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Results tagged with connections
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user 13268
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].
3
votes
Comparing holonomies along different connections?
The expression $\sigma_t=(1-t)\sigma+t\sigma'$ is a connection, for every $t \in \mathbb{R}$, giving a path between the holonomies.
- 26.3k
2
votes
Accepted
Definition of Connection as G-invariant splitting of a sequence which is a pulled back seque...
(Not how I think about principal bundles and connections, but certainly something Atiyah always does.) …
- 26.3k
4
votes
Existence of connections making a bundle endomorphism parallel
No. To be parallel under a connection, $N$ would have to have the same type at each point. In other words, for any two points, there must be a linear map identifying tangent spaces which identifies $N …
- 26.3k
5
votes
Two ways to differentiate a section of vector bundle
Yes: just write it out in a local basis: $\sigma_i = \sum f_{ij} E_j$ in some local choice of basis $E_j$, and $\sigma_1(p)=\sigma_2(p)$ means that $f_{1j}(p)=f_{2j}(p)$. Then $d(\sigma_1-\sigma_2)$ i …
- 26.3k
4
votes
Affine connections as equivariant maps
So I don't think you really don't get an action of the structure group of $FM^1 \to M$ on $S^2 \mathbb{R}^{n*} \otimes \mathbb{R}^n$ arising in this way: you are not making a difference between connections …
- 26.3k
3
votes
Flat covariant derivative
Michael Spivak, A Comprehensive Introduction to Differential Geometry, volume II, p. 383, theorem 19, proves that the vanishing of curvature tensor of a Riemannian manifold implies flatness. But in fa …
- 26.3k
3
votes
Accepted
Splitting of higher order jet sequence
Look at the paper P. Jahnke and I. Radloff, Splitting jet sequences. They classify such splittings on compact Kaehler manifolds. Those which admit a vector bundle with splitting jet sequence are preci …
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3
votes
Accepted
Torsion free (1,0)-connections on the holomorphic tangent bundle?
Therefore if the $h_a$ form a partition of unity, for local choices of torsion-free $(1,0)$-connections, then $\gamma$ is a torsion-free $(1,0)$-connection. …
- 26.3k
6
votes
Why torsion is only defined for linear connection on TM?
So all connections look the same at a point, to first order. You can only feel that the connection is not flat at second order. …
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10
votes
Flat connection, finite-dimensional space of covariant constant one forms
The covariant constant 1-forms are parallel. The value of a covariant constant 1-form is determined throughout each connected component by its value at any one point: just take that value and parallel …
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1
vote
Accepted
Integrability condition for flat connections
On the manifold $X=U\times \operatorname{GL}_r$, with points written $x=(m,a)$, each tangent space $T_x X$ contains a linear subspace $V_x$ consisting of tangent vectors on which $a^{-1}da=-\omega'$. …
- 26.3k
11
votes
When are geodesics straight lines?
See Kobayashi and Nagano, On projective connections, Journal of Mathematics and Mechanics, vol. 13, no. 2, 1964. …
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4
votes
Accepted
Confusion about complex differential forms
Forget about vector fields $\partial_{z^{\mu}}$ and $\partial_{z^{\bar\mu}}$. Just think about 1-forms: $\omega^j_i = \Gamma^j_{i\mu}dz^{\mu}$, with $C^{\infty}$ functions $\Gamma^j_{i\mu}(z)$. Then i …
- 26.3k
3
votes
Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?
The geodesics of Euclidean geometry in the plane are straight lines, i.e. curves that do not change direction. Their Cartan geometry is $$\omega=\begin{pmatrix}\gamma&\xi\\0&0\end{pmatrix}$$ on the or …
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28
votes
What is the Levi-Civita connection trying to describe?
I will try to help with the title question. I think that the real motivation for the Levi-Civita connection comes from looking at surfaces in Euclidean 3-space. Differentate one tangent vector field $ …
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