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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The Levi-Civita Connection on the Hyperbolic Plane

In this question here, I asked about computing the Levi-Civita connection matrix on the Hyperbolic Plane, defined as $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = ...
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Linear Connection on the Hyperbolic Plane

For the upper half-plane $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = \frac{1}{y^2}(dx^2+dy^2)$, I computed the Christoffel symbols as follows: ...
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How to define gradient of an affine connection

I heard somewhere (and just read on a physics forum) that the gradient of a smooth function $f$ on a manifold $M$ can be defined when $M$ is equipped with an affine connection on its tangent bundle, ...
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Local Representation of Euclidean Connection

I'm trying to understand how connections are locally represented, and the definition I have to work with is this: Let $(x^1,\dots,x^n)$ be local coordinates defined in some chart $U \subset M$ ...
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Is this computation of the Christoffel coefficients on a Kähler manifold correct?

Let $M$ be a Kähler manifold (in truth, I am only interested in $\Bbb C \Bbb P^n$). Is it possible to express the Christoffel coefficients of the Levi-Civita connection in terms of the coefficients of ...
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Why is the backwards curve needed for the inverse of a parallel transport?

Dearh math.stackeschange-community, I'm at a loss with the following problem: Let $I=(a,b)$, then the inverse of the parallel transport $P_{s,t}^\gamma$ from $s \in I$ to $t \in I$ along the curve ...
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Connection on $T\mathbb{R}^n$

Let $\nabla$ be a connection on the tangent bundle $T\mathbb{R}^n$. Now, I need to show that there exist smooth function $C_i: \mathbb{R}^n\to \mathbb{R}^n\times \mathbb{R}^n$, $i=1,\dots ,n$ such ...
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What is the mathematical understanding behind what physicists call a gauge fixing?

I'm learning fiber bundle from my poor physicist point of view. I understand that a gauge transformation (physicist language) corresponds to the transformation of the connections built from an ...
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Levi Civita connection intuition and motivation

Can someone explain why need we the Levi-Civita connection and what it does intuitively?
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Connection with zero torsion and curvature

I remember encountering the following theorem many, many time ago: if a connection $\nabla$ has $R = 0$ and $T = 0$ then... The problem is that I cannot remember the conclusion. Was it that the ...
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What is a pseudo-Kähler manifold?

I am reading a text which says that if a symplectic manifold is pseudo-Kähler, then there exists a unique symplectic connection on it. Since this a side remark without significance to the core of that ...
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Is $0 = \nabla_Xg(Y, -)(Z) = \nabla_X(g(Y, Z)) - g(Y, \nabla_XZ)$ if $\nabla g= 0$?

Suppose a manifold $M$ with a metric $g_{\mu\nu}$. I know that the covariant derivative (I'm assuming the connection induced by the metric) of a covector is given by: $(\nabla_X \eta)(Y)=\nabla_X ...
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Equivalence of definitions of Killing vector field

I read in wikipedia the following two definitions of Killing Vector $X$: $$\nabla_{\mu}X_{\nu}+\nabla_{\nu}X_{\mu}=0$$ $$ g(\nabla_Y X,Z)+g(Y,\nabla_Z X)=0$$ I have problems deducing the ...
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Christoffel symbols for the Poincaré ball model

The metric tensor $g_{ij}$ of the Poincaré ball model is $$ g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2} $$ where $\delta_{ij}$ is the Kronecker delta and $x^k$ are the ambient Cartesian ...
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The Levi-Civita and the Covariantly Constant Tensors in Kahler Manifold?

Please scroll down to the bold section if you are too bored to read the whole details. Aiming to explain the mathematical structure of Kahler manifolds, Freedman and Van Proeyen, in their book ...
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Definition of dual connection in riemannian geometry

If D is a connection on a vector bundle E, we define the dual connection D* so that $$d(v^*,w)=(D^*v^*)(w)+v^*(Dw)$$ I understand why this seems the natural thing to do. Why is the following not ...
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5 geometric shapes all touching each other

I was playing aroud with shapes, which all connected. I managed to get 3 and 4 (http://i.imgur.com/MjOnY3e.png) shapes all connected to each other, but I can't get 5 to work in 2D. Does anyone have ...
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Relation between curvature and sectional curvature

Let $(M,g)$ be a Riemannian manifold and $ h = c.g$ for some $c > 0$ . Then the Levi-Civita connections of $g$ and $h$ are same. From the above deduce the relation between corresponding curvature ...
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Flat $G$-structures have Hol=Id

Exercise I've been given the task to show, given a flat $G$-structure, we have that $\text{Hol}=\text{Id}$ (here "Hol" is the holonomy group; furthermore a flat $G$-structure is defined be one such ...
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Example of Skew-Symmetry of Connection Forms

As is commonly known, the connection 1-forms of a Riemannian manifold are skew-symmetric: $\omega^i_j=-\omega^j_i$. Until now, I have not actually thought to hard on this, but I think I've hit a snag. ...
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Understanding the Ehresmann connection

I am trying to understand the concept of an Ehresmann connection on a fibre bundle $B$. Am I correct in saying that the connection gives the decomposition of every vector in $TB$ into the sum of a ...
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Covariant derivative from an Ehresmann connection on a fibre bundle

Given an Ehresmann connection on a fibre bundle, is it possible to define a covariant derivative that measures the rate of change of a section of the fibre bundle as you move through the base ...
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When do parallel sections exist?

I suspect that this is a "trivial" question, but I don't have enough background to know the answer immediately: Suppose $\pi : E \to M$ is a trivial real line bundle on a smooth manifold $M$, and ...
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Every vector bundle has a metric connection?

Let $(E,g)$ be a vector bundle with a metric over a manifold $M$. Does $(E,g)$ always admit a compatible (metric) connection? If so, are there examples where there exists only one such metric ...
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Divergence as trace of Levi-Civita connection

In "Problems and Solution in Mathematics" by Ta-Tsien, 2nd Edition, exercice 3314, question b Exercise question For a vector field $X$ define the divergence of $X$, $\text{div}(X)$ as the trace of ...
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How does the Torsion of two vector fields act on their corresponding flows?

Let $X$ and $Y$ be vector fields defined on an open neighborhhod of a smooth manifold $M$ endowed with an (arbitrary) affine connection $\nabla$ (i'm not assuming anything apart from it being a ...
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Reference needed: Exterior covariant calculus

I would like to understand Cartan's formalism of exterior covariant calculus. I think it could be useful for some calculations in physics (But If I am wrong here and it's only good for abstract ...
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Metric compatibility, Ricci rotation coefficients & non-coordinate bases

I'm currently working through chapter 7 on Riemannian geometry in Nakahara's book "Geometry, topology & physics" and I'm having a bit of trouble reproducing his calculation for the metric ...
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construct a connection such that a given tensor is parallel wrt it

Let $\omega$ be a symplectic form on a smooth manifold $M$. How does one construct a connection on $TM$ such that $\omega$ is parallel to it? It's easy to construct a connection on a dual bundle ...
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Different definitions of covariant derivative on a vector bundle.

Given a connection on a principal $G$-bundle $P$ over a smooth manifold $X$, I have seen several ways of defining a covariant derivative on an associated vector bundle $E = P \times_{\rho} V$: One ...
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Connections on principal bundles: Local and Global Formulations.

The standard definition of a connection on a principal $G$-bundle $\pi : P \to X$ is a smooth family of subspaces $H_{p}$ of $T_p P$ such that for every $p \in X$ we have a splitting of vector spaces ...
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A diffeomorphism which maps geodesics to geodesics preserves the connection?

Let $(M,\nabla^M),(N,\nabla^N)$ be two smooth manifolds with given (affine) connections on their (tangent bundles). We say a diffeomorphism ,$\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ is an ...
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Equivalent conditions for a linear connection $\nabla$ to be compatible with Riemannian metric $g$

I am reading John M. Lee's Riemannian Manifolds: An Introduction to Curvature. In Lemma $5.2$, it is said that the following conditions are equivalent for a linear connection $\nabla$ on a Riemannian ...
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Parallel transport along a 2-sphere.

I'm currently learning about parallel transport and connections and we were considering the parallel transport of a tangent vector along a sphere as given in the picture below. From my ...
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Difference between types of connections [closed]

For my background, I am familiar with the basics of differential geometry, especially Riemannian geometry, and in some more advanced topics relevant to physics, especially general relativity. Lately ...
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Looking for a reference that explains connections and curvature by double tangent space

I'm looking for a book or a set of lecture notes on differential manifolds that explain connections (Levi-Cevita connection, prinicipal connections) and curvature on an abstract manifold from the ...
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If $\nabla_1$ and $\nabla_2$ are Levi-Civita connections for a metric on the smooth sphere, then their curvature tensor would recover the radius…?

I am a little confused by an idea suggested to me: putting a connection on a sphere doesn't specify a metric geometry - it remembers notions like straightness of paths, but not length of paths. Let ...
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Difficulty proving gauge invariance on an SU(N)-valued potential

Say we have a four-dimensional spherically symmetric $\mathfrak{su}(N)$ gauge potential in standard Schwarzschild co-ordinates which can be written \begin{equation} ...
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The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ bilinear … so is it tensor like?

The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ linear in both components... so is it a tensor of some kind? I know (I think) it is not a ...
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Decomposition of tangent space of principal bundle

A connection on a principal bundle $\pi:P\rightarrow M$ is a choice of horizontal subspace $H_p$ at each $p\in P$, such that $T_p P = H_p + V_p$ where $V_p = \ker((\pi_*)_p)$. It is very common to ...
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Defining an Ehresmann Connection Using Linear Connection

In some places that I have seen, given the covariant derivative $\nabla$ of a linear connection on a vector bundle $\pi: E\rightarrow M$ (dim(M)=k), an Ehresmann connection is defined in the following ...
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Transformation behavior of connection on vector bundle.

Using the notation from Jost's various books on geometry, let $$ D=d+A $$ be a connection on a vector bundle $\pi:E\rightarrow M$ with structure group $GL(n,\mathbb{R})$. Also let $\{U_\alpha\}$ be ...
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Question of well-definedness of the Levi-Civita connection?

On page $55$ of Do Carmo's Riemannian geometry, he proves that there is a unique symmetric affine connection compatible with a given metric on a manifold M. He defines it by a formula $\langle Z, ...
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Connection giving an identification of the double-tangent bundle

Let $M$ be a manifold, and let $\nabla$ be a (possibly torsion-free) connection on $M$. Then I am pretty sure that $\nabla$ induces an isomorphism between the double-tangent bundle and the Whitney sum ...
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Is there a unique preferred connection on a general manifold?

I was wondering whether, for a given finite-dimensional manifold, the connection $\nabla$ exists and is uniquely defined? Afais for Riemannian manifolds, there exists always exactly one Levi-Civita ...
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Connectedness of circle without center line across it

Using a definition I saw in an old Russian book, a set in $\mathcal R^{n}$ is said to be connected if it cannot be represented as a disjoint union of two nonempty, separated sets. Separated, meaning ...
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Is this operation meaningful or it is a mistake in the book?

I've been reading Nakahara's "Geometry, Topology and Physics" and found something quite strange on the section 10.3.3 which discusses the geometrical meaning of the curvature of a connection. It is ...
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80 views

Manifold,affine connection,vector field

An affine connection on $M$ is a differential operator, sending smooth vector fields $X$ and $Y$ to a smooth vector field $∇_X Y$ , which satisfies the some conditions.I would like to know the ...
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Applying $\bar \partial$ to a tensor product of holomorphic vector bundles bundles

Reading Griffiths and Harris, it is stated as an "obvious" fact that for a tensor product $E \otimes E'$ of holomorphic vector bundles, $(D_E \otimes 1 + 1 \otimes D_{E'})'' = \bar \partial$ where ...
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Rigorously what is this integral?

I've been studying some gauge theories approach to problems in mechanics in order to get a better understanding of the ideas from gauge theories and to see some applications of fibre bundle theory. ...