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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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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 ...
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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 $\...
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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. ...
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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 ...
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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 ...
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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 ...
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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 $...
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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{...
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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 ...
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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 $\nabla=d+A$....
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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 ...
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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 ...
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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 ...
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Why is $\Pi(X,Y)=\frac{\partial}{\partial t} \nabla_XY$?

Let $X,Y$ be two vector fields. Then the book "Lectures on the Ricci Flow" says $\Pi(X,Y)=\frac{\partial}{\partial t} \nabla_XY$. I don't understand how this is the case. The second fundamental form ...
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Covariant derivative vs. Ehresmann connection

I know about Ehresmann connections on fiber bundles and covariant derivatives as an (equivalent) way to define linear Ehresmann connections on vector bundles. My question is: Is there any notion of ...
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Horizontal Subbundles and Connection Maps

I'm trying to see the equivalence between the Ehresmann connection and the connection map, and having trouble getting the setup to be correct. Suppose $M$ is a smooth manifold. Let $\pi:TM\to M$ ...
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Confusion about connection, covariant derivative and definition of divergence of a vector in terms of covariant derivative

Let $V \in T_p M$ be a vector at the point $p$ in a manifold $M$. The connection $\nabla$ is defined such that $\nabla V$ is a $(1,1)$ tensor: \begin{align*} \nabla V: T_p M \rightarrow T_p M \otimes ...
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Equivalent definitions of Hyperkahler manifolds.

I am reading the paper HYPERKAHLER METRICS ON COTANGENT BUNDLES OF ¨ HERMITIAN SYMMETRIC SPACE by OLIVIER BIQUARD AND PAUL GAUDUCHON. Suppose $M$ is a manifold with a triple $(g,I,J)$ where g is a ...
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Flat sections of flat vector bundles

Let $E\to X$ be a vector bundle with flat connection $\nabla$. Is there a canonical way to construct a vector subbundle of $E$ from the flat sections ($\nabla \sigma=0$)? More specifically, I would ...
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Looking at the connection 1-form on a principal G-bundle in coordinates

I'm reading this paper, and I'm confused about something. Let $A$ be a connection on a principal $G$-bundle $P$ over $\mathbb{R}^4$, and $F(A)$ its curvature. Let $\mathrm{ad}(P)=P\times_G\mathfrak{g}...
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affinely flat manifold

I would like to prove that the existence of an affine atlas (smooth atlas with all transition maps are affine) on a given smooth manifold $M$ is equivalent to the existence of a flat and torsion free ...
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A question about the definition of $\overline{\nabla}_X \langle Y,Z\rangle$

Lee's "Riemannian Manifolds" has the following theorem: The Euclidean connection on $\Bbb{R}^n$ has one very nice property with respect to the Euclidean metric: it satisfies the product rule $$\...
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An Orthonormal Frame $(X_i)_i$ which satisfies $\bigtriangledown _{X_i} X_j =0$ at a point

I am an undergrad student learning Riemannian geometry. My question is about whether you have a nice orthonormal frame in the following sence. Let $(M, g)$ be a Riemannian manifold, with $\...
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Local construction of a map on a manifold

Assume $(M,J)$ is a smooth manifold equipped with an almost complex structure. Let $p$ be a point of $M$, and define the linear map $\Theta_p \colon T_pM \to T_pM$ so that $$\Theta_p = -\frac{1}{2}\...
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What is the curvature form $\Omega$ associated with the Levi-Civita connection for the $n$-sphere $S^n$ with respect to the standard metric?

What is the curvature form $\Omega$ associated with the Levi-Civita connection $\nabla^{\text{L.C.}}$ for the $n$-sphere $S^n$ with respect to the standard metric, i.e. what is $\Omega=d\theta+\frac{1}...
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Relation between gradient of a function with respect two linear connection

Let $\bar{\nabla}$ and $\nabla$ be two linear connection over $(M,\bar{g})$ and $(M,g)$ resp. and suppose that this two connection are related by the following equality: $$\bar{\nabla}_XY=\nabla_XY+\...
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A question about *A new proof of a theorem of Narasimhan and Seshadri*

I am reading S. K. Donsaldson's paper A new proof of a theorem of Narasimhan and Seshadri. Now $M$ is a compact Riemann surface and $E$ is a holomorphic bundle over $M$ with a Hermitian metric. There ...
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Two approaches to the definition of a connection

A connection can be defined as a map $$D:\Gamma(E)\to\Gamma(E)\otimes\Gamma(T^*M)$$ satisfying the usual conditions or, equivalently, $$F:\Gamma(TM)\otimes \Gamma(E)\to \Gamma(E).$$ My question is ...
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Why is a connection on the bundle $SO(M)$ metric compatible?

If we have an orientable manifold $M$ with a metric $g$ and signature $(r, s)$, we can define the principal-$SO(r, s)$ bundle $SO(M)$, the bundle of orthonormal frames of $TM$. This is a subset of the ...
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characteristic classes arising from connections are well-defined

I am reading an excerpt from Morita's book. The story is that if you have a connection $\nabla$ on a vector bundle $E$ over some manifold $M$ and a function $M_n(\mathbb{R})\to \mathbb{R}$ which is a ...
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A question about Narasimhan-Seshadri Donaldson theorem.

I see a statement of the theorem in Jonathan Evans' lecture note 13 An indecomposable Hermitian holomorphic vector bundle $\mathcal E$ on a Riemann surface $M$ is stable if and only if there is a ...
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induced Levi-Civita connection on exterior power

Let $M^n$ be a Riemannian manifold and $\nabla$ the Levi-Civita connection of $M$. Let \begin{align*} \Lambda^2(M) := \coprod_{p \in M} \Lambda^2(T_p M). \end{align*} My question is, how one can ...
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A question about holomorphic structure in Atiyah Bott's paper.

I am reading Atiyah & Bott's The Yang-Mills equations over Riemann surfaces. I meet a question about the holomorphic structure. Let $E\to M$ be a complex vector bundle over a compact Riemann ...
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Why is the interpolation between two connections related via a gauge transformation still a connection?

I am studying the theory of anomalies in gauge field. Let $A$ be a gauge field (or a connection for mathematicians). Let $A_{U}$ be an equivalent gauge related via a local gauge transformation $$A_{...
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Surjective geodesic exponential map on a Lie group

Is it possible to put a left invariant connection on an arbitrary Lie group whose associated geodesic exponential map is surjective?
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Sign issue when computing identity involving curvature for a connection $\nabla + A$ of a bundle $E\to M$

I'm trying to prove the well-known formula which relates the curvature associated to the connection $\nabla$ to the one associated to $\nabla + A$, namely: $$F_{\nabla +A} = F_{\nabla}+d_{\nabla}A+A\...
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Consider symplectic vector fields $X,Y$ and a symplectic connection $\nabla$. Is $\nabla_{X}Y$ symplectic?

Consider a symplectic manifold $(M,\omega)$, together with a symplectic connection $\nabla$, i.e. a torsion-free connection such that $\nabla{\omega} = 0$. Fix two symplectic vector fields $X$ and $Y$....
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How to arrive at frame bundle objects from chart calculation on the base manifold

In this lecture by Fredric Schuller it is said that the wave-function is not a wave-function. He attempts to find an appropriate coordinate independent derivative using "chart calculations" on the ...
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How to derive the Leibniz rule from this definition of a connection of a vector bundle?

Suppose $M$ is a Riemannian manifold and $\pi\colon E\to M$ is a vector bundle of rank $n$. Let the vertical subbundle of $TE$ be \begin{equation} V=\ker(d\pi),\quad d\pi\colon TE\to \pi^*(TM) \end{...
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Proof for the exterior derivative formula in terms of the covariant derivative.

SETUP Everything is smooth. Consider the sheaf $\Omega^{p}(M)$ of sections of the exterior powers of the cotangent bundle $\Lambda^{p}(T^{*}M)$ over a manifold $M$. Purely in terms of the smooth ...
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Parallel transport in two different polar coordinates

I need help with the basics of parallel transport. So I will write down what I have done in the plane $\mathbb{R}^2$ with non-cartesian coordinates, mixed with some small questions. First, I use ...
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Pushforward of a linear connection under a diffeomorphism

Let $\varphi : M \longrightarrow N$ be a diffeomorphism of smooth manifolds and let $\nabla$ be a linear connection on $M$. Define a connection on $N$ by $$\widetilde{\nabla}_X Y = \varphi \left( \...
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Exterior Covariant Derivative - Proof of Structure Equation for general r-form

Let $(P,M,\pi,G)$ be a principal bundle and $\omega \in \Omega^{1}(P,\mathfrak{g})$ a principal connection. Given a representation $\rho : G \to GL(V)$ and an equivariant form $\eta \in \Omega^{r}(P,V)...
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About the antisymmetry of the curvature matrix in vector bundle

Suppose we have a vector bundle $\pi: E \rightarrow M $ with fiber $F_p = \pi^{-1}(p) = \mathbb{R}^d$. $$D: Γ(E) \rightarrow Γ(TM^* \otimes E) $$ is connection in the bundle. $K = D \circ D : Γ(E) \...
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Covariant derivative and orthogonal projection

Let $V$ be a smooth vector field along a curve $c$. The covariant derivative of $V$ along $c$ at the point $c(t)$ us given by $\frac{D_cV}{dt}(c(t))=\lim_{s\to t}\frac{P_{c(t),c(s)}V_{c(s)}-V_{c(t)}}...
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Covariant derivative and the product rule

Let $(M,g)$ be a Riemannian manifold with a connection $\nabla$. It is required to satisfy the product rule: $$\nabla_X(fY) = f\nabla_X(Y) + \nabla_X(f)Y$$ where $\nabla_X(f)$ is the directional ...
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Classification of flat connections on the $TS^3$

How to classify all flat connections on $TS^3$. We know that they are exactly as the tangent bundle is trivial $TS^3 \simeq S^3 \times \mathbb{R}^3$. 1) Define connectivity is equivalent to ...
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Is the tangent bundle of a $S^2$ flat?

A vector bundle is said to be flat if it is endowed with an linear connection with vanishing curvature, ie. a flat connection. Is the tangent bundle $TS^2$ of a $S^2$ flat? My question is also about ...
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Does this condition on the curvature implies existence of a parallel section?

Let $E$ be a smooth vector bundle over a manifold $M$ ($\dim M>1$), equipped with a metric. Let $\nabla$ be a metric connection on $E$. Suppose there exist locally a non-zero section $\sigma \...
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A particular connection on the group

Let $e_1, \ldots, e_n$ be a basis in an Lie algebra $\mathfrak{g}$, regarded as a left-invariant movable frame, and $w^i -$ a dual movable left-invariant frame of 1-forms on $G$. $C_{i\, j}^k$ are the ...