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0
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1answer
48 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 ...
2
votes
1answer
20 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 ...
0
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1answer
23 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
25 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 ...
0
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3answers
28 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 ...
7
votes
1answer
103 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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0answers
24 views

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 ...
0
votes
1answer
28 views

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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0answers
17 views

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: ...
1
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0answers
25 views

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 : ...
2
votes
1answer
43 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 ...
1
vote
0answers
35 views

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 ...
3
votes
0answers
33 views

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 ...
0
votes
0answers
27 views

Connection defined by its geodesics

I have a question related to the definition of a connection (in the sense of Koszul) by its geodesics. I know that a torsion free connection is uniquely determined by its geodesics. Now, let $M$ be a ...
4
votes
0answers
58 views

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, ...
4
votes
1answer
56 views

Canonical connection on $CP^n$

I have heard something along the lines of "There is a canonical $U(1)$ connection on $CP^n$" and I am trying to understand what that means. First I suppose that the sentence refers to a line bundle ...
3
votes
1answer
95 views

Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
5
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0answers
55 views

Exact sequence of tangent spaces of principal $G$-bundles

Let $P$ be a smooth manifold, $G$ a Lie group, $\alpha:P\times G\to P$ a smooth action and $p:P\to P/G$ a smooth principal $G$-bundle. Then, we have the sequence $$ G \xrightarrow{\alpha_a} P ...
2
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0answers
66 views

Splitting of the tangent bundle of a vector bundle and connections

Let $\pi:E\to M$ be a smooth vector bundle. Then we have the following exact sequence of vector bundles over $E$: $$ 0\to VE\xrightarrow{} TE\xrightarrow{\mathrm{d}\pi}\pi^*TM\to 0 $$ Here $VE$ is ...
0
votes
0answers
26 views

Terminologies for induced connections

Given a Riemann manifold with a Kozul/Affine connection, if you take any subbundle of the tangent bundle there is an induced connection given by applying the ambient Kozul connection and projecting to ...
4
votes
2answers
83 views

What is a “connection” in algebraic terms?

It seems that I read this somewhere else, but I did not find the correct reference now. We know that a vector bundle $E\to M$ is a (projective or locally free) module of $C^\infty(M)$. Then how to ...
2
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0answers
51 views

Prove the existence (or well-definedness) of the induced connection in tensor bundle

Given a connection $\nabla$ on a vector bundle $E$ over a smooth manifold $M$, we know there is a unique extension of $\nabla$ to all tensor bundles of $E$ that satisfies Leibniz rule and contraction. ...
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0answers
48 views

Almost complex structure compatible with Levi-Civita connection of immersed submanifold?

Suppose we have Riemannian manifold $M$ isometrically immersed in a Kähler manifold $\tilde{M}$. Let $\tilde{g}$ be the metric and $\tilde{\nabla}$ the Levi-Civita connection on $\tilde{M}$ , we know ...
0
votes
2answers
71 views

Connections in GL(2,R)

I want to prove that there exists a unique affine connection on $GL(2,\mathbb{R})$ such that all left-invariant vector fields are parallel, and find its torsion.
1
vote
1answer
78 views

Can I construct an affine connection on a Riemannian manifold from arbitrary Christoffel Symbols?

The question is rather simple. All my definitions are as in Do Carmo's "Riemannian Geometry". If $M$ is a Riemannian Manifold, can I construct an affine connection $\nabla$ on it by setting, for all ...
7
votes
2answers
161 views

Geometric meaning of symmetric connection

If $(M, g)$ is Riemannian manifold, there is unique connection $\nabla$, called Levi-Civita connection, satisfying the following: 1) Compatibility with Riemannian metric, i.e. $\nabla(g)$=0 2) ...
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0answers
15 views

The time evolution of levi-Civita connection

Assume a smooth one-parameter family of Riemannian metrics $g_{t}$. Write $h:=\frac {\partial}{\partial t}g$. In addition, assume that the Levi-civita connection on the Riemannian manifold ...
1
vote
0answers
102 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
2
votes
0answers
45 views

Frenet formulas for curves in arbitrary Riemann manifold

As far as I understand, to have Frenet formulas one would need a curve, embedded in $\mathbb{R^n}$ and, desirably, naturally parametized. But there are homonomic notions of curvature and torsion of ...
0
votes
1answer
46 views

Orthogonal connection on tangent bundle

What does orthogonality of connection mean in coordinate way? As I understand, a connection $\nabla: \Lambda^1M \rightarrow \Lambda^1M \otimes \Lambda^1M$ is torsion-free iff in any local coordinates ...
3
votes
1answer
58 views

Recovering a frame field from its connection forms

I have a faced a research problem where I would need to recover a frame field given its connection forms. More precisely, I begin with an orthonormal frame field (given by data) in $\Re^3$ written as ...
2
votes
1answer
57 views

Connection vs path-connection

Let $(X,\tau)$ be a topological space. We say that $X$ is locally connected if there is a basis of $\tau$ consisting of open connected sets; we say that $X$ is locally path-connected is there is a ...
-1
votes
1answer
56 views

Space of all connections on a torus

As the title asks, what is the space of all connections on a torus $\mathbb{T}^2 $ defined in the complex plane $\mathbb {C}^2 $?
1
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0answers
29 views

Proving a global property of the connection from a property of local connection matrices

Let $D$ be an $m$-dimensional distribution on an $n$-dimensional manifold $M$. Let $U\subset M$ be an arbitrary open subset such that on $U$ we can define vector fields $e_i$ such that for each $x\in ...
4
votes
0answers
41 views

Can one exchange fibre and base space in a fibre bundle?

The first trivial example of a fibre bundle $E$ is a product bundle $E=F \times B$, with fibre $F$ and base space $B$. Of course in this trivial example, one can exchange base space and fibre and ...
0
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0answers
34 views

Holonomy of Maurer-Cartan 1-form

I am studying the book Sternberg (2012): Curvature in Mathematics and Physics; I am also doing research on LQG. I was wondering: if on a 4-dimensional spacetime one defined the Maurer-Cartan 1-form, ...
3
votes
0answers
47 views

Natural Action of Killing vector fields on space of connections

I try to understand some mathematical aspects of supersymmetric Yang Mills theory following the book "Quantum fields and strings - a course for mathematicians". In this context the following question ...
7
votes
0answers
57 views

Connection in fibre bundle from discontinuous group action

I am trying to understand connections in fibre bundles. I thought of the following problem: Let $\Gamma$ be the discrete group generated by \begin{pmatrix} 1 & 3 & 0 \\ 0 & 1 & 0 \\ ...
0
votes
0answers
29 views

Homeomorphism between the space of all Ashtekar connections and spacetime?

This is a question I've asked in physics.stackexchange: Excerpt from an essay of mine: Let $\Psi(\varsigma)$ be the wavefunction in the loop representation, where $\varsigma:[0,1]\to\mathcal{M}$, ...
2
votes
1answer
63 views

Superspace as the Hilbert Space for Quantum Gravity

This is a question I've asked in physics.stackexchange, but have obtained no answers: Let $\mathcal{A}$ be the Ashtekar connection. Since $^{(3)}g_{AB}=i\frac{\delta}{\delta\mathcal{A}^{AB}}$ (see R. ...
0
votes
0answers
15 views

Normalizing the value of a principal connection at a point

Let $P_0 := G \times X \to X$ be the trivial principal $G$-bundle, and consider a principal connection on it, defined by a 1-form $\alpha$ on $P_0$ with values on the Lie algebra $\mathfrak{g}$ of ...
2
votes
1answer
106 views

Why is the space of all connection on a vector bundle an affine space?

I think this result is very well known, but I don't understand its proof. Let E a vector bundle over a manifold M, and $\Omega^i(E):=\Gamma(\Lambda^iT^*M\otimes E)$ the space of E-valued differential ...
3
votes
1answer
60 views

A connection over a 1-dim manifold is flat

Let $M$ be a 1-dimensional manifold and let $E$ be its vector bundle. I want to show that every connection $D$ on this vector bundle is flat. A connection $D$ is flat means that we have $$D_v D_w ...
5
votes
1answer
56 views

When can a connection be lifted?

Let $P \rightarrow X$ be a principal $G$-bundle, and $P' \rightarrow X'$ be a principal $G'$-bundle. Let $(f',f'')$ be a morphism from $P'$ to $P$, i.e., a pair of maps $f': P' \rightarrow P$ and ...