The connections tag has no wiki summary.

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### Gauge theory on a trivial bundle [migrated]

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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### Unifying Geometry for Characteristic Classes

When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches:
Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ...

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### Parallel Ricci condition - Status report and bibliography

First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's ...

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### semi flat connections

Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity.
For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in ...

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### Manifolds admitting flat connections

For each Riemannian manifold one can construct the Levi-Civita connection. While this connection is unique, we can call a (Riemannian) manifold flat if the Levi-Civita connection is flat. However when ...

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### Symplectic form on moduli space of connections

Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it.
Is there any known construction of a bundle with a ...

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120 views

### The “Rolle theorem” for sections of a vector bundle

1)Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...

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228 views

### Analogy between connections and $\ell$-adic sheaves: what happens with the residue?

There are many analogies between $\ell$-adic sheaves on varieties over finite fields and vector bundles with connections on varieties over fields of characteristic zero. I would like to know what is ...

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### reference for Levelt-Turritin

Can anybody recommend a good reference to learn the Level-Turritin decomposition theorem of formal connections? An intuitive description of what it says would also be very appreciated.

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### Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?

Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...

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### Decomposition of Lie subspaces

If $M=G/H$ is a reductive homogeneous space then we can write $\frak{g}=\frak{m}+\frak{h}$
where $[\frak{h}, \frak{m}]\subset \frak{m}$. Here $\frak{g}$ and $\frak{h}$ are the Lie algebras of $G$ and ...

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### A cohomology group which depends on the connection

Warning: I am not a differential geometer, so some of the following might not make sense.
Background:
Let $w: (T\Omega)^k \to \mathbb{R}$ be a $k$-tensor on $\Omega$, an open subset of ...

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### Decomposing connections on extensions of the frame bundle

I have posted this question on math.stackexchange, without success. I'll make it brief:
Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, ...

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120 views

### A 3-connected graph property by Tutte

Tutte （1961): A graph $G$ is $3$-connected if and only if there exists a sequence
$G_0, ...,G_n$ of graphs that have the following two properties
1) $G_0 = K_4$ and $G_n = G$
2) $G_{i+1}$ has an ...

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### connections on principal bundles over $S^1$

Suppose $G$ is a compact connected Lie group and $P$ is a $G$-bundle over $S^1$, $A$ is a connection. Then we can choose a frame such that $A = a d\theta$ where $a\in \mathfrak{g}$ is constant. My ...

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### a technical question on the definition of connections with regular singularities

Let $X$ be a quasi-projective variety over a field $k$ of characteristic zero. A good compactification of $X$ means a projective variety $\overline{X}$ containing $X$ as the complement of a simple ...

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### regular singularities and comparison isomorphism

Let $X$ be a smooth variety over some field $k \subset \mathbb{C}$. By a theorem of Grothendieck, one has a canonical isomorphism of complex vector spaces
$$
\mathbb{H}^j(X, \Omega_{X/k}^\bullet) ...

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### Connections on a Lie Group

A Lie group $G$ can be considered as a reductive homogeneous space in at least two different ways; $G/\{e\}$ and $G\times G/G^*$. In the first case, the canonical connection associated with the ...

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### Two ways to differentiate a section of vector bundle

Let $\pi:E\rightarrow M$ be a vector bundle, and $D$ a connection on it. Suppose $\sigma_1,\sigma_2\in\Gamma(E)$, $p\in M$, $V\in T_pM$ such that $\sigma_1(p)=\sigma_2(p)$. Are the following two ...

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### Normalizing the value of a principal connection at a point

Let $\nabla$ be a symmetric, linear connection on a smooth manifold $X$.
If $p \in X$ is any point, on a normal chart for $\nabla$ around $p$ it holds:
$$ \Gamma_{ij}^k (p) = 0 \ , $$ where ...

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### how is the dual connection defined?

Let $E$ be a vector bundle (i.e. locally free $\mathcal{O}_X$-module) on some smooth algebraic variety $X$ and let $\nabla: E \to E \otimes \Omega^1_X$ be an integrable connection.
I have seen that ...

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### Isomorphism of connections on a complex line bundle

Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact?
Theorem. Let $E ...

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### Construction of a classifying map from a connection 1-form

From a connection 1-form on $M$, I can construct a parallel transport from which in turn I can construct a classifying map $M \to BG$.
Is there a construction of such a classifying map directly from a ...

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### Obstruction to this gauge choice of the connection of a vector bundle

Let $M$ be a compact manifold with a nowhere-vanishing vector field $R$. Consider principal $G$-bundle $P$ over $M$, and $\mathcal{A}$ being the space of irreducible connections.
Let me denote a ...

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### Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...

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### General form of a metric affine connection with zero curvature

I have read somewhere that the general form of a metric affine connection whose Riemann curvature is zero is given by
$$\Gamma^{i}_{ij}=A^{i}_{\alpha}\partial_{j}A^{\alpha}_{k}$$
where ...

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### Existence of connections making a bundle endomorphism parallel

Let $M$ be a manifold, $TM$ its tangent bundle, and $N:TM\to TM$ a vector bundle morphism. It is possible to find a torsionless linear connection $\nabla$ on $TM$ such that $\nabla N=0$?

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### Formula for the curvature of an induced connection

Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles ...

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### Kernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundle

I'm looking for references for the following fact, to pass on to a grad student. I think I can prove it, but it is a bit sloppy and I'd rather have something which is already written up.
Let $X$ be a ...

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### choices of connection in prequantization

In the definition of pre-quantization of representation $f\to \hat{f}$, (here $\hat{f}$ is Hermitian operator)of $C^{\infty}(M)$ on $L^2(M,L,\mu)$ where $\mu$ is Hermitian form, suppose that there ...

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### Anti-self-dual connections on CP^2

I'm learning Yang-Mills theory and its applications on 4-manifold.
I want to know that have someone computed all the anti-self-dual connections on principle
$SU(2)$ bundles over complex projective ...

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### comparison theorem for connections with regular singularities

This is something I've never understood.
Let me first recall the classical case, in which you start with a smooth algebraic variety $X$ over $\mathbb{C}$. One has the algebraic de Rham complex ...

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### locally free sheaves and logarithmic connections

I know that, when $\mathcal{F}$ is a coherent sheaf on a smooth algebraic variety $X$ over a field $k$ equipped with a connection
$$
\nabla: \mathcal{F} \to \mathcal{F} \otimes \Omega^1_X,
$$ then ...

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### Conjecture on homotopy groups of moduli space of self-dual connections

In the paper Stability in Yang-Mills Theories (1983), Taubes puts a (topological) bound on the Hessian of the YM-action on $S^4$. He consequently conjectured:
"The inclusion ...

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### Connections on tangent bundles and double tangent bundles

This can be viewed as a sequel to my previous question on double tangent bundle. Where I learned that the double tangent bundle $TTM$ is not natural diffeomorphic to $\oplus^3 TM$.
Recently, I also ...

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### Chern class of a logarithmic connection

Let $X$ be a smooth complex projective algebraic variety and $E$ a line bundle on $X$. It is a classical result that if $E$ carries an integrable connection, then the first Chern class $c_1(E)$ ...

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### Lefschetz hyperplane section theorem for connections

Let $X$ be a projective, smooth, algebraic variety over a subfield of the complex numbers, and let $Y \hookrightarrow X$ be a smooth hyperplane section of $X$. The classical Lefschetz theorem claims ...

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### The (-)-Connection on a Lie Group

Is the geodesic exponential map for a Lie group with the (-)-connection a diffeomorphism? This connection is one of two flat connections introduced by Cartan and Schouten on a Lie group and has ...

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### How many flat connections has a line bundle in algebraic geometry?

Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how ...

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### Flat connection, finite-dimensional space of covariant constant one forms

hallo,
I have the following question: Let $U\subset \mathbb{R}^{n}$ be an open subset. Furthermore, let $\nabla$ be a flat connection on $U$ (not necessary Levi-Civita). How can one show that the ...

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### Why is it important that partial derivatives commute?

I am asking this in the context of differential geometry (specifically Riemannian).
When the Levi-Civita Connection is defined, we require that the torsion tensor is 0, which in local coordinates ...

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### flat connection

Suppose $p:P \to X$ is a projective bundle and $O(1)$ is the line bundle on $P$ restricting to $O(1)$ on each fibre. When is $p_*(O(1))$ flat on $X$?

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### Why do we use the less simple convention for the definition of a vector bundle connection?

For a (smooth) vector bundle $F$ over a manifold $M$, one normally defines a connection to be a linear map
$$
\nabla:\Gamma^{\infty}(V) \to \Omega^1(M) \otimes \Gamma^{\infty}(V),
$$
satisfying ...

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### connections with regular singularities

Let $k$ be a field of characteristic zero, $X=\mathbb{G}_{m, k}=\mathrm{Spec}\ k[t, t^{-1}]$ the multiplicative group over k and $E=\mathcal{O}_X$ the trivial line bundle.
Consider the connection ...

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### regular singularities

Hi friends,
Let me ask you about connexions having regular singularities. So imagine $X$ is some smooth algebraic variety over a subfield $k \subset \mathbb{C}$ and you have a locally free ...

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### Tensor contraction and Covariant Derivative

What is the importance and intuition behind the the contraction operator on tensors (or the trace of a matrix, for that matter)?
In addition, I see that one of the requirements for a covariant ...

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### cohomology of the Gauss-Manin connection

Let $U$ be a smooth algebraic variety defined over $k \hookrightarrow \mathbb{C}$. Let $\mathcal{E}$ be a locally free sheaf on $U$ equipped with an integrable connection
$\nabla: \mathcal{E} \to ...

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### decompose a connection

Is there any theorem which states any general linear connection can be decomposed into another linear connection plus the contortion tensor ?
i didn't find any References

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### Connection Transformation Formula; Degree 3 Cech Cohomology

While reading through Brylinski, as in all of my posts, I am trying to understand the following equation:
$ g_* \tilde{\theta} = \tilde{\theta} - g^{-1} dg$
Setting
I have a principal ...

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### Intuition for Levi-Civita connection?

Levi-Civita connection is usually defined as the unique connection which is torsion free and preserves metric.
Question Is there some intuitively transparent constructive way to define it (or ...