p M$. Show that if sections $s$ and $t$ agree on a curve through $p$ with ...
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“Simple” formula, besides Cartan’s equation, for exterior covariant derivative of vector-valued forms on principal bundle

Let $P$ be a principal bundle with structure group $G$, equipped with a principal connection, whose connection $1$-form we denote $\omega$. Suppose we also have a linear representation $\rho:G\to GL(V)...
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Ray Tracing In Mathematical Spaces

I really enjoyed the Not Knot video, but I don't fully understand the mathematics which is going on there. They are animating the space of the complement of the Borromean rings, but you can't just ...
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Marked points on fibers of $TS^2$ can trace out a helix during (parallel) transport.

It is claimed here that a mark on a rod parallelly transported by an observer moving along a geodesic $\gamma(t)$ on a smooth manifold $M$ (with a metric $g$) will trace out a helix, instead of a ...
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Classifying Vector Bundles arising from Induced Representations and Connections on those bundles

Suppose $G$ is a Lie group and $H$ is a (closed) subgroup. There is a natural $H$ principal bundle $H \to G \to G/H$. This master bundle seems to lead to the following vector bundle $$ V \to E \to G/H....
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Recover a connection from torsion

Let $\nabla$ be a linear connection on $M$, let $\{e_i\}$ be a local frame on some open subset $U \subset M$, and let $\{\omega^i\}$ be the dual coframe. We know that there is a uniquely determined ...
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Connections of Sections in terms of Connections of frame

Tu's book "Differential Geometry" makes the claim on p79 that for a $C^{\infty}$ vector bundle $\pi : E \to M$, over a trivializing open set U any connection $\nabla_Xs$ for X a vector field ...
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Can the units of an arbitrary differential operator be completely arbitrary on a smooth manifold?

On a smooth manifold, given any two derivative operators $\tilde{\nabla}_{a}$ and $\nabla_{a}$, there exists a connection $C^{c}_{ab}$ such that, acting on a metric tensor $g_{bc}$, we have: \begin{...
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Confusion with covariant derivatives with vielbeins

I have some confusion regarding how the covariant derivative is defined for one forms on a manifold in the context of frames/vielbeins. I am a physics student and my reference is Sec 4.3 of the ...
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The Covariant Derivative as a Derivation on $\mathrm{TFA}^{\infty}({M})$

Let ${M}$ be a smooth manifold, then let $\mathrm{TFA}^{\infty}({M})$ be the smooth tensor field algebra, which is defined as \begin{align*} \mathrm{TFA}^{\infty}({M})=\bigoplus_{{m}={0}}^{\infty}\...
DeVoyd's user avatar
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For some $X\in \Gamma TM$, does there exist a parallel transport that preserves it?

Given a nowhere vanishing vector field $X\in\Gamma TM$ on a smooth manifold $M$, is it always possible to find a connction $\nabla$ on $M$ so that the parallel transport induced by $\nabla$ preserves $...
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Is there a notion of Hurewicz fibration with additional structure equivalent to UNIQUe path lifting

Is there a notion of Hurewicz fibration with additional structure equivalent to UNIQUe path lifting? or even a UNIQUE homotopy lifting?cf parallel transport in diff geom
jim stasheff's user avatar
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exterior derivative vs connection: $d= \theta^i\wedge\nabla_{e_i}$

Let $\nabla$ be the Levi-Civita connection of a Rimannian manifold $M$, let $e_1,\dots,e_n$ denote an orthogonal local flame field of $TM$ and let $\theta^1,\dots,\theta^n$ be the its dual coflame. I ...
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Understanding the coefficients of a pullback connection

Following the last piece of the accepted answer to this question, we have the following. Let $f : M \to N $ be a smooth function between smooth manifolds, $x^i$ be local coordinates on $M$, $\\{y^\...
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What is the intuition (geometric picture) of Bianchi's identity

The Bianchi's identity reads $\textrm{d}\Omega = \Omega \wedge \omega - \omega\wedge \Omega$, where $\Omega$ is the curvature 2-form, $\omega$ is the connection 1-form. Is there an intuitive picture ...
Alex's user avatar
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Horizontal distribution on $S^3 \to S^2$ and non torsion free connections

Thanks in advance to everyone! I am a little bit confused regarding connection on principal bundles. I explain my problem. Consider $S^2$ with the standard metric and the Levi Civita connection and ...
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$\nabla_X(AY)=A\nabla_XY$ for all vector fields $X$ and $Y$ and $A$ is constant invertible matrix.

I've been working with Riemannian manifolds $(M,g)$ of dimension $n$, with Riemannian connection $\nabla$, and I've encountered a problem that I can't seem to solve. I would appreciate any help you ...
Curious student's user avatar
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Orientation bundle and its flat connection

Let $M$ be a smooth $n$-manifold (which is not assumed to be orientable), and write $o(TM)\to M$ for its orientation bundle. Equivalently, it is the top exterior bundle $\Lambda^n(TM)\to M$. In any ...
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Covariant derivative of orthonormal frames

Given a $2$ dimensional riemannian manifold with a local orthonormal frame $e_1,e_2$ I want to evaluate the covariant derivatives $\nabla_{e_1}e_1$, $\nabla_{e_2}e_1$, $\nabla_{e_1}e_2$ and $\nabla_{...
ebenezer's user avatar
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Riemannian submersion that preserves the Levi-Civita connection where the fibers are not totally geodesic submanifolds?

I am not an expert in Riemannian geometry. I have been told that if $f: (M,g) \rightarrow (N,h)$ is a Riemannian submersion, then if the fibers of $f$ are totally geodesic submanifolds of $(M,g)$, ...
Functorial Nonsense's user avatar
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Curvature tensor and Levi-Civita connection

Assuming $M$ is a Riemannian manifold and $p$ is a point on $M$, consider a local orthonormal frame around $p$ denoted by $\{e_1,\dots,e_n\}$. Suppose that we are given the components $R_{ijk} = R(e_i,...
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Christoffel symbols of Riemannian metric with asymptotic decay

Say we are given an oriented and complete Riemannian manifold $(M,g)$ such that for a compact set $K \subset M$ we have that $ M \setminus K $ is diffeomorphic to the complement of the closed ball $\...
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Which book construct the concept philosophy of Riemannian geometry as Arnold does but more elaborated?

In Riemannian geometry, the concept of covariant derivative/connection have at least 5 different equivalent definitions, as listed in Spivak's book. Most text books, like do Carmo' or S.S.Chern's et, ...
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Exponential of a spin connection belongs to what group?

This is probably a silly question but please humor me. In an orthonormal tetrad formulation of GR we define (co)frame fields $e^{a},e_{a}$ that everywhere obeys $e^{a}e_{b}=\delta_{b}^{a}$. Choosing a ...
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Does the covariant derivative of the unit normal vector field vanish (in this special case)?

Assume we have a Riemannian manifold $(M,g)$, and then define the "cylinder" $M' := M\times\mathbb{R}$. Then $M'$ together with the product metric $g'_p : (v_1,r_1),(v_2,r_2)\mapsto g(v_1,...
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An Compact Expression for the Tensor Laplacian

Let us have a second rank tensor $T$. So how the components of tensor Laplacian of $T$ be computed? I mean if I expand $\nabla^\lambda \nabla_\lambda T_{\mu\nu}$, that would be a monstrous expression! ...
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Are normal coordinates a unique property of the Levi-Civita connection?

It is a well known fact that given a Riemannian manifold $M$, then for all $x\in M$, there exists a local orthonormal frame $\{e_i\}$ on an open neighborhood of $x$ such that: $$(\nabla e_i)(x)=0$$ ...
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Proof that $(\nabla\nabla s)(X,Y)=\nabla_X\nabla_Y s-\nabla_{\nabla_XY}s$

Let $M$ be a Riemannian manifold and $\nabla$ a covariant derivative on a vector bundle $E$. In Heat Kernels and Dirac Operators the connection Laplacian $\Delta$ is defined by $$\Delta=-\mathrm{Tr}(\...
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How is the connection 1 form defined?

I am studying connection on principal bundles and the definition I've encountered is the following "A connection over a principal G-bundle p:P$\rightarrow$B is a splitting of the Altiah sequence $...
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