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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Torsion and curvature of a linear connection

Could you help me to solve the following problem ? Let $M$ a parallelizable manifold of dimension $n$, {$E_1$,...,$E_n$} a global frame of $M$. Let $X$,$Y$ a vector fields on $M$ with $Y= \sum_{i=1}^...
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How different definitions of connections fit together?

I'm working my way through Ivey and Landsberg, Cartan for Beginners, and I'm working on 2.6.13.2(b). After defining, for $X \in \Gamma(TM)$ a vector field, with section $s:M \rightarrow \mathcal{F}_{...
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Quick question: Extension of vector bundles on a compact Riemann surface

Given the following short exact sequence of holomorphic vector bundles on a compact Riemann surface: $0\rightarrow M\rightarrow E \rightarrow N\rightarrow 0$ Fix a hermitian metric on $E$ and $n=...
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Invariant connection form

Let $\pi : E \to \mathbb{R}^3$ define a vector bundle with a connection form $\nabla$ on $\mathbb{R}^3$. The text that i'm am reading then goes on to say that $\nabla$ is $\mathbb{R}$-invariant in ...
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Construction of connections given in “From Calculus to Cohomology”

I'm struggling to understand part of the descripition of a connection on page 167 of Madsen, Tornehave "From Calculus to Cohomology". Immediately after defining a connection $\nabla$ on a bundle $\xi$ ...
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Covariant derivative fullfils Levi-Civita in Euclidean space

$\newcommand{\Reals}{\mathbf{R}}$In our lecture, when we introduced the Levi-Civita connection, we had as an example the directional derivative of a vector field $X$ in direction of another ...
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computational insight behind why connections fix the shape of surface

Based on a video lecture, I had some queries. If we just have a manifold [M-set,O-topology,A-atlas] say $S^2$, this manifold represents a football or a potato equally. But once we choose a connection $...
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Metric non symmetric connection in the tangent bundle?

Let $\Sigma$ be a surface endowed with a Riemannian metric $g.$ According to the fundamental theorem of Riemannian geometry, there exist a unique $\nabla$ symmetric connection ( i.e. torsionless) in ...
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Covariant derivative in $\mathbb{R}^n$

I am studying my lecture notes on covariant derivative, and is having difficulty to do a computation: Suppose $X,Y$ be smooth vector fields in $\mathbb{R}^n.$ Consider the integral curve $c_p:(-\...
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Quick question: Curvature form of a connection on the trivial bundle

Let $L=\mathbb{R}^2\times U(1)$ be the trivial $U(1)$-bundle over $\mathbb{R}^2$. Define a connection $\nabla=d+A$ where $A=fdx+gdy$ is an $\mathbb{R}$ valued $1$-form on $L$. That is, $\nabla$ gives ...
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Can we define flat connection on any given smooth manifold?

For example, a sphere $S^2$ in $\mathbf{R}^3$ is apparently not flat with respect to the Euclidean connection, but can we define a flat connection and thus with affine charts on $S^2$?
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Compute Christoffel symbols of a rotating cartesian coordinate system

Suppose we have a smooth manifold $(\mathbb{R}^3, \mathcal{O}_{\mathbb{R}^4}, \{(\mathbb{R}^3,x),(\mathbb{R}^3,y)\},\nabla,t)$ where $t:\mathbb{R}^3\rightarrow\mathbb{R}$ is such that $t(a,b,c,d)=a$, $...
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Definition of formal adjoint of covariant derivative

I read in Einstein Manifolds, L. Besse that the covariant derivative $D: \mathcal{J}^{(r,s)}(M)\to \Omega^1(M)\otimes\mathcal{J}^{(r,s)}(M)$ admit an formal adjoint $D^*:\Omega^1(M)\otimes\mathcal{J}^{...
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Bott's topological obstruction to integrability

Let $M$ be a smooth manifold, $E\subseteq TM$ be an integrable distribution and $\pi:TM\to Q=\frac{TM}{E}$ the quotient bundle with $\dim Q=q$. Bott showed that \begin{equation} Pont^k(Q)=0 \qquad \...
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Geodesics, isometries and connections.

I am trying to prove with the differential definition that isometries preserve geodesics. That is: Let $c(t)$ be a geodesic in a $n$-dimensional semirriemannian manifold $M$ and $g$ an isometry of $M$ ...
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local representation of “logarithmic connection”

Let X be a Riemann compact surface, $D\subset X$ be a finite subset, and (E,$\nabla$) be a logarithmic connection. And let $z$ be a local coordinate at $p\in D$, why $\nabla $ can be written by: $\...
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Computing Curvature of a Connection (Dirac Monopole)

I'm trying to verify some computations in a paper I'm reading and am feeling a little lost. In particular I haven't been able to properly compute curvature of a connection acting on a line bundle. ...
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How are the extended de Rham differential and the covariant derivative related by Cartan's 2nd structural equation?

I am reading Prof. N.Poncin's notes on fibre bundles and connections. You can find it here:https://orbilu.uni.lu/bitstream/10993/14274/1/MM4-9November2011.pdf In $\it{Section \ 4.5.2}$ the author ...
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Given a $1$-form $\omega$ on $\Bbb R^n$, is there a connection whose torsion is $T(X,Y)=\omega(X)Y-\omega(Y)X$?

Consider $(R^n, g_0 )$, where $g_0$ is the Euclidean metric, and a differential $1$-form $\omega$ on $R^n$. Can this differential form define a connection on $M=R^n$ such that its torsion is $$T(X,Y)=...
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Induced connection on an oriented regular surface $S\subset\mathbb{R}^3$

Let $S\subset\mathbb{R}^3$ be an oriented regular surface - then we have an embedding $\iota:S\rightarrow\mathbb{R}^3$ where $\iota$ is the inclusion of $S$ into $\mathbb{R}^3$. We also have a smooth, ...
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Example of a degenerate metric which doesn't have the Levi-Civita connection

The proof of existence of the Levi-Civita connection for pseudo-Riemannian manifolds uses heavily the fact that the metric is non-degenerate - so that $\nabla_XY$ is characterized by all the values $\...
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Connection planes associated to differential 1-forms

In textbook in differential geometry such an idea appears: there is a tangle bundle $\pi:TM\rightarrow M$ and we are actually looking at the trivialization $\pi^{-1}(U)=U\times \mathbb{R}^n$. We are ...
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Choice of a horizontal tangent space of a principal bundle

Let $\pi:P\to M$ be a principal bundle with group $G=\pi^{-1}(p)$, and let $u\in P$ and $p=\pi(u)$. As I understand it, the choice of the vertical tangent space $V_uP=\mathrm{ker}(\pi_*)$ is natural, ...
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Two definitions of the first Chern class

there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome. So, first approach: say that I have a complex vector bundle $E\to M$; I can ...
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32 views

Definition of Levi-Civita connection map?

Does anyone know definition of Levi-Civita connection map that defined as $TTM\to TM$. I would appreciate if you could give a good reference. Thanks.
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Exterior derivative for functions with values in a parallelizable manifold

In Sharpe's text on Cartan geometry, he explains in section 1.5 on page 52 how to define an exterior derivative for maps into a parallelizable manifold $N$. Let $f: M \to N$ be a smooth map, and $\...
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39 views

How many distinct flat connections are there on a flat bundle?

Given a flat smooth vector bundle (i.e. with constant transition functions), how many distinct flat connections could we put on it? If the flat connection is not unique, is it unique up to gauge ...
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1answer
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Effect of mapping of principal fiber bundles on a principal connection

Let $\lambda = (P,\pi,M,G)$ and $\lambda' = (P',\pi',M',G')$ be two principal fiber bundles, $\lambda$ being equiped with a principal connexion whose connexion form is $\omega$. If $(f,k,\rho)$ is a ...
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Index attached to derivative operator

So in "General Relativity", Wald introduces a derivative operator $\nabla$ on a smooth manifold $M$ that sends $(k, l)$ tensors to $(k, l + 1)$ tensors. One of their properties he says is that if $f \...
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definition of covariant derivative (along curve)

An affine connection on a smooth manifold $M$ is a map $\nabla: \mathcal{V}(M) \times \mathcal{V}(M) \to \mathcal{V}(M)$ satisfying several properties, where $\mathcal{V}(M)$ denotes the set of smooth ...
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How to reparametrize a geodesic such that $\nabla_V V = 0$

In Nakahara's section on geodesics he says that $\nabla_V V = fV$ can be reparametrized to give $\nabla_V V = 0$ if we use the reparametrize the tangent vector components $t \rightarrow t'$ such ...
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Question about homogeneity of the geodesic

I have the following questions about the Homogenity of geodesic: Let $\gamma:(-\delta,\delta)\to M$, where $t\to\gamma(t,q,v)$ is a geodesic, then $\gamma:\left(-\dfrac{\delta}{a},\dfrac{\delta}{a}\...
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Affine manifolds which are *not* locally symmetric

Let $M$ be a manifold equipped with an affine connection $\nabla$. By standard theorems of differential geometry we have convex normal coordinate balls around every point in which geodesics are axis. ...
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Proving the Ricci identity

I'm trying to prove the Ricci identity Let $Z^a$ be a vector field, $R^a_{\,bcd}$ the Riemann curvature tensor and $\nabla$ a torsion-free connection. Then: $\nabla_c\nabla_dZ^a-\nabla_d\nabla_cZ^...
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1answer
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Are curvature forms in complex line bundles symplectic

I know that the curvature form $F_\nabla$ of a connection $\nabla$ in a complex line bundle $L \to B$ is presymplectic (i.e. antisymmetric and closed). Does it also have to be non-degenerate, i.e ...
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The tangent space of the moduli space of connection?

I'm reading one of Floer's paper. (An Instanton-Invariant for 3-Manifold). Let $M$ be a $3$-manifold. A principal $SU_2$-bundle P over $M$ must be trivial. Fixed a trivialization $P \cong M \times ...
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What is the skew-symmetric part of the covariant derivative of a one-form?

This is a followup question to here. Let $E \to M$ be a vector bundle with connection $D := \nabla$. Extend $D$ to $E^*$ and $\text{Hom}(E, E)$. Let $E = TM$ here, and suppose that the torsion is $0$:...
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Equivalent definition of metric compatibility for a connection: what does $\nabla g$ mean?

So the definition I know for metric compatibility is: $$Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),$$ which make sense, as $g(Y,Z)$ is a smooth function from the manifold to reals and we think of $X$ as ...
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Equivariant flat U(1) bundle on a torus

I am trying to understand equivariant flat $U(1)$ bundles on a torus, $(S^1)^N$. By equivariant, I mean equivariance with respect to the natural action of $(U(1))^N$ on $(S^1)^N$. $G$-equivariant ...
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1answer
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Differing definitions of a connection on a vector bundle

My lecture notes define a connection on a vector bundle $\pi:E\rightarrow{M}$ to be an $\mathbb{R}$-linear map: \begin{equation} \nabla:\Gamma(E)\rightarrow\Gamma(T^*M\otimes{E}) \end{equation} ...
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52 views

Proving that Levi-Civita connection is preserved by isometries

I am trying to prove that given two Riemannian submanifolds $S,S'$ with Levi-Civita connections $\nabla , \nabla'$ and an isometry $f$, then $$ Df(\nabla_XY)=\nabla'_{X'}Y' $$ where, $X',Y'=Df(X),Df(...
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Chern Weil theory-independence from the choice of connection

In Milnor-Stasheff's book about characteristic classes there is an appendix about Chern-Weil theory. Suppose that $E \to M$ is a (complex) vector bundle with connection $\nabla$: this connection can ...
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Designing a Formula?

QUESTION: The half-life of a drug is the time it takes for a dose to reduce to half its initial amount. A doctor prescribes a 4mg sleeping pill with a half-life of 24 hours. Write down a ...
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Gauss formula for a 3-sphere embedded in $\mathbb{R}^4$

Given connections $\nabla$ and $\bar{\nabla}$ as connections on $\mathbb{R}^4$ and the 3-sphere of radius $r$: $\mathbb{S}^3(r)$, the vector fields $X,Y$ tangent to $\mathbb{S^3}(r)$, how do I obtain ...
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Understanding the notion of a connection and covariant derivative

I have been reading Nakahara's book "Geometry, Topology & Physics" with the aim of teaching myself some differential geometry. Unfortunately I've gotten a little stuck on the notion of a ...
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1answer
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connection and change of local trivialization

$E\to B$ is a vector bundle. $\nabla$ is a connection of this bundle. Choose a local trivialization of $E$, then we can write a formula of $\nabla$ with respect to this local trivialization: $\nabla(s)...
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Derivative along a curve

Suppose $M$ is a hypersurface of the sphere $S^n \subset \mathbb{R}^{n+1}$, and denote the riemannian connections of $M$, $S^n$ and $\mathbb{R}^{n+1}$ by $\nabla, \overline{\nabla}$ and $\tilde{\nabla}...
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Prove that the curvature of a connection on a line bundle is a global two form

For a connection $\nabla$ on a line bundle, in a local trivialisation, the connection looks like a one form $\nabla s=ds + sa$ but this is not a proper one form cause it depnds on the choice of local ...
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Relationship Between Connections on a Vector Bundle and a Riemannian Base

I'm starting to get acquainted with how to define affine connections on a vector bundle. Suppose $\pi: E \to M$ is a rank $k$ vector bundle over a Riemannian manifold $M$ with metric $g$, where we ...
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1answer
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Implicit formula for the Levi-Civita connection

Let $(M, g)$ be a Riemannian manifold and $X, Y, Z$ smooth vector fields on $M$. Let $\theta_X$ be the $1$-form defined as $\theta_X(Y) = g(X,Y)$ and let $d\theta_X$ be its exterior derivative. Let $...