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Results tagged with connections
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user 4572
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].
4
votes
Accepted
Let $G \subset \mathrm{GL}(n)$ be a matrix Lie group. Does there exist an affine connection ...
Yes, take the the trivial connection with respect to the left trivialisation of the tangent bundle. Then, all of your curves $\gamma$ are geodesics, but there are no further geodesics.
Some more deta …
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9
votes
When do flat holomorphic connections exist?
The comment of HYL should be an answer. Since the OP has asked for explicit counterexamples, I will give an example that 1) does not imply 2):
Consider a compact Riemann surface $\Sigma$ of genus $g\ …
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1
vote
Curvature as infinitesimal holonomy 2
For example, there exists flat $SU(2)$-connections on the 1-holed torus with non-trivial monodromy along the boundary. …
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2
votes
Flat connection of a degree zero line bundle on curve
Still, there is something to say: I guess you are interested in meromorphic connections with first order poles. … So you have a natural map from the space of meromorphic connections with integer residues to the Jacobian. …
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1
vote
Connections in the setting of algebraic geometry
You should also have a look into Atiyah's 'Complex analytic connections in fibre bundles'. Even though he considers complex manifolds you can gain a lot insights from reading this paper. …
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3
votes
Vector field along an immersion whose covariant derivative is the differential
Your differential $df\in\Omega^1(\Sigma,f^*TM)$ satisfies the integrability condition $$d^\nabla df=0$$ where $d^\nabla$ is the induced exterior derivative from the (pull-back of) the Levi-Civita conn …
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1
vote
Accepted
Flatness as an integrability condition without invoking bundles
Consider the manifold $X=M\times G$ (where $G=GL(n)$ for simplicity). Define a distribution $\mathcal H$ on $X$ as the image of your connection 1-form, i.e.:
$$\mathcal H_{p,g}:=\{ (X_p,\omega_p(X) g) …
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4
votes
Accepted
Do "associative" connections exist / arise naturally in some context?
No, such a connection cannot exist: Consider a function which vanish to first order at a point $p\in M$, i.e., $f(p)=0$ and $d_pf=0$, but assume that there are vector fields $X,Y$ with $(X\cdot (Y\cdo …
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2
votes
Does "symmetry" of a pullback connection should be obvious?
Here is a proof which is not very elegant, but avoids coordinates. First of all, you define $$\hat T(X,Y)=\nabla_X^{\phi^*TN}d\phi(Y)-\nabla_Y^{\phi^*TN}d\phi(X)-d\phi([X,Y])$$ and observe that this i …
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6
votes
Can we define exterior derivatives using pushforwards and connections?
An affine connection induces the exterior derivative by taking the ant symmetrization of the covariant derivative if and only if the torsion of the connection vanishes. This can be computed directly.
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2
votes
A connection on $Hom( E,E)$ whose parallel transport is compatible to parallel transport of $E$
.$
The construction is compatible with metrics, as the same standard arguments for tensor products, dual bundles and corresponding connections carry over to the Riemannian/hermitian situation, i.e., the …
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4
votes
Accepted
Definition of Levi-Civita connection map and a theorem about it?
After the comment of Jez, here is the corrected answer:
There is the well-defined vertical bundle $VTM\subset TTM\to TM$ given as the kernel of the (differential of the) projection.
Moreover, for an …
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7
votes
Generalized Dirac operators
A very good place to read about this is the 3. chapter of the book "Heat kernels and Dirac operators" by Nicole, Getzler & Vergne.
Up to the wrong sign in your second definition, 1. and 2. are equiva …
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5
votes
A question about flat connection
More concretely (after Anton Petrunin's comment):
Consider the case of $S^1=\mathbb R/2\pi\mathbb Z$ and the trivial line bundle over it with connections $\nabla_0=d$ and $\nabla_1=d+d\varphi.$
Consider …
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1
vote
Accepted
Normalizing the value of a principal connection at a point
Yes,
consider a chart centered at $p$ and the rays emitting from $p$ with respect to this chart. Now, take a frame at $p$ and consider the parallel transport along
the rays. This gives you a local sec …
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