Holonomy of the sphere

Question

I saw an example in which the holonomy of $\mathbb{S}^n$ with the standard metric is calculated. I'm just starting to get familiar with holonomy groups and I wanted to know what could one do by knowing that the holonomy of the sphere is $SO(n)$. Does it have topological consequences? or maybe something about the differentiable structure? I would like to know some results in this direction just to stimulate me and to generate some intuition and a notion of what kinds of things can be done with holonomy (if it is possible specifically on the sphere).

Answer 1

MathWorld and Wikipedia both have very good articles on this.

• http://mathworld.wolfram.com/HolonomyGroup.html
• https://en.wikipedia.org/wiki/Holonomy

Perhaps, of most potential interest, might be Berger's classification:

• https://en.wikipedia.org/wiki/Holonomy#The_Berger_classification

This addendum completes Berger's classification:

• http://arxiv.org/pdf/math/9911266.pdf