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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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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28 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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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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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 ...
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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: ...
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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 ...
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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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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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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, ...
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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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63 views

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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31 views

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: ...
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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 ...
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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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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 ...
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When parallel transport determines a connection

Consider a vector bundle $E \to M$. Given a connection $\nabla$, it induces a parallel transport, which (in particular) is a choice of isomorphism $T_{\gamma} : E_{\gamma(0)} \to E_{\gamma(1)}$ for ...
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Explicitly constructing the Total Metric on a line bundle

Suppose that I'm given a Riemmanian manifold $<B,g_B>$ and a real line bundle $\pi:E\rightarrow B$, such that each fiber above any $b\in B$ comes equipped with a Riemmanian metric $g_b$. Then ...
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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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102 views

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 ...