During my Master's thesis I encountered the theory of holonomy for the first time. Unluckily it was only tangentially related to the topic of my thesis, so I couldn't dive into it. The book I was using is Differential Geometry - Cartan's Generalization of Klein's Erlangen Program which talks a bit about the argument.

I wonder which prerequisite (apart from the elementary differential/Riemannian geometry) are necessary to understand the topic and which book do you think is more appropriate for studying the subject. I was thinking to something that drives me to the Berger classification.

By a rapid google search the first book I found is Submanifolds and Holonomy. Is this a good book?


1 Answer 1


Dominic Joyce has two relevant books: Compact Manifolds with Special Holonomy and Riemannian Holonomy Groups and Calibrated Geometry for the study of holonomy groups of Riemannian manifolds, the Berger classification, and the associated special submanifolds.

Sharpe's book that you mentioned discusses holonomy of flat Cartan geometries, which is quite far from Riemannian holonomy, where one is interested in non-flat connections but arising as the Levi-Civita connection of a Riemannian manifold. One can also consider holonomy groups of Cartan geometries which are perhaps not flat, but there is no book which provides a good reference, as this subject is in its infancy.


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