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Results tagged with connections
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user 6818
Ehresmann connections; covariant derivatives; connections on vector bundles, principal bundles, ∞-bundles, submersions, bundle gerbes; holonomy and higher holonomy; parallel transport; torsion; curvature. See also the tags [principal-bundles], [vector-bundles], [gerbes], [curvature], [geodesics], [characteristic-classes], [torsion].
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Accepted
Does the holomorphic curvature determine the connection?
So if two connections are equal (or isomorphic in appropriate sense) then they must have the same holonomy. … As has been already mentioned in the comments, point singularities of connections pretty much behave as punctures in $M$. …
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11
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Symplectic connections are (locally) Levi-Civita connections
For any Riemannian metric $g$ on a symplectic manifold $(M, \omega)$ there exists (canonical) almost complex structure $J$ making these threetensors compatible (any one can be defined by other 2). com …
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Riemann-Hilbert correspondence for non-flat connections
I'm a little bit familiar with the smooth category version. Fixing a basepoint $b$, you can parallel transport the fiber from the base point to any other point $p$. The transfer is path-independent if …
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4
votes
Do "associative" connections exist / arise naturally in some context?
Consider the associativity for $fZ$ for any $f\in \mathcal{C}^\infty(M)$. Then on the right hand side of the associativty condition you are differentiating $f$ twice, while on the left hand side you a …
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1
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Holonomy map on a connected manifold determines the connection and the bundle
See Group of loops, holonomy maps, path bundle and path connection by Jerzy Lewandowski and related question on MO.
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1
vote
Accepted
What is the group of symmetries of $\mathbb{R^n}$ with the flat projective structure?
Category of manifolds with an equivalence class of torsion free affinne connections that have the same unparametrized geodesics is equivalent to (properly normalized) Cartan geometry modeled on a homogeneous …
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Kernel of a non-integrable connection
In (real) differential geometry the kernel of a connection $\nabla$ is called the space of parallel sections. These form a bundle whose fiber over a point $p$ is the space of fixed points of the defin …
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