# Questions tagged [connections]

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

43 questions
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### Flat connections with logarithmic poles and p-adic parallel transport under choice of branch of logarithm

Consider the following simple situation: We work over the ring $R=\mathbb{Z}_p[[t]]$, over which we consider a rank $2$ free module $M$ with basis $(e,f)$. On $M$, we define a flat (topologically ...
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### Equivalent notion of Curvature of a connection on a Principle bundle

We know given a Connection 1-form $\omega$ on a Principle bundle $P(M,G)$ we can define a curvature 2-form $\Omega$ of $\omega$. We also know that given a Connection 1-form $\omega$ we can define a ...
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### Basic questions about crystals and Grothendieck connections

I have a few basic questions about Grothendieck connections and crystals. I know it's bad practice to ask a bunch of different questions at once, however I feel they naturally come bundled together. ...
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### Existence of sections of a fibre bundle which are covariantly constant along certain directions

Given a vector bundle $\pi\colon E \rightarrow B$ equipped with a connection $\nabla$, it is well known that a basis of flat sections $s_i$ ($i=1,\dots,\text{rank}(E)$) (i.e. $\nabla_X s_i = 0$ for ...
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### Do we have classical Riemann-Hilbert correspondence for infinite dimensional flat vector bundles?

Let $E$ be an $n$-dimensional vector bundle on a manifold $M$ and $\nabla: \Gamma(E)\to \Omega^1(M,E)$ be a flat connection on $E$. Classical Riemann-Hilbert correspondence tells us that ker$\nabla$ ...
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### Maximally symmetric affine manifold

As a physicist who knows (something) about General Relativity, I'm accustomed to the term "maximally symmetric space" being an $n$-dimensional manifold with $\frac{n(n+1)}{2}$ Killing vectors. A ...
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### Applications of Ambrose-Singer theorem on holonomy

I am planning to introduce to a group of Graduate students the notion of connections on Principal bundle, curvature of connection, Holonomy. I want to conclude with the statement of Ambrose-Singer ...
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### Existence of connections in a vector bundle whose parallel transport preserves a function on a total space

Let $p:E \to M$ be a vector bundle over a smooth manifold $M$, $M\times 0$ be the image of its zero section of $p$, $\mathcal{X}(M)$ be the space of vector fields on $M$, and $\Gamma(E)$ be the space ...
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### Connections on bundles over algebraic variety

I am asking for a reference for the theory of connections on vector bundles over algebraic varieties. I am particularly interested in the notion of Gauss-Manin connection, the notions of torsion, p-...
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### Is there a ''simple'' formula for the inverse of the Drinfeld associator?

The Drinfeld associator $\Phi(x_0, x_1)$ encodes the parallel transport of the Knizhnik-Zamolodchikov (KZ) connection $\nabla$ on the bundle $\mathbb{C}\langle\langle x_0, x_1\rangle\rangle$ of formal ...
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### A question about a paper of Bismut and Lebeau

Let $X$ be a Riemannian manifold, and $Y\hookrightarrow X$ be a closed submanifold of $X$ with normal bundle $N$ with the induced metric. Then near $Y$, we have $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...
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### Invariant Lagrangians of a connection and its derivatives: how do they look like?

Let $$L=L(\Gamma,\partial\Gamma,\ldots,\partial^n\Gamma)$$ be a Lagrangian depending on a linear symmetric connection $\Gamma$ on the tangent space of a manifold $M$ together with its derivatives up ...
For a Hermitian vector bundle E over a Riemannian $(X,g)$ and a unitary connection A we may define the sup norm of the curvature by: $$\|R_A\|=\sup \{\| tracefree (R_A (v))\|_{op} \mid v \in \Lambda ^... 0answers 252 views ### Connection and reduction of the structure group I apologize for this question which is not really research-level, but I don't get any answer on mathStackExchange, and I asked a professor in my university... which couldn't find the solution. I am ... 0answers 214 views ### 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 ... 0answers 255 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 ... 0answers 761 views ### Gauss--Manin connection for de Rham realisation Let X and S be smooth schemes of finite type over a field k and let \pi:X\to S be a smooth morphism of finite type. The relative de Rham cohomology H^i_{dR}(X/S) of X over S is a ... 0answers 85 views ### Weil homomorphism In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal G bundle. In this book, they use connection on a principal bundle to ... 0answers 71 views ### The dimension of the subspace of flat spin connections I am interested in the the flat spin connections in a Riemann spacetime of dimension 4. They appear in the context of the frame formalism of metric gravity theories. I believe that they form a ... 0answers 113 views ### Does the monodromy of such VHS have to be trivial Consider a variation of polarized Hodge structure on a punctured disk. Suppose that connection preserves Hodge filtration (which is much stronger, than Griffiths transversality). Moreover assume that ... 0answers 219 views ### 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 ... 0answers 214 views ### Hermitian connections on real hypersurfaces of \mathbb C^{n+1} I would like to find some non-trivial and "geometric" examples of hermitian connections (that is compatible with a give hermitian metric) on complex hermitian vector bundles over a smooth real ... 0answers 81 views ### Compatibility of Kirillov-Kostant-Souriau form and Killing form Let \mathfrak{g} be a real semisimple Lie algebra. We know that a coadjoint orbit \mathcal{O} \hookrightarrow \mathfrak{g}^* carries a natural symplectic form \omega, namely the Kirillov-Kostant-... 0answers 489 views ### Splitting Short exact sequences of vector bundle with connection Let F\to M be a vector bundle and E\subseteq F a subbundle. Suposse that \nabla is a connection on F s.t. preserves E, i.e. \nabla_X(e)\in \Gamma E \quad \forall e\in \Gamma E, \ X\in\Gamma ... 0answers 45 views ### How to find \beta^\prime_e(t) where \beta_e(t)=\textrm{Hol}^\sigma_{\gamma_{1, t}}(e)? Let p:E\longrightarrow B be a surjective submersion and \sigma: p^*(TB)\longrightarrow TE a complete connection. Given a path \gamma: [a, b]\longrightarrow B and s, t\in [a, b] such that s<... 0answers 163 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 ... 0answers 312 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 ... 0answers 249 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? 0answers 284 views ### Non-Existence of a Principal Connection for the Sphere over Projective Space? As of the Wikipedia article on principal bundles connections: Let \pi: P \to M be a smooth principal bundle, a principal G-bundle over a smooth manifold M. Then a principal G-connection on ... 0answers 82 views ### The algebraic connectivity of P_n, the path on n vertices does not exceed \frac{12}{n^2-1} The algebraic connectivity of P_n, the path on n vertices does not exceed \frac{12}{n^2-1}. Let algebraic connectivity of P_n be denoted by \mu. I have proved a result that if G is a ... 0answers 233 views ### residue formula for connections on curves Let X be a smooth, projective curve over a field k (characteristic zero is enough for me) and E a line bundle on X. Assume that E is equipped with an integrable logarithmic connection \... 0answers 195 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 $A^{i}_{\alpha}$...
(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 $x \in X$. ...