What does holonomy measure? - Mathematics Stack Exchange
2021年7月13日 · The holonomy group constrains the possible values of the Riemann curvature tensor. Roughly speaking, the smaller the holonomy group, the flatter your manifold. Conversely, by the Ambrose-Singer Theorem, the Riemann curvature determines the holonomy
Holonomy group of Poincaré metric - Mathematics Stack Exchange
2020年2月11日 · You correctly calculated that the Identity is in the holonomy group. But other loops will give you other holonomies. To show that the holonomy group is SO(2), you need to show that the angle of any two vectors and the length of any vector is preserved under parallel transport along any curve. $\endgroup$ –
About two notions of holonomy - Mathematics Stack Exchange
The first holonomy does not obviously give a representation of the fundamental group: for example it is not invariant under homotopy of the paths. My question is: are these two notions somehow related (at least in some well-behaved case), or it just happens that we use the same name and we'd better call the second one "monodromy".
Berger's theorem on holonomy - Mathematics Stack Exchange
The restricted holonomy group is the subgroup of the full holonomy group generated by null-homotopic loops, and is the identity component. The above is a 'topology free' formulation, because it makes sense using only information in any …
basic question about holonomy - Mathematics Stack Exchange
So to use the holonomy principle, start with non-zero $\alpha_p\in \Lambda^n(T^\ast_pM)$. We want $\alpha_p$ to be preserved under Hol$(g)$ so that it can be extended, by the holonomy principle, to a nowhere vanishing form. Again, we just need $\alpha_p$ to be fixed by the holonomy group.
How does the holonomy act on the tangent space at a point?
2017年4月25日 · The point is that by definition the holonomy group IS a subgroup of U(n). In particular the action or representation is "faithfull". So to say that the holonomy is U(n) is to say both that the group as abstract group is U(n) and that the action on the tangente space is the standard representation of U(n).
differential geometry - Why is the restricted holonomy the identity ...
Since loops are compact (these are the loops you work with in holonomy), it is almost guaranteed that you only need a finite number of steps. This probably doesn't work if there are too many "holes", meaning, if the fundamental group of our base manifold is not finitely generated.
Possible holonomy groups of flat manifolds? - Mathematics Stack …
2022年8月13日 · Your intuition about curvature is right because of the Ambrose-Singer theorem. The holonomy group of a orientable Riemannian manifold is always a subgroup of ${\rm SO}(n)$ and, when this holonomy group is a Lie group, the holonomy algebra is generated by curvature endomorphisms. So, no curvature means "no holonomy". $\endgroup$ –
Holonomy group as a Lie group - Mathematics Stack Exchange
2021年4月3日 · the holonomy of the connection can be identified with a Lie group, the holonomy group. Why is this true?
Holonomy of a curve in case of principal $U(1)$ bundle
2017年6月18日 · Given a loop $\gamma: S^1 \to M$, we can form the pullback bundle $\gamma^*P$ with pullback connection $\gamma^*A$.