# 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.

185 questions
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### 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 ...
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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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### Connections in terms of tangent ($\infty$-)categories?

Given a commutative ring $k$ and a commutative $k$-algebra $A$, we know that the Kähler differential $\Omega_{A/k}^1$ could be described through machineries of tangent categories (see, for example, ...
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### When are geodesics straight lines?

Suppose I have a global coordinate system on a manifold, which is affine with respect to an affine connection on that manifold. The connection is flat and torsion free, and the connection coefficients ...
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### Do “associative” connections exist / arise naturally in some context?

Here is a little bit of curiousity that's been itching me, let's hope it doesn't get me killed, meow. Definition: Let $M$ be a smooth manifold. A connection $\nabla$ on $TM$ is called associative ...
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### Metric connection on $\mathbb{R}^4$ that is locally Kähler but not globally Kähler

in a comment to this question When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection? Robert Bryant mentions that it is possible to construct a metric connection ...
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### Locally Riemannian Connection

Let $\Gamma^a{}_{bc}=\Gamma^a{}_{cb}$ be a symmetric connection whose curvature is R^a{}_{bcd}=\partial_c\Gamma^a{}_{bd}-\partial_d\Gamma^a{}_{bc}+\Gamma^a{}_{ec}\Gamma^e{}_{bd}-\Gamma^a{}_{ed}\...
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### Riemannian vs Non-Riemannian curvature

If you neither know the metric nor the holonomy group, how do you recognize a curvature tensor is Riemannian? I assume a curvature, by definition, satisfies Bianchi identities. I know it is ...
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### Is torsion of a connection always an obstruction to some kind of integrability? [closed]

Let $E$ be a vector bundle over a smooth manifold $M$ equipped with a linear connection $\nabla : \Gamma(E) \to \Omega^1(M;E).$ I say $(M,E,\nabla)$ is flat if it admits trivial local models; i.e. if ...
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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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### References on principal G bundle and connections

I am trying to understand about principal G bundle given a Lie group $G$. For that I started with action of Lie groups on manifold $M$ and convinced my self that if the action is smooth, proper, free ...
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### Most natural connection on Lie group: comparison of different pictures

Let $G$ be a Lie group (not necessarily compact). One can equip $G$ with the left invariant metric (or right invariant but in general there is no biinvariant metric in the noncompact case). Once the ...
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### Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?

I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...
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### Riemann-Hilbert correspondence for non-flat connections

First of all, let me warn that my knowledge of the correspondence is rather superficial, and I apologize for any technical inaccuracies below. Setting Let $X$ be a smooth complex algebraic variety, ...
The Riemann-Hilbert correspondence states that the kernel of an integrable (zero curvature) connection is a local system. Here, a connexion on a vector bundle $E$ over a manifold $X$ is a morphism of ...
### Relation between the geodesics of Finsler norms $F(V)$ and $F(-V)$
I am trying to solve this exercise. Let $(M,F)$ be a Finsler space and define $\tilde{F}(x,y):=F(x,-y)$. Then $(M,\tilde{F})$ is a Finsler space and given a geodesic $t\mapsto \gamma(t)$ of $F$, \$t\...