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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: http://en.m.wikipedia.org/wiki/Connection_(vector_bundle))

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Show that $[\widetilde{X},\widetilde{Y}]^H=\widetilde{[X,Y]}$ and that $[\widetilde{X},W]$ is vertical if $W$ is vertical.

This is problem 9 from chapter 5 in Riemannian Manifolds: An introduction to Curvature. Suppose $p:(\widetilde M,\widetilde g)\rightarrow (M,g)$ is a Riemannian submersion. We denote $\widetilde X$ ...
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Geodesics on a Riemannian manifold under non-Levi-Civita connections

I'm a beginner on this topic—so please comment if anything is ambiguous, unclear, or wrong. In particular, I'm trying to figure out how to think of geodesics under arbitrary connections. Background ...
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Question about a proof on Spivak's Comprehensive Introduction to Differential Geometry

On the Addedum to the Chapter VI, vol. II, the first proposition states that to connections have the same geodesics if and only if their difference tensor is antisymmetric. When proving that if the ...
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Relation between Levi-Civita connection and any another metric connection.

On a Riemannian manifold $(M,g),$ we have $D^g,$ Levi-Civita connection, the only connection that is metric (i.e. $D^g g=0$) and without torsion (i.e. $T^{D^g}=0$). My question is that if I have say $...
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Connection does not depend on entire vector field

Editted to add Lemma 3.4. The book "An Introduction to Differentiable Manifolds and Riemannian Geometry" by William Boothby has the following definition of connection. (3.1) Definition. A $C^\...
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Non-flat connection on trivial bundle?

From what I have read it seems like there exists non-flat connections on trivial vector/principal bundles. However I can't find any notes on it or examples. Can anyone confirm that such connections ...
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Is a curve whose derivative has constant length a geodesic?

Given a (complete) riemannian Manifold $(M,g)$ and a curve $\gamma \colon [0,1] \to M$ with $$g\big( \dot{\gamma} (t) , \dot{\gamma}(t) \big) = C \quad \forall t \in [0,1]$$ where $C \neq 0$ is some ...
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Isometric Immersions: equivalent condition for $R^\perp=0$

In Dajczer's Submanifolds and Isometric Immersions, a couple paragraphs before presenting the Fundamental Theorem for Submanifolds, the author presents Ricci's equation: $$(\widetilde{R}(X,Y)\xi)^\...
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Connection from a covariant derivative

It is a basic question on connections of vector bundles, more preciously how to obtain a connection of a vector bundle starting from a covariant derivative. Suppose $B \to M$ is a smooth vector ...
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Pullback bundle of the tangent bundle through a diffeomorphism and an application to parallel global sections

It is probably a stupid question and it is probably poorly written. Hopefully, it is not already answered somewhere else. (1) Let $f:M \to N$ be a diffeomorphism. Is it true that the pullback ...
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Geodesics of null surface and induced connection

Although this is a question related to GR this is purely a math question, so I think this is the right place to ask it. Let $(M,g)$ be a Lorentzian manifold. Let $\phi : \Sigma\to M$ be one null ...
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to which intrinsic object corresponds connection's hypersurface

Given a complex manifold $M$ and an hypersurface $S$, and some connection on the line bundle associated to $S$, to which intrinsic object of $S$ corresponds the connection ? (more specifically, same ...
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What is $\nabla X$ in Riemannian geometry?

What exactly is $\nabla X$, for a vector field $X$? Can we write it in coordinates? I'm familiar with $\nabla f$, and also $\nabla_Y X$, where both $X$ and $Y$ are vector fields. However, what would $...
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Constructing the flat vector bundle associated to a given linear representation of the fundamental group

I'm reading this notes, and I have some questions about the contruction on page 18. Let $M$ be a connected manifold and $E\rightarrow M$ a flat vector bundle over $M$. Consider $\{(U_\alpha, \phi_{...
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What constitutes a gauge theory? Help me understand electromagnetism as the prototype of all gauge theories

In The Geometry of Physics and Knots, Sir Michael Atiyah says that electromagnetism is the prototype of all gauge theories: The prototype of all gauge theories is electromagnetism. From the ...
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Prove that the covariant derivative commutes with musical isomorphisms

Suppose I have a covector field $\omega$ and a covariant derivative $\nabla_{X}$ for some vector field $X$ on a Riemannian manifold $(M, g)$. Define $X^{\flat} \in \mathfrak{X}^{*}(M)$ as $X^{\flat}(...
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In differential geometry, to what extent does the curvature tensor determine its associated connection?

In the absence of a metric, it is not clear to me to what extent does knowing the curvature tensor determine its associated connection? I would be satisfied knowing this for zero torsion. I'd like to ...
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Topology on the affine space of connections

What is the natural topology we generally define on the Affine space of Connections? I am not able to find any literature where this topology is explicitly described. It would be really helpful for ...
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Does existence of a parallel section on $E$ imply local existence of a parallel section on $\bigwedge^k E$?

Let $E$ be a smooth vector bundle equipped with an affine connection $\nabla$. Suppose that $(E,\nabla)$ admits a non-zero parallel section. I think that $(\bigwedge^k E,\bigwedge^k \nabla)$ does ...
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Literature suggestion for understanding Gauge theory from the perspective of a Mathematician.

Can anyone please suggest some good literature or references for understanding Gauge theory from the perspective of a mathematician (from the point of view of differential geometry)? Being a ...
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Does $\nabla g=\omega(\cdot) g$ imply $\nabla$ is metric w.r.t a conformal rescaling of $g$?

Let $E$ be a smooth vector bundle over a manifold $M$, where $\text{rank}(E) > 1,\dim M > 1$. Suppose that $E$ is equipped with a metric $g$ and an affine connection $\nabla$, such that $\...
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Example on Local Systems: Flat Connections

I am reading about the local system, and a local system $\mathcal{L}$ on $X$ with values in $\mathcal{C}$ is defined as a functor $$ \mathcal{L}:\Pi(X)\to \mathcal{C} $$ where $\Pi(X)$ is the ...
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How to read the expression of an affine connection: $\nabla_X Y$?

I am studying Riemannian Geometry from the textbook Riemannian Geometry by do Carmo (English edition). In section 2 of chapter 2, page 50, he defines an affine connection as follows: 2.1 Definition....
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Existence of a parallel vector field implies a splitting of the metric

Where can I find a proof of the following claim: Existence of a parallel vector field on a Riemannian manifold implies that the metric splits locally as a product of a one-dimensional manifold and $n-...
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Lie derivative of Almost complex structure $J$ along a vector field $X$.

I'd like some guidance in computing the Lie derivative of an almost complex structure $J$ on a smooth closed Riemannian manifold $M$ in terms of the Levi-Civita connection $\nabla$. The formula I'm ...
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Relationship between Carnot-Caratheory Distance and Levi-Civita Connection

Suppose that $G\cong H\times K$ is a nilpotent Lie group, where $H,K$ are Lie-Subgroups for which $H$ and $K$ are Lie-Subgroups such that $H$ is Commutative $K$ is Compact Then $G$ admits a bi-...
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Ordinary and covariant derivative inequality: $ \| u\|_1 \leq C\left( \| \frac{Du}{dt} \|_0 + \|u\|_0 \right) $

Let $I=[0,1]$. Define: $H_0 = L^2(I,\mathbb{R}^3)$ with inner product $\langle u,v \rangle_0 = \int_0^1 \langle u(t), v(t) \rangle \text{ dt}$ $H_1 = W^{1,2}(I, \mathbb{R}^3)$ with inner product $\...
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Local frame inducing a map of principal bundles

Let $V \rightarrow M$ a vector bundle. $P \rightarrow M$ a principal $G$-bundle. Let $\phi:G \rightarrow GL(V)$ be a representation. A local section $s$ for $P$, frame bundle for $V \rightarrow M$ , ...
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Geometric way to view affine connection and parallel transport.

Given a parametrized curve $\gamma$ on a manifold $M$ with metric $g$ and some affine connection on it, we can transport vectors of tangent spaces $T_{p} M$ and $T_{q}M$ to each other (when $p, q \in \...
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Transformation of local connection form in a representation

In this question I am trying to build up the theory of connections but without actually using fiber bundles, working solely with local objects on the base, however I am stuck on something. For the ...
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Existence of connection on dual bundle

I quote the construction given in Madsen's Calculus to Cohomology. This is more or less the construction explained here defining connection on dual bundle. For a vector bundle $\Omega^i(\xi) := \...
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Coordinate independence of connections

So I am trying to prove the following: Let $V \rightarrow M$ be a vector bundle $\nabla$ a connection on $V$. Then there is a unique sequence of linear maps $$ \Omega^0(M;V) \xrightarrow{\...
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Flat connection: holonomy is invariant under homotopy of loops

I am looking for an elementary proof of the fact (and probably not any broader results) that when the connection is flat, holonomy does not change under homotopy of the loop. Although this is a first ...
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covariant derivative and metric compatibility

The requirement that parallel transport preserve the length of vectors is equivalent to requiring that $ \nabla_Z ( g(X,Y) )$ vanish for all vector fields $Z$ and all vector fields $X,Y$ that are ...
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Connection matrix in orientable 2-bundle is skew-symmetric

In the notes I am following to learn about connections, there is the following lemma: whose proof is natural and I understand. Later in the text the author writes the following (referring to a metric ...
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Geodesics in $\mathbb{R}^n$ with the trivial connection

Define geodesic as follows: Given a tangent bundle $TM\rightarrow M$ with connection $\nabla$, a geodesic is a curve $\gamma:I\rightarrow M$ such that $(\gamma^*\nabla)\dot \gamma = 0$. (Notice ...
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Connecting pipes

Suppose we have three pipes in 3D and one end of a pipe mark as inlet and other end marks as outlet. We have to connect outlet edge with other pipe inlet edge. Here is the visual representation of ...
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Connection $1$-form acting on vector fields

I'm reading this paper about the c-map between special Kähler manifolds and Hyperkähler manifolds and in the introduction the authors talk about the cotangent bundle as a certain associated bundle of ...
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Connection/Curvature as a matrix of Real valued forms

Let $P(M,G)$ be a principal $G$ bundle. Let $\omega$ be a connection $1$ form on $P(M,G)$. This is a $\mathfrak{g}$ valued $1$ form on $P$ i.e., for each $p\in P$, we have $\omega(p):T_pP\rightarrow ...
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Given a map $X \to \text{GL}_2(\mathbb{R})$ how do I determine a flat connection on this Riemann surface?

I need help determining the Euler class of this vector bundle $\phi:E\to X$. The base space is the torus $X = \mathbb{R}^2/\mathbb{Z}^2$ and the fiber over each point, $f^{-1}(x) \simeq \mathbb{R}^2$....
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Use of Leibniz rule in a proof envolving a metric connection in a vector bundle

Let $E\rightarrow M$ be a vector bundle with metric $g$ and metric connection $\nabla:\Omega^0(M,E)\rightarrow \Omega^1(M,E)$. I am trying to understood this short proof: I do not understand the ...
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Integral curves are horizontal

Suppose we're given an Ehresmann connection on a submersion (or a fiber bundle, but I don't think it's needed) $\begin{smallmatrix}X\\ \downarrow\\ Y \end{smallmatrix}$. Given a curve $\gamma$ in the ...
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Questions regarding Connections, and in particular, Hermitian Connections

I have been reading Chapter 0 of Griffiths' and Harris' Principles of Algebraic Geometry, in particular, the section on Vector Bundles, Connections, and Curvature. I have three questions: A ...
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Christoffel symbols of the second kind transformation law

We want to show that the Christoffel symbols of the second kind transform like a connection. the Christoffel symbols of the second kind are given by: $$\begin{Bmatrix}a \\ bc\end{Bmatrix} = \frac{1}{...
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Non-trivial explicit example of a flat connection

We all know that the exterior derivative on the trivial bundle forms an example of a flat connection. Can anyone provide an explicit example of a flat connection that is not just the exterior ...
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Showing that a given vector bundle with connection is not trivial

Given the following exercise: where d is the trivial connection. We defined an isomorphism between two vector bundles with connection in the following way: I'm not sure what I have to show. Can I ...
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Levi-Civita connection for a metric in $\mathbb{R}^{3}$

Let $g$ be a metric in $\mathbb{R}^{3}$ defined as $\partial_{x}, \partial_{y}, \partial_{z}$ are orthogonal everywhere, and $g(\partial_{y},\partial_{y})=1, g(\partial_{z},\partial_{z})=f(x), g(\...
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Different almost-complex structures $\Rightarrow$ different complex structures?

Let $(M,J)$ be an almost-complex structure. By definition, if $M$ admits local holomorphic coordinates for $J$ around every point and these patch together to form a holomorphic atlas (complex ...
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Symplectic connections are (locally) Levi-Civita connections

I was wondering... Is every symplectic connection $\nabla$ on some symplectic manifold $(M, \omega)$ the Levi-Civita connection of some metric $g$ on $M$? What about the local statement?
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Torsion-freeness of a connection and anti-symmetrization

Let $M$ be an Hermite manifold, and $\nabla$ be the Levi-Civita connection on $TM$ and extend it to $\Lambda^*_{\mathbb{C}}(M)$. Then $\nabla$ is torsion-free by definition. But I read from a paper ...