Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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))

2
votes
1answer
46 views

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 ...
2
votes
1answer
33 views

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}(...
3
votes
0answers
41 views

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 ...
0
votes
1answer
51 views

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 ...
5
votes
2answers
76 views

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 ...
2
votes
1answer
55 views

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 ...
2
votes
0answers
51 views

Non-conformal metrics on vector bundles where $\nabla g=\omega(\cdot) g$

Let $E$ be a smooth vector bundle over a manifold $M$ ($\dim M \ge 2$), equipped with a metric $g$ and a connection $\nabla$, such that $\nabla_X g=\omega (X) g$ for every vector field $X$ on $M$. ($\...
0
votes
0answers
12 views

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 ...
2
votes
1answer
37 views

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....
0
votes
0answers
13 views

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-...
2
votes
1answer
40 views

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 ...
2
votes
0answers
9 views

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-...
5
votes
1answer
71 views

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 $\...
0
votes
0answers
23 views

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 $V \rightarrow M$ , ...
1
vote
1answer
19 views

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 \...
0
votes
0answers
18 views

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 ...
1
vote
1answer
52 views

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) := \...
2
votes
1answer
31 views

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{\...
0
votes
0answers
12 views

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 ...
0
votes
0answers
33 views

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 ...
0
votes
0answers
18 views

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 ...
1
vote
0answers
34 views

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 ...
0
votes
0answers
53 views

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 ...
0
votes
0answers
33 views

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 ...
0
votes
0answers
21 views

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 ...
2
votes
0answers
20 views

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, $f^{-1}(x) \simeq \mathbb{R}^2$....
0
votes
1answer
11 views

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 ...
2
votes
1answer
36 views

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 ...
2
votes
1answer
35 views

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 ...
0
votes
0answers
48 views

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}{...
0
votes
1answer
22 views

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 ...
2
votes
1answer
50 views

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 ...
2
votes
1answer
61 views

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(\...
2
votes
0answers
51 views

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 ...
0
votes
0answers
23 views

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?
0
votes
1answer
25 views

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 ...
2
votes
1answer
45 views

Show that any affine connection $\nabla$ on $\mathbb{R}^n$ is of the form $\nabla=D+\Gamma$.

Show that any affine connection $\nabla$ on $\mathbb{R}^n$ is of the form $\nabla=D+\Gamma$, where $D$ is the Euclidean connection and $\Gamma:\mathcal{X}(\mathbb{R}^n) \times \mathcal{X}(\mathbb{R}^n)...
2
votes
0answers
54 views

When (if ever) does a connection on a Lie manifold G define that groups Lie algebra?

Note I study physics so I apologize in advance for poor notation terminology. admittedly I'm very new to Lie algebras and groups (at least in the general sense, I've encountered specific groups many ...
1
vote
1answer
26 views

Uniqueness of horizontal lifts

Consider a fibre bundle $E$ and a certain connection, $TE=H\oplus V$. A path $\gamma(t)\in B$ can be horizontally lifted to a path $\gamma(t)\in E$ according to $\pi\circ\tilde\gamma=\gamma$ and $\...
0
votes
1answer
73 views

How to look at the covariant derivative along a curve?

Let $ \Gamma(M,TM) $ be the space of smooth sections from $M$ to $TM$ and $ (M,g) $ be a Riemannian manifold, $ I \subset \mathbb{R} $ be an interval and $\gamma: I \rightarrow M$ be a $C^1$ curve. ...
0
votes
0answers
23 views

what is the monodromy of an isolated singularity of an affine structure?

Let $\nabla$ be a torsion free flat connection on the tangent bundle of a punctured disc (otherwise known as affine structure). Are there any restrictions on the monodromy of $\nabla$? Is it unipotent ...
0
votes
0answers
41 views

General connections on fiber products

Let $p_1:F_1 \longrightarrow M$ and $p_2:F_2 \longrightarrow M$ two smooth fiber bundles over $M$ and let's suppose that we have horizontal bundles $HF_1$ and $HF_2$ (connections on the bundles). What ...
2
votes
0answers
26 views

a torsion-free connection that preserves a complex structure

Let $(M,I)$ be a complex manifold with a complex structure $I$, i.e. an endomorphism $I$ of the tangent bundle such that $I^2 = -Id$ and such that the subbundle $T^{1,0}$ of eigenvectors of $I \otimes ...
0
votes
0answers
33 views

connections that preserve a given tensor

I am trying to come to grips with the notion of an linear connection and here are two things that I have trouble understanding. Let $M$ be a compact smooth manifold and let $\alpha$ be a section of $...
0
votes
1answer
37 views

Upper bound on distance between trajectories that share same initial position and velocity

Let $\mathcal{M}$ be a manifold. Let $d : \mathcal{M} \times \mathcal{M} \rightarrow \mathbb{R}_{\geq 0}$ be the geodesic distance defined by $$d(x,y) = \inf \left\{ \int_0^1 \|\gamma'(t)\| \,\mathrm{...
1
vote
1answer
34 views

Connection on the cotangent bundle

I'm reading "Differential forms and connections" by R. Darling and I must have made a mistake in problem 4 in section 9.4. It states: Prove that $\nabla_X \omega := \iota_X\, d\omega$ is not a ...
2
votes
0answers
46 views

Isn't $∇^{0,1}=\bar\partial_E+A^{0,1}$?

A connection ∇ on a holo bundle $E$ is called compatible with holo structure if $∇^{0,1}=\bar\partial_E$. And such a connection is called a Chern connection. (reference) p.17 And we know $\nabla=d+A$....
0
votes
1answer
59 views

Connection on the dual vector bundle

(Note: I looked at the other questions about defining a connection on the dual bundle, but the answers do not apply to my case since I use a slightly different definition of connection). I am ...
1
vote
0answers
32 views

Defining a connection in $\mathbb{R}^2$ using a connection $1$-form

I'm reading Hitchin's paper Self-duality Equations on a Riemann Surface (Hitchin, self duality). In the first chapter on pages 63/64 he considers a principal $G$-bundle $P$ over $\mathbb{R}^4$ and a ...
0
votes
0answers
20 views

Fibred charts adapted to principal bundle structures

If $\pi_E:E\rightarrow M$ is a rank $k$ vector bundle (let's assume everything in this question to be real for simplicity), it is the most common to use fibred charts adapted to the vector bundle as ...