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Results tagged with connections
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user 13972
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
Accepted
Are non-linear connections with linear holonomy, linear?
The answer to your literal question is 'no', you can even have the holonomy be trivial (in which case, it is certainly linear) and yet the connection not be linear. This is a local
question, so you …
- 108k
6
votes
Accepted
Connection, compatible with type (1, 1) tensor field
N.B.: I'm fixing my answer, which was off for two reasons: First, I didn't correctly interpret the OP's notation. (Thanks, Sebastian, for pointing that out!) Second, I didn't check the case when the …
- 108k
2
votes
complex vector bundles and curvature
An obvious necessary local condition is that $d\omega=0$. On the open set $U\subset X$ on which $\omega^2\not=0$, one has the further condition that the only ${\frak su}(2)$-valued connection that co …
- 108k
6
votes
Accepted
connections on principal bundles over $S^1$
Yes. Let $g\in G$ be the holonomy of the connection $A$; since $G$ is compact and connected, one can choose a frame (i.e., trivialization of $P$) such that $A = a\,\mathrm{d}\theta$ where $g = \exp(2 …
- 108k
20
votes
Geometrical meaning of the Ricci Tensor and its Symmetry
NB: I'm combining my previous comments into an answer, because I believe that this is better than leaving them scattered.
As another commenter has pointed out, the skew-symmetric part of the Ricci t …
- 108k
6
votes
Accepted
Left invariant connections on a Lie group
Yes. Just take the Levi-Civita connection of any left-invariant Riemannian metric on the Lie group. The metric is complete, so any two points can be joined by a geodesic (Hopf-Rinow). Thus, the geo …
- 108k
7
votes
Accepted
The automorphism group of a symplectic symmetric space
The affine group of $(M,\nabla)$ is a Lie group $G$ by Kobayashi's theorem that shows that the automorphism group of any affine connection is a Lie group (see Kobayashi and Nomizu's Foundations of Dif …
- 108k
9
votes
Accepted
A consequence of Ambrose-Singer theorem on holonomy
Your first question is a bit ambiguous. Are you asking whether, for each $p\in U$, the matrices $S_k(p)$ span the Lie algebra of $\mathrm{Hol}^0_p(\nabla)$ or are you asking whether, after taking the …
- 108k
3
votes
Obstructions to the existence of a flat connection on a vector bundle
A slightly different point of view for answering this question is the following one:
First, if $M$ is simply connected, then $E\to M$ admits a flat connection if and only if $E$ is trivial, so in this …
82
votes
Accepted
When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?
Even though they are linear ODE, for most connections given explicitly by some functions $\Gamma^i_{jk}$ on a domain, one cannot perform their integration. …
- 108k
32
votes
Accepted
Can a manifold have a curvature-free connection that is not torsion-free?
Many manifolds have curvature-free (i.e., flat) connections on their tangent bundles. … For example, any orientable $3$-manifold $M$ is parallelizable, i.e., its tangent bundle is trivial, so it carries a flat connection (in fact, many flat connections). …
- 108k
5
votes
Accepted
Metric, torsion free connections on principal bundles
I don't know what you mean by 'without using at all the picture on the tangent bundle $TM$', but here is how one normally does it:
First, one shows that there are canonical $1$-forms $\omega^i$ on $F …
- 108k
6
votes
Accepted
How large can the cone of $\nabla$-compatible metrics be?
The $\nabla$-compatible metrics on $E$ are the positive-definite $\nabla'$-parallel sections of $S^2(E^*)$, where $\nabla'$ is the connection on $S^2(E^*)$ induced by $\nabla$. When $M$ is connected …
- 108k
8
votes
Interpretation of Curvature and Torsion
Élie Cartan proposes such interpretations in his fundamental paper Sur les variétés à connexion affine et la theorie de la relativité généralisée (Ann. Ec. Norm. 40 (1923), 325–412 and 41 (1924), 1–25 …
- 108k
7
votes
Accepted
Is there such a connection on the punctured plane?
Yes. Take the Levi-Civita connection of any conformal metric $g = e^{2u}(dx^2+dy^2)$ of positive curvature, say. Then, by (local) Gauss-Bonnet, the holonomy around any smooth closed loop $\gamma$ is …
- 108k