# 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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### Is this figure interpretable in terms of Holonomy Group and connections with different metrics?

Given the following picture I interpret the two loops as moving orthonormal frames along Levi-Civita connections. In particular, based on the paragraph about parallel transport, I recognize that the ...
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### Defining geodesics in differential geometry and reparametrizing

In a differential geometry course we were given the following two definitions of a geodesic on a manifold $M$ with $TM$ the tangent bundle on $M$: Definition 1: Let $V \in \Gamma(\gamma, TM)$ for an ...
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### Levi-Civita connection for euclidian spaces and surfaces

One approach to construct the Levi-Civita connection for a surface $S \subseteq \mathbb{R}^3$ is to construct the connection first for $\mathbb{R}^3$ and then apply a suitable orthogonal projection to ...
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### What does holonomy measure?

I have difficulty understanding conceptually what holonomy measures. it can return a phase shift of the vector transported parallel along the connection. If there is no phase shift, it means that the ...
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### Algebraic intuition of the Levi-Civita connection

I was wondering about the difference between the algebraic approach to the Levi-Civita connection and the purely geometric one. Let me explain myself a little better. So, I'm just entering the world ...
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### Flat connection form [closed]

If $\omega^i_j$ is a flat connection form, then does it necessarily mean $d\omega^i_j = 0$, (i.e. its exterior derivative equal to zero) ?
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### Defining Connection in Hom Vector Bundle without Coordinates, using Dieudonne's Definition

In Dieudonne's Treatise on Analysis (Volume III Section 17.16) the following definition for a linear connection in a vector bundle is given: A linear connection $C$, in a vector bundle $(E,\pi, M)$ ...
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### Constructing a connection $1$-form from local forms.

I am following Section $10.1.3$ of Geometry, Topology and Physics by Nakahara, and have ran in to an issue regarding local connection forms. Consider a principal $G$-bundle, $P(M,G)$, and an open ...
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### Explicit example of an Ehresmann connection on the trivial bundle $\mathbb{R}^4 \times S^1$

$\mathbb{R}^4 \times S^1$ is of course a principal $U(1)$-bundle and trivial. However, I cannot find an explicit example of the Ehresmann connection one-form on this bundle in terms of the coordinates ...
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### Computation of the pullback of connection along a curve and horizontal lift with respect to the pullback connection

A connection on a vector bundle $\pi : E \to M$ is a distribution $\Delta \subset TE$ such that for every $v \in E$, the map $$D\pi(v)_{|\Delta_v} : \Delta_v \rightarrow T_{\pi(v)}M$$ is a smooth ...
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### Second fundamental form of an immersion

I cannot quite replicate the following calculation from Aubin's Some Nonlinear Problems in Riemannian Geometry (page 349). The setting is as follows. Let $(M^n,g)$ and $(\tilde{M}^m,\tilde{g})$ be two ...
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### About the curvature of a Cartan connection

Let $M$ be a manifold, $L$ a Lie group and $L_0 \subset L$ a closed subgroup such that $\text{dim } M = \text{dim }L/L_0 = n$. Let $P$ be a $L_0$-bundle over $M$ and let $\omega$ be a Cartan ...
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### Why we define connection on a vector bundle in such an unnatural way? [closed]

From Wikipedia, we have: Let $E \to M$ be a smooth vector bundle over a differentiable manifold $M$. Let $\Gamma(E)$ be the space of all smooth sections. A connection on $E$ is an $\mathbb{R}$-linear ...
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### Local coordinates of one form on a principal bundle

I am reading Natural and Gauge Natural Formalism for Classical Field Theory by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates. Let's say we have a ...
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### What kind of connection is the directional derivative?

Given a scalar function $f(X)$ as a function of coordinates I can write it's directional derivative in direction $\hat{u}$ as: $$D_{\hat{u} } f = \nabla f \cdot \hat{u}$$ This is pretty easy to think ...
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### Connection on a bundle

Let $\nabla_A$ and $\nabla_B$ be two connections on a vector field $E$ over a compact manifold $M$, we know that $\nabla_B=\nabla_A+a$ for some $a\in\Omega^1(X;\text{End}(E))$. And for any smooth ...
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### On the proof about the dimension of the conformal group of a manifold

I have been reading the book "Transformation Groups in Differential Geometry" by S. Kobayashi. More concretely, I am trying to understand the proof of the Theorem 6.1 of Chapter IV. Theorem ...
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### Cylinder Parallel Transport

In general the parallel transport of a certain vector between two points p and q in a given surface may change depending on the curve used to go from p to q. However, in a cylinder the parallel ...
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### Questions about Connections on normal bundles & infinitesimal action of a vector field

I've just started reading a paper called "Formules de Localisation et Formules de Paul Lévy" by Bismut, and I didn't understand two things : 1). Let $M$ be a compact connected oriented ...
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### Relation between connection and connection form

In the smooth setting, we define a connection in a principal $G$-bundle $(P, \pi, M)$ as a smooth assignment to each point $p \in P$ of a subspace $H_p P$ of $T_p P$ such that the following two ...
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### Symmetry of a connection and Christoffel's symbols

To prove that the symmetry of a connection is equivalent to the symmetry of the Christofell's symbols in $i$ and $j$, i.e $$\nabla_XY-\nabla_YX-[X,Y]=0\iff \Gamma_{ij}^k=\Gamma_{ji}^k$$ I have thought ...
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### Tensor product used in this notation

A connection form is a Lie-algebra-valued one form $\omega\in\Omega^1(P,\mathfrak{g})$ on a principal bundle P. I have also seen this written this as $\Omega^1(P)\otimes\mathfrak{g}$, but I do not see ...
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### Covariant derivative of the parallel transport operator.

I know that it is possible to define a covariant derivative of a tensor field given the parallel transport operator, but I wonder whether the covariant derivative of the parallel transport operator ...
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### Definition of horizontal subspace using horizontal lift

Let $\pi: E \longrightarrow M$ be a vector bundle, $e \in E$ and $h: T_{\pi(e)}M \longrightarrow T_eE$, $X \mapsto \tilde{X}=\sigma_*(X)$, where $\sigma$ is any local section of $E$ which is parallel ...
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### Lie bracket of tangent vectors in a submanifold is tangent to the submanifold

Let $M$ be a smooth manifold and $N$ a submanifold of $M$. Let $X$ and $Y$ be vector field of $N$. We know that the differentials on $M$ of those vector fields $d_XY$ and $d_YX$ are not necessarily ...
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### Connections and Projection Maps

Let $M$ be a manifold and let $\pi: TM\rightarrow M$ be the projection map. Taking the pushforward of $\pi$ we obtain a bundle map $$\pi_\ast: T(TM)\rightarrow TM.$$ Let $$V=\ker \pi_\ast$$ be ...
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### Section 4.2 in Loring Tu's Differential Geometry

Section 4.2 in Loring Tu's Differential Geometry: My Question: Since $D_XY −D_YX = [X,Y]$, then why define the quantity $T(X,Y)=D_XY −D_YX - [X,Y]$? Isn't $T$ always equal to $0$? I got very confused,...
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### Skew-symmetry $\nabla_{[X,Y]}Z = -\nabla_{[Y,X]}Z$

I'm new to the theory of affine connections and have yet to become comfortable working with. I wanted to confirm to myself that the curvature of an affine connection is skew-symmetric which boils down ...