A connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.

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2
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87 views

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 ...
5
votes
1answer
684 views

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 ...
0
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2answers
339 views

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 ...
2
votes
1answer
141 views

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 ...
8
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3answers
737 views

Can a manifold have a curvature-free connection that is not torsion-free?

Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a ...
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0answers
146 views

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 ...
8
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2answers
819 views

Does every manifold have a flat connection?

Suppose I have a manifold and a vector bundle over it, but not a connection or a metric. Can I always find a connection on it that has a Riemann curvature tensor that is identically zero? If so, can I ...
10
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2answers
1k views

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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0answers
156 views

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 ...
4
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2answers
272 views

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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0answers
256 views

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 ...
3
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2answers
214 views

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 ...
1
vote
1answer
204 views

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 ...
5
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0answers
251 views

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 ...
3
votes
1answer
250 views

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 ...
4
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1answer
241 views

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 ...
6
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1answer
192 views

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 ...
0
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1answer
230 views

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 ...
4
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2answers
392 views

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 $c_1(E)$ ...
3
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0answers
177 views

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 ...
3
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1answer
472 views

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 ...
10
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2answers
951 views

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 ...
0
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1answer
170 views

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 ...
17
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4answers
2k views

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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0answers
207 views

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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1answer
551 views

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 ...
1
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1answer
297 views

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 ...
0
votes
1answer
220 views

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 ...
12
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6answers
4k views

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 ...
2
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1answer
417 views

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 ...
3
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2answers
336 views

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
4
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3answers
376 views

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 ...
10
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2answers
1k views

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 ...
3
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1answer
974 views

1-jet bundle on vector bundle with metric connection

Background I'm working to simplify the Lagrangian formalism of classical field theory for the situation of a vector bundle with a bundle metric and a metric connection. Particularly, I want to ...
4
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3answers
772 views

Open problems in sub-Riemannian geometry

What are some open problems in sub-Riemannian geometry? I am interested especially in problems concerning connections and curvature, but any contribution is welcomed.
3
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2answers
1k views

Interpretation of Curvature and Torsion

Dear all, When dealing with General Relativity one uses the Levi-Civita connection with is torsion-free. Thus the commutator of the covariant derivatives yields $[\nabla_\mu,\nabla_\nu]V^\rho = ...
9
votes
1answer
686 views

Is there a mathematical explanation for the Aharonov-Casher effect?

Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows. Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the first differential ...
7
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4answers
356 views

Hamiltonians which commute both as operators and as connections

This is something which I suspect is written up in introductory books on mathematical physics if I knew where to look. Suppose I have some parameters $t_1$, ..., $t_k$ ranging over a neighborhood in ...
6
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0answers
192 views

Does every stack with a connection admit an atlas with a connection?

Dear all, Let $S$ be a scheme in characteristic $0$, and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme ...
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vote
2answers
469 views

complex vector bundles and curvature

Let us suppose that $X$, with a 2-form $\omega$. Suppose $J$ is an element of $su(2)$ such that $J^2=-e$ for $e$ the identity. Is there a necessary and sufficient condition on $\omega$ which will give ...
1
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2answers
2k views

Line bundles with complex connection

Suppose that we have a complex manifold $X$, and a line bundle $L$ over $X$. It is known that the line bundles over $X$ are parametrized by their Chern class, the Chern class being the cohomology ...
3
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1answer
882 views

Terminology of “covariant derivative” and various “connections”

I apologize for the long question. My ultimate aim is to understand the existing terminology and find appropriate terminology for covariant derivatives on vector bundles which generalizes to ...
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0answers
163 views

Analytic Characterization of Parallel Transport of Fundamental Groups

(Note that I've edited the main body of the question to make it clear for other readers.) Fix a principal $G$-bundle $\rho: P \rightarrow X$ and fix a point $p \in P_x$ in the fiber above $x \in X$. ...
0
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1answer
335 views

A Follow-up About Connection Forms on Principal Bundles

In this question I asked about proving that a connection form $\alpha$ on a $\mathbb{C}^*$ bundle had to have $2\pi i(\alpha - \overline{\alpha})$ be exact. From the answer to that question I ...
3
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1answer
882 views

Principal bundle connections

It is my understanding that a connection form on a principal G-bundle over a manifold X is defined to be a Lie algebra-valued 1-form $\alpha$ which reproduces the Lie algebra generators of the ...
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2answers
334 views

Connections with compatible Hermitian products on complex line bundles

Let $X$ be a manifold, $L$ be a complex line bundle over $X$, and $L^{*}$ be the associated principal bundle. Suppose $\alpha$ is a connection form on $L^{*}$, with associated connection $D$ on $L$. ...
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1answer
706 views

Are bundle gerbes bundles of algebras?

The category of line bundles (possibly with connection) on a smooth manifold M can be defined in two different ways: The first definition uses transition functions that satisfy a cocycle condition ...
17
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3answers
3k views

Geometrical meaning of the Ricci Tensor and its Symmetry

Let $M$ be a smooth, pseudo-Riemannian manifold with $\dim(M) \ge 2.$ Let $\nabla$ be any affine connection on $M$. No reason for it to be the Levi-Civita connection. All we assume is that it has zero ...
5
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1answer
678 views

Symmetric Ricci Tensor

Let $M$ be a pseudo-Riemannian manifold. Assume that $M$ comes with a zero-torsion affine connexion $\nabla.$ There is no need for $\nabla$ to be the Levi-Civita connexion. Recall that the curvature ...
6
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1answer
587 views

Almost Flat Connections On Principal G-Bundles

Let $E \rightarrow F$ be a principal $G$-bundle. Let $\alpha$ be a connection 1-form with values in the Lie algebra of $G$. Let $\omega$ denote the curvature 2-form of connection $\alpha$. We know ...