Additional Structure
Many concepts of Riemannian geometry can be applied. There is only one obvious notion of parallel transport and only one natural connection. The concept of holonomy is strikingly simple in this case. The restricted holonomy group is trivial, and so there is a homomorphism from the fundamental group onto the holonomy group. It may be especially convenient to remove all singularities to obtain a space with a flat Riemannian metric and to study the holonomies there. One concepts thus arising are polyhedral Kähler manifolds, when the holonomies are contained in a group, conjugate to the unitary matrices. In this case, the holonomies also preserve a symplectic form, together with a complex structure on this polyhedral space (manifold) with the singularities removed. All the concepts such as differential form, L2 differential form, etc. are adjusted accordingly.
Read more about this topic: Polyhedral Space
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