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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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Do covariant derivatives uniquely define connections?

Consider a smooth vector bundle $\pi:E \to M$ with a connection \begin{equation} \nabla: \Gamma(E) \to \Gamma(E \otimes T^*M). \end{equation} For any vector field $X \in \Gamma(TM)$, we can define the ...
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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 ...
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Show the complex one-form defines a connection

Let $E$ be the complex line bundle for $\mathbb {CP}^1$, $E=\{(L,v)\mid L\in Gr_1(\mathbb C^2),v\in L\}$. I know that this with zero section removed is diffeomorphic to $\mathbb C^2-\{0\}$. Now I want ...
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Mathematical challenge for unification of gravity and electromagnetism in classical theory?

I am trying to better understand the mathematical foundations of a possible reconciliation between quantum field theory and gravity in general relativity. However, before the application of the ...
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Commutation of the covariant Hodge Laplacian with the covariant derivative

Let $(M, g)$ be a Riemannian manifold, $E$ a Hermitian vector bundle and $A$ a unitary connection over $E$ (i.e. the covariant derivative $d_A$ respects the inner product). The action of $d_A$ is ...
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Differential geometry of line bundles

I am trying to understand concepts of Gerbes and their differential geometry as generalisation of line bundles and their differential geometry using Hitchin’s notes. I am familiar with concepts of ...
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Space of principal connections is affine modelled on $\Lambda^1(M;\mathfrak{g})$?

I'm working within the jet-formulation espoused by Saunders in "The Geometry of Jet Bundles" and am struggling to prove the stated result. I would like to stay in this context and understand the ...
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1answer
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Motivation for the generalization of the de Rham isomorphism to twisted vector bundles

I am reading a paper that makes the following comment: Let $(V,\nabla^V,h^v)$ be a vector bundle with a flat connection $\nabla^V$ and metric $h^V$. Then we can form the twisted de Rham complex $\...
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Question concerning covariant derivatives on vector bundles

I am currently reading the book "Lectures on Kähler Geometry" by Andrei Moroianu and I try to understand the following theorem : The proof of this theorem is given in the book, but unfortunately the ...
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How to interpret $\nabla_i = \frac{\partial}{\partial x_i} +A_i$ in terms of a connection $A$ on a principal bundle.

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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Recovering a principal connection from its monodromy

Given a principal bundle $P \to M$ with structure group $G$ ($M$ and $G$ are connected), it is well known that one can recover the data of $P$ and a flat connection $\Gamma \in \Omega^{1}(P, \mathfrak{...
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Does the trace of a Levi-Civita connection always vanish?

If $\Gamma^a_{bc}$ are the Christoffel symbols of the Levi-Civita connection corresponding to the metric tensor $\boldsymbol{g}$, i.e., if $$\Gamma^a_{bc} = \tfrac{1}{2} \, g^{ad} (g_{cd,b} + g_{bd,c} ...
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Extension of a vector bundle

I've got two isomorphic holomorphic vector bundle $E,E' \to X$, where $X=X'-\{x_0\}$ where $X'$ is a complex variety. I know that $E'$ is the restriction of an holomorphic bundle over $X'$. Can I ...
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Show that the number of edges of $G$ is at least $m\mu$

(1) Let $G$ be a connected graph with $n = 2m$ vertices. Let $V_1$ and $V_2$ be disjoint subsets of $V(G)$ with $|V1| = |V2| = m$. Let $\mu$ be the algebraic connectivity of $G$. Show that the number ...
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Are there holonomic $\mathcal{D}$-modules besides flat connections?

Question: are there interesting holonomic $\mathcal{D}$-modules on smooth variety $X$ except those coming from flat meromorphic connections? $$\text{}$$ Since $M$ is holonomic iff $\dim \text{ch}(M)...
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Name of the relation between LC connections in isometric immersions

Let $(\tilde{M},\tilde{g})$ be an $(n-1)$-dimensional Riemannian manifold isometrically embedded in the $n$-dimensional $(M,g)$. The L-C connections are respectively $\tilde\nabla$ and $\nabla$. $N$ ...
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derivative of curvature tensor on vector bundle

Let $E \to M$ be a riemannian bundle and $D$ a linear connection compatible with the metric. For every pair of vector fields $X$ and $Y$ on $M$ and every section $\xi$ of $E$, the curvature is given ...
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Difference Quotients on Riemannian Manifolds

I want to define "the difference quotient of a differential form $ \omega $ in the direction of the vector field $ X $" on a Riemannian manifold. Let's call this object, if it can be defined, $ \...
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Is the determinant of the holonomy gauge invariant / significant?

I am currently reading the book Baez & Muniain - "Gauge Fields, Knots, and Gravity". Chapter 2 of part II defines the holonomy on a bundle $E\to M$ with a gauge group $G$ and connection $D$. They ...
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A smooth map of manifolds that preserves linear connections.

Let $D$ and $D'$ be two linear connections on manifolds $M$ and $M'$ respectively. If $f:M\to M'$ is a smooth map from $M$ into $M'$, what is meant by "$f$ is a connection preserving" ?. Thanks in ...