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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 ...
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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 ...
Aidan Beecher's user avatar
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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 ...
Armando Patrizio's user avatar
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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. ...
Jeff's user avatar
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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 ...
Richard Muniz's user avatar
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1 answer
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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 ...
Alex Pawelko's user avatar
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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-...
Gao Minghao's user avatar
2 votes
1 answer
58 views

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 ...
Zoudelong's user avatar
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Constructing a $1$-form that vanishes iff the connection is compatible with the $G$-structure

I'm studying $G$-structures from Crainic's lecture notes. I'm stuck on the proof of Proposition $4.18$ at page $122$. Let $\mathscr{N}(\mathfrak{g}_S)$ be the normal vector bundle, i.e.: $$\mathscr{N}(...
Armando Patrizio's user avatar
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Covariant derivative defined by Maurer-Cartan form

Let $G$ be a matrix Lie group with Lie algebra $\mathfrak{g}$ and let $H$ be a closed Lie subgroup with Lie algebra $\mathfrak{h}$ embedded in $\mathfrak{g}$. Suppose we have chosen a complementary ...
lolabol's user avatar
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1 answer
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A question about dual connection in bundle AdE

I am currently studying Jost's book Riemannian Geometry and Geometric Analysis seventh edition, Chapter 4 Section 2 Metric Connections. The Yang-Mills Functional. he defines an operator $D^*$dual to ...
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Why $\omega|_S\in\Omega^1(S,\mathfrak{g})$?

Let $S$ be a $G$-structure on $M$ and let $\nabla$ be a connection on $M$ compatible with $S$, i.e. the parallel transport preserves $S$. Consider now the induced principal bundle connection on $Fr(TM)...
Armando Patrizio's user avatar
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2 answers
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Is the $k$th covariant derivative of a function $f\colon M\to\mathbb R$ symmetric?

In the flat case ($\mathbb R^N$ with the Euclidean metric) it is true that the $k$th covariant derivative $\nabla^kf$ is symmetric because its coordinates are: $$(\nabla^kf)_{i_1\ldots i_k}(x)=\dfrac{\...
Raoní Cabral Ponciano's user avatar
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2 answers
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Confusion of definition of covariant derivative along curve

I am currently studying Riemannian geometry, and have come across the following proposition: Proposition. Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$ and let $\gamma:I\...
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Determine the matrix valued one-form in Chern connection on $(E,h)$

In Proposition 4.2.14 of Huybrechts complex geometry, it is stated that there exists a unique Hermitian connection $\nabla$ that is compatible with the holomorphic structure, known as the Chern ...
領域展開's user avatar
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1 answer
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Connections on principle G-bundles via parallel transport

Suppose we are given a principle $G$-bundle $P\overset{\pi}{\to}M$ and let $\mathfrak{g}$ denote the Lie algebra of $G$. Normally one defines a connection on the principle $G$-bundle $P\overset{\pi}{\...
Matthew Lou's user avatar
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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 ...
user615345's user avatar
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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" ...
Tob Ernack's user avatar
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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 ...
GeometriaDifferenziale's user avatar
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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 ...
kahlerian's user avatar
1 vote
1 answer
108 views

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 ...
Tomás's user avatar
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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 ...
N00BMaster's user avatar
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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 ...
Amit Vishwakarma 's user avatar
3 votes
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Computing the connection form $\omega_1^2$ for $ds^2 = dt^2 + f(t)^2d\theta^2$

For $S^1 \times I$ where $I \subset \mathbb{R}$ consider the following metric $$ds^2 = dt^2 + f(t)^2d\theta^2.$$I want to compute $\omega_1^2$, the connection form. I let $\theta^1 = dt$ and $\theta^2 ...
user57's user avatar
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Ehresmann connections on general G-bundles?

There are two definitions of connections on bundles in terms of horizontal bundles: In the case of a principal $G$-bundle $P$, a connection is a subbundle $HP<TP$ such that $HP\oplus VP=TP$ and it ...
Alex Bogatskiy's user avatar
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Coframe on coset space using Maurer-Cartan form

Suppose $G$ is a Lie group and $H$ is a Lie subgroup. Then, consider the principal $H$-bundle, $\pi:G\longrightarrow G/H$ where $G/H$ is coset manifold and $\pi$ is the canonical projection. Then, we ...
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Doubts about alternative definition to connection in differential geometry

We consider a variety M and coordinate bases of the tangent space ${∂_i}^N_{i=1}$, and of the tangent space ${dx^i}^N_{i=1}$. We consider the following definition of connection ∇ on the variety, as an ...
Guillermo Fuentes Morales's user avatar
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Connection with constant Christoffel symbols

Assume that $(\mathcal{M},\nabla)$ is a manifold diffeomorphic to $\mathbb{R}^n$ equipped with an affine connection $\nabla$. Assume that there is a chart in which the Christoffel symboles, defined as ...
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Computing the tangent space of the orbit of a gauge group action at a connection

Let $E\to M$ be a smooth real vector bundle, and let $\mathfrak{G}$ be the group of smooth bundle automorphisms. (The Lie algebra of $\mathfrak{G}$ is the space $\Omega^0(\text{End}(E))$.) For a ...
blancket's user avatar
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1 vote
1 answer
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Two $SO(2)$ connections on a smooth $SO(2)$-vector bundle

Let $L\to M$ be a smooth $SO(2)$-vector bundle. We can identify the Lie algebra $\mathfrak{so}(2)$ with $i\Bbb R$. Suppose $\nabla, \nabla'$ are two $SO(2)$-connections on $L$ such that $\nabla=\nabla'...
blancket's user avatar
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1 answer
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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 ...
John Doe's user avatar
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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 ...
Zoudelong's user avatar
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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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Linearly independence of parallel vector fields at a point implies globally linearly independence.

Here is one of my homework question: Let $(M^m,g)$ be a connected Riemannian manifold, $\{X_1,\dots,X_r\} \ (r\leqslant m)$ be parallel vector fields. Show that (i) If for some $p\in M$, $\{X_1|_p,\...
Zoudelong's user avatar
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0 answers
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Why is the torsion of a connection in the frame bundle of a manifold $0$ on vertical vectors?

I briefly recall the context of a connection on a frame bundle. Let $M^n$ be a smooth manifold, $p: Fr(M) \rightarrow M$ its frame bundle where a frame $z \in Fr(M)$ is seen as an isomorphism $z:\...
rosecabbage's user avatar
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2 votes
1 answer
50 views

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 ...
blancket's user avatar
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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 ...
markusas's user avatar
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2 votes
1 answer
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Two connections on a smooth $SO(3)$-vector bundle

Let $E\to M$ be a smooth oriented real vector bundle of rank 3, so that its structure group can be reduced to $SO(3)$. Suppose $E\to M$ is given a Riemannian metric and a connection $\nabla:\Omega^0(E)...
blancket's user avatar
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4 votes
1 answer
54 views

Self duality of a connection is invariant under a gauge transformation

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $P\to M$ be a (smooth) principal $G$-bundle over an oriented Riemannian smooth 4-manifold $(M,g)$. Let $E=P\times_{\text{Ad}}\mathfrak{g}...
blancket's user avatar
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6 votes
0 answers
177 views

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 ...
N00BMaster's user avatar
1 vote
1 answer
45 views

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 $G$. (...
blancket's user avatar
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1 vote
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Construction of a connection 1-form

Let $E \to (M,g)$ be a vector bundle over $M$ a compact riemannian manifold and consider the structural group $G$ of bundle. We can define a linear connection over $E$ as a map $\nabla: \Gamma(E) \to \...
Bri3.1's user avatar
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Connection on spinor bundle induced by Levi-Civita Connection

Given a Levi-Civita connection $\nabla$ on the tangent bundle with bundle metric $g$, a spinor representation $(\rho, S)$ and a spin structure $P$, what is the induced connection on a spinor bundle $...
anonymous250's user avatar
4 votes
1 answer
96 views

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 ...
anonymous250's user avatar
1 vote
2 answers
265 views

Relation between curvature form and the Riemann tensor

Let $ (M, g) $ be a Riemannian manifold and $ \omega $ a connection on the tangent bundle of $ M $ at a given point. Then the curvature form is the $\mathfrak{g} $-valued 2-form $ \Omega \in \Omega^2(...
Tomás's user avatar
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Characterization of Derivations

I am reading the article ON THE BATCHELOR TRIVIALIZATION OF THE TANGENT SUPERMANIFOLD by O.A Sánchez Valenzuela. Right at the beginning, the following two statements appear: 1.- Let $E\rightarrow M$ ...
Dr. Bain's user avatar
1 vote
1 answer
47 views

Connection $1$-forms and the local expression $d + A$

Let $M$ be a smooth manifold and $E \to M$ a vector bundle over $M$ with a connection $\nabla$. Locally on an open set $U \subset M$ with a frame $(E_1,\dots,E_k)$, we can write any section $s$ of $E|...
Johannes's user avatar
  • 137
8 votes
1 answer
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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 ...
B.Hueber's user avatar
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3 votes
1 answer
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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 $E=M\times \Bbb R^k$ ...
Johannes's user avatar
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