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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Is this figure interpretable in terms of Holonomy Group and connections with different metrics?

Given the following picture I interpret the two loops as moving orthonormal frames along Levi-Civita connections. In particular, based on the paragraph about parallel transport, I recognize that the ...
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Defining geodesics in differential geometry and reparametrizing

In a differential geometry course we were given the following two definitions of a geodesic on a manifold $M$ with $TM$ the tangent bundle on $M$: Definition 1: Let $V \in \Gamma(\gamma, TM)$ for an ...
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Levi-Civita connection for euclidian spaces and surfaces

One approach to construct the Levi-Civita connection for a surface $S \subseteq \mathbb{R}^3$ is to construct the connection first for $\mathbb{R}^3$ and then apply a suitable orthogonal projection to ...
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What does holonomy measure?

I have difficulty understanding conceptually what holonomy measures. it can return a phase shift of the vector transported parallel along the connection. If there is no phase shift, it means that the ...
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Geodesics on product manifold

I need to prove this: Let $(M_{1}, g)$ $(M_{2}, h)$ riemannian manifolds, consider $(M_{1} \times M_{2})$ with the product metric, and $\nabla = \nabla^{1} + \nabla^{2}$. Then $\gamma=(\gamma_{1}, \...
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Levi Civita Connection of the product manifold with product metric

I’m trying to prove exercise 1a) down here that I took from Do Carmo "Riemannian Geomtetry", chapter 6, but I’ve been stuck into it for a while. Particularly, if $X_{1} \in \Gamma(TM_{1}), ...
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Algebraic intuition of the Levi-Civita connection

I was wondering about the difference between the algebraic approach to the Levi-Civita connection and the purely geometric one. Let me explain myself a little better. So, I'm just entering the world ...
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Flat connection form [closed]

If $\omega^i_j$ is a flat connection form, then does it necessarily mean $d\omega^i_j = 0$, (i.e. its exterior derivative equal to zero) ?
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Defining Connection in Hom Vector Bundle without Coordinates, using Dieudonne's Definition

In Dieudonne's Treatise on Analysis (Volume III Section 17.16) the following definition for a linear connection in a vector bundle is given: A linear connection $C$, in a vector bundle $(E,\pi, M)$ ...
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Constructing a connection $1$-form from local forms.

I am following Section $10.1.3$ of Geometry, Topology and Physics by Nakahara, and have ran in to an issue regarding local connection forms. Consider a principal $G$-bundle, $P(M,G)$, and an open ...
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Explicit example of an Ehresmann connection on the trivial bundle $\mathbb{R}^4 \times S^1$

$\mathbb{R}^4 \times S^1$ is of course a principal $U(1)$-bundle and trivial. However, I cannot find an explicit example of the Ehresmann connection one-form on this bundle in terms of the coordinates ...
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Computation of the pullback of connection along a curve and horizontal lift with respect to the pullback connection

A connection on a vector bundle $ \pi : E \to M $ is a distribution $\Delta \subset TE$ such that for every $v \in E$, the map $$ D\pi(v)_{|\Delta_v} : \Delta_v \rightarrow T_{\pi(v)}M$$ is a smooth ...
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Second fundamental form of an immersion

I cannot quite replicate the following calculation from Aubin's Some Nonlinear Problems in Riemannian Geometry (page 349). The setting is as follows. Let $(M^n,g)$ and $(\tilde{M}^m,\tilde{g})$ be two ...
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About the curvature of a Cartan connection

Let $M$ be a manifold, $L$ a Lie group and $L_0 \subset L$ a closed subgroup such that $\text{dim } M = \text{dim }L/L_0 = n$. Let $P$ be a $L_0$-bundle over $M$ and let $\omega$ be a Cartan ...
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Why we define connection on a vector bundle in such an unnatural way? [closed]

From Wikipedia, we have: Let $E \to M$ be a smooth vector bundle over a differentiable manifold $M$. Let $\Gamma(E)$ be the space of all smooth sections. A connection on $E$ is an $\mathbb{R}$-linear ...
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Slight confusion about projection operator in principal bundle

Let $P(M,G)$ be a principal bundle with base space $M$ and structure group $G$. Let $\Gamma$ be a connection on $P$ so that the horizontal subspaces are denoted by $\Gamma_u$ for $u\in P$. Let $\pi:P\...
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Pullback connection from a local diffeomorphism under an integral.

Suppose $(M^n, g)$ and $(N^n,h)$ are Riemannian manifolds, and for each $p\in M$ there is a neighborhood $U\subset M$ and a diffeomorphism $\phi:(U, g|_U) \to (N,h)$ such that $g|_U = \phi^*h$. I ...
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Connection defined by parallel transoport

Let $E$ be a vector bundle. I'm facing a problem to prove that a given defined isomorphism between fibers $E_{\gamma(t)}$ is a parallel transport corresponding to a given connection defined on $E$. ...
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Connection on isomorphic principal bundle

I’m trying to find a natural induced connection $1$-form on the isomorphic principal bundle, more explicitly, given a principal bundle isomorphism $(P:P_1\rightarrow P_2$, $\Phi:G\rightarrow H)$ ...
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Reference request on Ehresmann connections

I've been asked read up on Ehresmann connections. I have experience in smooth manifold theory, vector bundles, and Riemannian geometry, but I feel in unfamiliar territory after briefly looking into ...
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Why does $\nabla_{X}Y$ depend only on $X_{p}$? Lee's book on Riemannian geometry

I am trying to understand the following theorem from Lee's book about Riemannian geometry. I had a look at several books but unfortunaletly I am still not quite sure if I get it. Question 1: Why does ...
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Calculating the variation of a scalar field along the two parameter variation $\alpha(t; r, s) = \exp_{x(t)}(rV(t) + sW(t))$

I'm currently reading about a problem regarding second variations of some functional defined on a Riemannian Manifold $M$ equipped with the Levi-Civita connection, and am confused about how to express ...
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Meromorphic connection on simply connected manifold

Let $(\mathcal{E},\nabla)$ be a logarithmic connection of rank $n$, with poles on $D$, on a complex manifold $M$. Suppose $M$ is simply connected. If $M=D(0,1)^{n-1}\times D(0,1) $, $D= D(0,1)^{n-1}\...
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Local coordinates of one form on a principal bundle

I am reading Natural and Gauge Natural Formalism for Classical Field Theory by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates. Let's say we have a ...
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What kind of connection is the directional derivative?

Given a scalar function $f(X)$ as a function of coordinates I can write it's directional derivative in direction $\hat{u}$ as: $$ D_{\hat{u} } f = \nabla f \cdot \hat{u}$$ This is pretty easy to think ...
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Connection on a bundle

Let $\nabla_A$ and $\nabla_B$ be two connections on a vector field $E$ over a compact manifold $M$, we know that $\nabla_B=\nabla_A+a$ for some $a\in\Omega^1(X;\text{End}(E))$. And for any smooth ...
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On the proof about the dimension of the conformal group of a manifold

I have been reading the book "Transformation Groups in Differential Geometry" by S. Kobayashi. More concretely, I am trying to understand the proof of the Theorem 6.1 of Chapter IV. Theorem ...
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Cylinder Parallel Transport

In general the parallel transport of a certain vector between two points p and q in a given surface may change depending on the curve used to go from p to q. However, in a cylinder the parallel ...
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Questions about Connections on normal bundles & infinitesimal action of a vector field

I've just started reading a paper called "Formules de Localisation et Formules de Paul Lévy" by Bismut, and I didn't understand two things : 1). Let $M$ be a compact connected oriented ...
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Why is the covariant derivative defined as $\nabla_X Y = P((X\otimes I)Y)$ in the Wikipedia article “Riemannian connection on a surface”?

In the Wikipedia article Riemannian connection on a surface, I am struggling to interpret the following: "Since $\mathcal X(M)$ is a submodule of $C^\infty(M,\mathbf E^3)=C^\infty(M)\otimes \...
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Connections on vector bundles: how they allow us to differentiate sections and to identify fibers? [duplicate]

A connection $\nabla$ in a vector bundle $E$ over manifold $M$ is a (globally defined) map $$ \nabla : \Gamma(E) \to \Gamma(T^* M) \otimes \Gamma(E) $$ that satisfies the Leibniz rule: $$ \nabla(f \, ...
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Is there a natural extension of Levi-Civita connection to include torsion?

The Levi-Civity connection described in https://en.wikipedia.org/wiki/Levi-Civita_connection specifically for torsion-free. The question is: do we have connection on the tangent bundle which takes ...
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Levi-Civita Connection in $\mathbb{R}^n$

I'm trying to work myself through Riemannian geometry and I believe I still don't have a firm grasp on what the Levi-Civita connection actually defines. Specifically, if $\gamma(t)$ is a geodesic this ...
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Question about covariant derivative on a vector bundle

I'm reading about vector bundles from somme lecture notes written by Matvei Libine. He say that If $E \rightarrow M $ is a vector bundle, a covariant derivative on $E$ is a differential operator $$\...
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Second Bianchi identity on tangent bundle

I'm having a hard time on proving the second Bianchi identity in the case of tangent bundle without choosing of metric. I already know that on a general vector bundle, the second Bianchi identity ...
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Why $\frac{DV}{dt}= \sum\limits_j \frac{dv^j}{dt}X_j +\sum\limits_{ij}\frac{dx_i}{dt} v^j \nabla_{X_i}X_j$ means the $\frac{DV}{dt}$ is unique?

Picture below is from pages 50-51 of do Carmo's Riemannian Geometry. I can't understand why the red line means that $\frac{DV}{dt}$ is unique when $V$ is fixed. In my view, there is not any proof to ...
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Connection on principal $G$-bundle v.s. connection on associated vector bundle

I'm confused with the relation between these two. I'll summarize what I know so far and explain my questions. So for a principal $G$-bundle $P$, we can define a principal $G$-connection on it, we can ...
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Geodesics under Levi-Civita connection

I read somewhere that minimum energy paths are geodesics under the Levi-Civita connection, on a Riemannian energy landscape. The Levi-Civita connection is the "unique connection on the tangent ...
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A horizontal bundle provides a connection one-form for a principal $G$-bundle

Let's have a principal $G$ bundle $P \xleftarrow{\triangleleft G} P \xrightarrow{\pi} M$. At some point $p \in P$, let's consider the tangent space $T_p P$. We define the vertical tangent space as $...
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Relation between connection and connection form

In the smooth setting, we define a connection in a principal $G$-bundle $(P, \pi, M)$ as a smooth assignment to each point $p \in P$ of a subspace $H_p P$ of $T_p P$ such that the following two ...
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Symmetry of a connection and Christoffel's symbols

To prove that the symmetry of a connection is equivalent to the symmetry of the Christofell's symbols in $i$ and $j$, i.e $$\nabla_XY-\nabla_YX-[X,Y]=0\iff \Gamma_{ij}^k=\Gamma_{ji}^k$$ I have thought ...
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Tensor product used in this notation

A connection form is a Lie-algebra-valued one form $\omega\in\Omega^1(P,\mathfrak{g})$ on a principal bundle P. I have also seen this written this as $\Omega^1(P)\otimes\mathfrak{g}$, but I do not see ...
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Covariant derivative of the parallel transport operator.

I know that it is possible to define a covariant derivative of a tensor field given the parallel transport operator, but I wonder whether the covariant derivative of the parallel transport operator ...
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Definition of horizontal subspace using horizontal lift

Let $\pi: E \longrightarrow M$ be a vector bundle, $e \in E$ and $h: T_{\pi(e)}M \longrightarrow T_eE$, $X \mapsto \tilde{X}=\sigma_*(X)$, where $\sigma$ is any local section of $E$ which is parallel ...
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Connection form takes value in Lie algebra of structure group

Consider a vector bundle with structure group $G$, for any connection, we can have its local expression using the connection form matrix. If the connection form matrix takes value in the lie algebra $\...
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Lemma on covariant derivatives (Lectures on Kähler manifolds)

I am trying to understand a proof from "Lectures on Kähler Geometry" by Andrei Moroianu. I still have lots of trouble working with covariant derivatives. Lemma 5.2 states the following: Let $...
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Lie bracket of tangent vectors in a submanifold is tangent to the submanifold

Let $M$ be a smooth manifold and $N$ a submanifold of $M$. Let $X$ and $Y$ be vector field of $N$. We know that the differentials on $M$ of those vector fields $d_XY$ and $d_YX$ are not necessarily ...
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Connections and Projection Maps

Let $M$ be a manifold and let $\pi: TM\rightarrow M$ be the projection map. Taking the pushforward of $\pi$ we obtain a bundle map $$ \pi_\ast: T(TM)\rightarrow TM. $$ Let $$ V=\ker \pi_\ast $$ be ...
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Section 4.2 in Loring Tu's Differential Geometry

Section 4.2 in Loring Tu's Differential Geometry: My Question: Since $D_XY −D_YX = [X,Y]$, then why define the quantity $T(X,Y)=D_XY −D_YX - [X,Y]$? Isn't $T$ always equal to $0$? I got very confused,...
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Skew-symmetry $\nabla_{[X,Y]}Z = -\nabla_{[Y,X]}Z$

I'm new to the theory of affine connections and have yet to become comfortable working with. I wanted to confirm to myself that the curvature of an affine connection is skew-symmetric which boils down ...

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