What is the difference between holonomy and monodromy?


And what is the simplest example in which one is trivial and the other is not?



What is the difference between holonomy and monodromy?
Holonomy= monodromy iff the bundle is flat. In general, monodromy group is the quotient of holonomy group by the normal subgroup formed by parallel transports along homotopically trivial loops. One of the simplest examples when two groups are different is the holonomy of the tangent bundle of the standard Riemannian metric on the 2-sphere. Then monodromy is trivial since sphere is simply-connected, while holonomy group is $SO(2)$. 


Two more remarks. First, the conundrum: What is the holonomy of a complete hyperbolic surface $S$? The answer: It depends who you ask. A differential geometer (like Robert Bryant) would think of the tangent bundle and his answer would be $SO(2)$ or $O(2)$ (depending on orientability). A hyperbolic geometer (like William Thurston) would think of the hyperbolic structure as a special $(X,G)$-structure and answer: $\pi\_1(S)\subset PSL(2, {\mathbb R})$. 
(An $(X,G)$ structure could be regarded as a flat $X$-bundle with a section transversal to the flat connection, so holonomy of the flat bundle is the holonomy of the structure.) If you were to ask me, I would say "It depends ..." 


Second remark: For cultural, historic, etc. reasons, given a flat bundle, differential geometers and topologists tend to use the word "holonomy," while people in algebraic geometry, complex analysis, singularity theory, tend to use the word "monodromy." 


