# 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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### 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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### 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}\...
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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 ...
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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 ...
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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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### 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}(\...
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 \$...