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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Some incorrect terms in generalizing Bochner's Formula

I am interested in Bochner's Formula but for slightly more general applications. In particular, I am interested in $\Delta g(U,V)$ where $U$ and $V$ are vector fields on a manifold. This involves ...
1 vote
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Clarification regarding the transformation law of the Christoffel symbols

I'm learning about general relativity from Sean M. Carroll's textbook. I recently encountered the transformation law for the Christoffel symbols, and I'm confused, as it seems like I'm seeing two ...
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(Necessary and sufficient) Conditions for the Ricci tensor of an affine connection to be symmetric.

Let $\nabla$ be an affine connection on a smooth manifold $M$. It is widely known, that if $\nabla$ is torsion-free, then its Ricci tensor is symmetric iff there exists a volume form $\omega$ on $M$ ...
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Why $\omega|_S\in\Omega^1(S,\mathfrak{g})$ iff $\omega(\nabla,e)\in\Omega^1(U,\mathfrak{g})$?

Let $M$ be a smooth manifold, let $\mathcal{S}$ be a $G$-strucutre on $M$ and let $\nabla$ be a connection on $TM$. Let $\omega\in\Omega^1(\text{Fr}(TM),\mathfrak{gl}_r)$ be the connection 1-form ...
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Differential of a connection form of a Principal Bundle

In Hamilton's "Mathematical Gauge Theory", he defines the curvature as follows, $$F(X, Y) = dA(\pi^H(X), \pi^H(Y))$$ However, he hasn't defined what $d$ is for a vector-valued one form is. ...
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1 vote
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Sign discrepancy in covariant derivative

Suppose that $E\rightarrow M$ is a vector bundle over a differentiable manifold, equiped with a connection $\nabla$. The connection $\nabla$ induces connections on the various vector bundles ...
1 vote
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Equivalent forms of second Bianchi identity on $TM$

$\DeclareMathOperator{End}{\mathrm{End}}$ This question is already asked here Second Bianchi identity on tangent bundle but with no answer. Let $M$ be a smooth manifold, and $E \to M$ a smooth vector ...
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Construct a torsion-free connection on Lie group

Let $G$ be a Lie group, and consider a connection for left-invariant vector fields on $G$ defined as $$\nabla_X(\sum_j\alpha_jZ_j):=\sum_j(X\alpha_j)Z_j$$ where $\{Z_j\}$ is a global frame for left-...
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An equivalent condition of parallel mean curvature vector in normal bundle (in an orthonormal frame)

Question: Let $f:M\rightarrow (\bar{M},\bar{g},\bar{\nabla})$ be an isometric immersion of Riemannian manifolds, let $\{e_1,\dots,e_m,e_{m+1},\dots, e_{m+p}\}$ be a (local) orthonormal frame, with its ...
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Connection on $U(1)$-bundle vs. 1-form

My original idea of a $\mathfrak{u}(1)$-valued connection $\omega$ is that it's simply a normal $1$-form. But in the middle of page 3 of https://www.arxiv.org/abs/math/0511710 he says "a ...
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Understanding the rolling motion of a circle on $\mathbb{R}$ in terms of parallel transport and Ehresmann connections

I am trying to understand a toy example to help build my intuition about connections on fiber bundles and parallel transport. My main issue is trying to understand if and how the "no-slip" ...
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On the uniqueness of the induced connection on the restriction bundle

Let $M$ be a smooth manifold and let $E\xrightarrow{\pi} M$ be a vector bundle on $M$. Let ${\nabla}$ be a connection on $E\xrightarrow{\pi} M$. I'm trying to show that given any open subset ...
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Parallel transport of a hermitian form on the fiber $E_x$

I have the following problem: Let $E\rightarrow M$ be a complex vector bundle over a complex manifold $M$, and let $v_x$ a hermitian form defined on the fiber $E_x$ such that it is invariant under the ...
1 vote
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Derivation of a spin connection in general relativity

On my journey to understand the mathematical structure behind general relativity, I came across the concept of a spin connection. which (I understand) is a connection defined in a spinor bundle. The ...
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How to define twisted connection on vector bundle

I got motivation from twisted Cohomology where we twisted the derivative $d_\psi=d+\psi\wedge$ and find cohomology class $H_{\psi}^k(M)$ where $\psi$ is closed one form. I try to define twisted ...
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Question about the Riemannian connections and metric

Suppose for the Riemannian manifold $M$ we have the Levi-Civita connection as $\nabla^{LC}$ and $g$ be the Riemannian metric. Now my question is suppose for an arbitrary dual connection on the ...
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Understanding of connections differential geometry

I have some questions regarding connections and christoffel symbols. The definition of connections im working with is simpler than the more general definition on a vector/tensor bundle, it is the ...
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A symmetric affine connection is compatible with $g$ if it preserves length or angle of parallel vector fields.

Here is one of my homework: Let $(M^n,g)$ be a Riemannian manifold, $\nabla$ be a symmetric affine connection. Show that $\nabla$ is a Riemannian connection if it satisfies one of the following ...
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Connection 1-form associated to tensor product of connections

I'm not an expert on this topic so I need some guidance. I have a complex line vector bundle $$V\otimes \overline{\Bbb C}\to D,$$ where $\overline{\Bbb C}\to D$ is the trivial rank $1$bundle complex ...
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An action on the space of connections on a $SO(3)$-vector bundle

Let $M$ be a Riemannian manifold and $E$ a smooth $SO(3)$-vector bundle over $M$ with fiberwise metric. Suppose there is an action of a finite cyclic group $\Bbb Z_n$ on $M$ by isometries, and there ...
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Covariant derivative on a product manifold

Let $\mathcal M$ and $\mathcal N$ be smooth manifolds. Suppose $X_1$ and $Y_1$ are vector fields on $\mathcal M$ and $X_2$ and $Y_2$ are vector fields on $\mathcal N$. How does one go about defining ...
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Understanding complexified Levi-Civita connection in complex geometry

Proposition $3.18$ From this note we have, Let $(X,J,g)$ be a Hermitian manifold. If we denote the complexified Levi-Civita connection by $\nabla$. $\nabla$ is characterized as the only connection on ...
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Covariant derivative induced by pullback connection under an automorphism

Let $G$ be a Lie group and $\pi:P\to M$ a smooth principal $G$-bundle. Let $\omega$ be a connection on $P$; it is a $\mathfrak{g}$-valued 1-form on $P$ where $\mathfrak{g}$ is the Lie algebra of $G$. (...
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Smoothness of horizontal bundle defined by connection one-form

Let $G$ denote a Lie group and $\mathfrak{g}$ its Lie algebra. $P$ is a smooth principal bundle. Given a smooth $\mathfrak{g}$-valued one-form $\omega_p: TP_p \to \mathfrak{g}$, which fulfils the ...
1 vote
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Notational Ambiguity: Covariant Derivative

Let $M$ be a smooth manifold and $\nabla$ the Levi-Civita connection. Now, I am a bit puzzled by a serious notational ambiguity, namely for the second covariant derivative. To explain myself, let us ...
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Obtaining a connection on a trivial bundle by giving a matrix of $1$-forms
I'm new to connections and I'm going over the page (https://mathworld.wolfram.com/VectorBundleConnection.html) in which they state the following For example, the trivial bundle $E=M\times \Bbb R^k$ ...