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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How to obtain the metric tensor of a principal bundle total space given a connection (assuming it is metric compatible) in the total space?

The title says it all. In a principal bundle I know the connection defined in the total space. How can I calculate the metric that would be compatible with this connection.
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When do the Nakano identities hold?

In "Complex Geometry" by Huybrechts, he states the following version of the Nakano identity on page $240$: Let $X$ be a Kähler manifold and $(E,h)$ a holomorphic hermitian vector bundle on $X$. ...
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are extensions of flat connections flat?

Let $U$ be a smooth complex variety and $X$ a compactification by a normal crossings divisor $D$. Let $E$ be a vector bundle on $U$ (i.e. locally free $\mathcal{O}_U$-module), together with a ...
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Quick question about covariant derivative

Let $f$ be a function and define $\nabla_X f = X(f)\,\,(1)$, where $\nabla$ is the connection on a manifold and as far as I understand the r.h.s is a function and $X$ is a vector field. I am just a ...
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What is an infinitesimal automorphism of a connection?

Let $P\to M$ be a principal bundle over a differentiable manifold $M$, with fibre $G$, equipped with a connection $H\subset TP$. I have heard the term "infinitesimal automorphism of $H$" but I haven't ...
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Non-affinely parametrized geodesics

Consider a non-affinely parameterised geodesic, i.e., a geodesic whose tangent vector field obeys $\nabla_X X = fX$ for some function $f$. Prove that one may reparameterise the geodesic so the tangent ...
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Intuition behind the connection one-form

When we define a connection on one principal bundle $\pi: P\to M$ with structure group $G$ we define it as an association of one subspace $H_pP\subset T_pP$ for each $p\in P$ such that $T_pP = H_pP ...
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Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
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Why is the kernel of the connection one form a connection on a principal bundle?

Let $\pi:P\rightarrow M$ be a principal bundle and let $\omega\in \Omega(P;\mathfrak{g})$ be a one form satisfying $\omega(\sigma(X))=X$ and $R_g^*\omega=\text{Ad}_{g^{-1}}\circ\omega$ Then ...
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Difference of two connections is a tensor

I am currently reading through Jost's Riemannian Geometry and Geometric Analysis and am seeking clarification of the statement in the title. Jost abstractly defines a connection on a vector bundle ...
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65 views

Curvature of a principal bundle and the exterior covariant derivative

Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a connection on $P$, where $\mathfrak{g}$ is the Lie algebra of $G$. Associated to every ...
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Covariant derivative: The meaning

Many text books on differential geometry motivate covariant derivative more or less by saying that if you have a vector field along a curve on a manifold (that is a curve $\gamma(t)$ and an assignment ...
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66 views

Two definitions of curvature

my question is about the compatibility of two definitions of curvature of a Riemannian manifold. In particular I refer to the one from algebraic geometry and the one from differential geometry. ...
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Help! How to derive the result related to Darboux derivative?

First, define Darboux derivative. There is one Lie group $G$ and one manifold $M$. Let $\phi:M\rightarrow G$ be a smooth map. The Darboux derivative $\Delta(\phi):TM \rightarrow M\times \mathfrak g$ ...
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42 views

Connections and Ricci identity

Given $\nabla$ a torsionless connection, the Ricci identity for co-vectors is $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = -R^d_{\,\,cab}\lambda_d.$$ Prove $R^a_{[bcd]} = 0$ by ...
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60 views

Deriving Ricci identity for co-vector fields

Let $\nabla$ be the covariant derivative associated with a torsionless connection. Prove the Ricci identity for covectors: $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = ...
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68 views

Gauge fields and restrictions of the connection one form

I am working through some lecture notes on principal bundles and am stuck on the proof of a certain proposition. In the following, $\pi:P\rightarrow M$ is a principal bundle, $\omega$ is the ...
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53 views

Ask for an explicit proof of linear lemma in Riemannian geometry

Lemma Let $f:(M,g)\rightarrow(\bar{M},\bar{g})$ be an isometry between two Riemannian manifolds, then $df(\nabla_X Y)=\bar{\nabla}_{df(X)} df(Y)$ where $\nabla,\bar{\nabla}$ are Riemannian connections ...
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43 views

Vanishing of the Riemann tensor

The Riemann tensor in a coordinate basis is $$R^{i}_{\,jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \Gamma^m_{jl}\Gamma^i_{mk} - \Gamma^m_{jk}\Gamma^i_{ml}$$ Consider $\mathbb{R}^2$ ...
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88 views

Connections and tensor fields

Let $T$ be a $(1, 1)$ tensor field, $\lambda$ a covector field and $X, Y$ vector fields. We may define $\nabla_X T$ by requiring the ‘inner’ Leibniz rule, $$\nabla_X[T(\lambda, Y )] = ...
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64 views

preservation of the curvature tensor implies preservation of the connection?

For every connection on a smooth manifold there is a corresponding curvature tensor. Any diffeomorphism $\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ which preserves the connection (in the sense of ...
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Coding for a calculation in differential geometry using Maple

I am beginner in maple. And my field is Differential geometry. I've learnt lie brackets using maple help. But I am testing this calculation through maple. I have these vector fields $e1=z^2\ast ...
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37 views

Failure to close of horizontal lift on principal bundles.

When considering the geometrical meaning of the curvature on principal bundle $\pi: P \rightarrow M$, let us consider a coordinate system ${x_μ}$ on a chart $U$. Let $V = ∂/∂_{x_1}$ and $W = ...
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Existence of a holomorphic connection implies existence of a flat connection

Let $E$ be a holomorphic vector bundle on a complex manifold $X$, and let $D: E \to E \otimes \Omega_X$ be a holomorphic connection on $E$ i.e. $$ D(fs) = s\otimes \partial(f)+ fD(s), $$ for any local ...
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Induced connection is well-defined

Let $M=$Riemannian manifold,$N=$differentiable manifold, $\phi:N\to M$ be smooth map. If $v\in T_xM$, and $\{E_i\}_{i=1}^n$ is a frame field in a neighborhood $V$ of $\phi(x)\in M$, then $$\forall ...
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$D_vX$ is completely determined by $X$ restricted on any curve $r$ with $r'(0)=v,r(0)=p$.

I want to show given $v\in T_pM$, then $D_vX$ is completely determined by $X$ restricted to any curve $r$ with $r'(0)=v,r(0)=p$. I have shown that if $r_1'(0)=r'_2(0)=v,$ then ...
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Equivalence for Christoffel symbol and Koszul formula

I am trying to show to define a Levi-civita connection, it's equivalent to define Christoffel symbols or define Koszul formula. $$ 2g(\nabla_XY, Z) = \partial_X (g(Y,Z)) + \partial_Y (g(X,Z)) - ...
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71 views

Check Christoffel symbol defines Levi-Civita connection

I am trying to prove the existence of Levi-Civita connection. The hint says given $(U_\beta,\phi_\beta)$ be altas of $M$, for $X=x^i\partial_i,V=v^j\partial_j$, we define $$D_VX=v^i(\partial_i ...
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51 views

existence of affine connection on manifold

I am studying Riemannian Geometry following my professor's notes. On the proof of existence of affine connection on a $C^\infty$ manifold, the notes states: By partition of unity, a connection can be ...
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Gauge theory on a trivial bundle

I am learning gauge theory, so I tried to understand what happens in the case of a trivial principal bundle. However I have some problems understanding how a connection looks like in that case. Here's ...
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34 views

Finding the neighbors of a node/vertex in a 2D mesh

I have a 2D mesh defined by nodes and elements. Structure of a node: Node ID, X position, Y position Structure of an element: Element ID, Node 1, Node 2, Node 3, Node 4 Example of a 2x2 elements ...
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Connections between the solution of simple ordinary equation, normal distribution and heat equation

The solution to the following simple first-order linear ordinary differential equation: $$x'=-tx, x(0)=\frac{1}{\sqrt{2\pi}}$$ is the Standard normal distribution! One solution to another famous ...
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129 views

Puzzle: How Many Possibilities Are There Between Connected Points?

Puzzle Jenny drew on her page six points, as shown below: Jenny wants to build a cool match of her points. In a match , divide the six-point into pairs, so that each point has one partner exactly. ...
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102 views

Covariant derivative of vector field along itself: $\nabla_X X$

Consider a vector field $X$ on a smooth pseudo-Riemannian manifold $M$. Let $\nabla$ denote the Levi-Civita connection of $M$. Under which conditions can something interesting be said about the ...
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69 views

Reference for principal bundles and related concepts

I am looking for a good reference for fibre bundles, Ehresmann connections, principal $G$-bundles and principal Ehresmann connections (the $G$-equivariant version of Ehresmann connections). Could ...
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1answer
17 views

Multiple points in the parallel transport equation

Let $E \to M$ be a vector bundle with connection $\nabla$, $\gamma \colon [0,1] \to M$ a smooth curve. A section $s = \gamma^* \tilde s$, $\tilde s \in \Gamma(E)$ of $\gamma^* E$ is called parallel if ...
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89 views

Covariant derivative as a connection on a vector bundle

In the Wikipedia article Connexion (vector bundle), such a connection is defined as a function $\Gamma(E) \to \Gamma(E\otimes T^*M)$ . Then the definition of a covariant derivative is given as a ...
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34 views

Calculate days needed for the 4 workers to complete an 80 person-months job

The workload required to develop a system is estimated at 80 person-months if it is carried out by one person. When four staff members with the same productivity work together to develop the ...
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36 views

Calculate time needed to brute force a password

When the maximum time required to find an 8-character password consisting of only 26 lowercase alphabetic characters in a brute force fashion is 1, the maximum required time is ? if the ...
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3answers
164 views

Calculate how long does it take to complete a task by two workers

It takes 36 days for Mr. A to complete a certain task, and 18 days for Mr. B to complete the same task. When Mr. A performs the task together with Mr. B, approximately how many days does it ...
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571 views

Calculate time to fill an empty tank

An empty tank can be filled with water in 20 minutes by using Pipe A or in 30 minutes by Pipe B, and the tank filled with water can be emptied of water in 40 minutes by using Pipe C. When the ...
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243 views

Is there a codifferential for a covariant exterior derivative?

For forms on a Riemannian $n$-manifold $(M,g)$ there is a notion of a codifferential $\delta$, which is adjoint to the exterior derivative: $$\int \langle d \alpha, \beta \rangle \operatorname{vol} = ...
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Map between two sets of points that minimizes their total distance

Let's have a set of $N$ points at positions $\vec{R}_i$, where $i=1...N$. They are displaced into arbitrary positions $\vec{r}_{M(i)}$, where $M(i)=1...N$. The problem is to find a map $M: i \mapsto ...
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connection on a vector bundle. horizontal spaces canonically isomorphic to horizontal spaces in the projection

Let $E$ be a vector bundle over $M$. A connection on a vector bundle $E$ is a smooth field of horizontal spaces $v \in E \mapsto H_v$. Where the projection is $\pi:E \rightarrow M$, $V_v$ is the ...
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Metric on total space, connection and Hopf fibration

As is well known that the three sphere $S^3$ may be viewed as a $S^1$ bundle over base space $S^2$. And it is more interesting, but likewise confusing to me that the metric on $S^3$ may be written as: ...
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Invariant characterization of vector bundles associated to a principal bundle?

I have two related questions. Suppose I have a principal $G$-bundle $P\xrightarrow{\pi} M$. The usual construction of an associated vector bundle goes as follows. Fix some representation $\rho : ...
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102 views

Relation between geodesics and exponential map for Lie groups

I've been trying to find a clear explanation on the Internet but failed unfortunately, so I'm asking here. How does the exponential map relate to parallel transport and geodesics for Lie groups. If it ...
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Lecture notes on holomorphic Yang-Mills theory

Some time ago I've found these lecture notes on the gauge theory. In particular, in these lecture notes the author introduces and studies the Yang-Mills equations in the case of real bundles and ...
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How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...
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Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...