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."

I'm not sure the literature is entirely consistent on the use of these terms.

Here are some ideas I learned from Jean Pradines' explanations in 1981 in Toulouse of his note

• Pradines, J., Th\'eorie de Lie pour les groupoides diff\'erentiables, relation entre propri\'et\'es locales et globales, Comptes Rendus Acad. Sci. Paris, S\'er A, 263 (1966), 907-910.

the first of his 4 notes introducing the relation between Lie groupoids and Lie algebroids. The ideas for the first note were written up in detail in various work by research students at Bangor, with the full knowledge of Pradines. (Kirill Mackenzie worked quite independently on the succeeding theory of Lie groupoids and Lie algebroids, published in his 1987 book.)

Intuitively, non trivial holonomy may be explained as an iteration of local procedures which return to the starting point with a change of phase. This idea is related to hysteresis, and shows the nice relation with physics. The problem is to define rigorously all the terms used in this explanation!

A Monodromy Principle is enunciated in Chevalley's famous book ``Lie groups"; the Principle may be explained as giving an extension of a restriction of a local morphism to a morphism on a simply connected cover.

In foliation theory, it is usual to define a monodromy groupoid as the disjoint union of the fundamental groupoids of the leaves, with a topology reflecting the local structure of the foliation; and to define the holonomy groupoid as a quotient of the monodromy groupoid. However this does not easily yield a Monodromy Principle.

Pradines' idea for his Th\'eor`eme 2 in the Note was to use the Monodromy Principle as guiding the construction of a Monodromy Groupoid of a Lie groupoid, generalising the universal cover of a connected Lie group. So, given a neighbourhood $W$ of the identities of a Lie groupoid, one forms the groupoid $M(W)$ which is universal for all local morphisms of $W$ into groupoids. The problem is to define an appropriate topology on $M(W)$ and Pradines solves this using a notion of holonomy groupoid, although in this case the holonomy is trivial (!).

In the case of a Lie group $G$, the topology on $G$ is defined by a neighbourhood of the identity satisfying some reasonable conditions given in, for example, Bourbaki. Now one can define a local Lie groupoid to be a groupoid $G$ with a set $W$ containing the identities and satisfying a number of reasonable conditions. However it is no longer true that the topology of $W$ extends to a topology on $G$ making it a Lie groupoid. Instead there is, under reasonable conditions, a Holonomy Groupoid $Hol(G,W)$ which projects to $G$ and which has a Lie groupoid structure locally like $W$. The construction of Pradines is written up in:

• Aof, M.E.-S.A.-F. and Brown, R., The holonomy groupoid of a locally topological groupoid, Top. Appl. 47, 1992, 97-113

(with the agreement of Pradines). It really does use the idea of ``iteration of local procedures" where the local procedures here are given by Ehresmann's local admissible sections of $G$ with values in $W$. The holonomy groupoid $Hol(G,W)$ has a universal property for maps of Lie groupoids into $G$.

The application to the monodromy groupoid is written up in: Brown, R. and Mucuk, O., The monodromy groupoid of a Lie groupoid, Cah. Top. G\'eom. Diff. Cat. 36 (1995) 345-369.

See also: Mucuk, O., Kılı¸carslan, B., S¸ahan, T. and Alemdar N. Group-groupoid and monodromy groupoid, Topology and its Applications 158 (2011) 2034-2042.

So holonomy comes out as a kind of right adjoint, and monodromy as a kind of left adjoint, which explains one difference. But there seems still work to do to explain everything stated in the two Theorems of Pradines' first Note, and to apply these ideas more widely. This is a reason for advertising Pradines' ideas here.

May 29, 2019 In response to Arrow's request, here are some references, The statements on adjoints are made in Pradines' cited paper, but very briefly, and with no proofs.

One idea behind holonomy is that of "an iteration of local procedures returning to the starting point but with a change of phase"; an example is moving a pencil at the North pole of a globe down a line of longitude, then along the equator, then back up to the pole, when it is pointing in a different direction. Pradines formalised this in terms of a "locally topological groupoid". An old theorem is that a locally topological group is extendible to a topological group (Bourbaki). For groupoids this is no longer true; the precise statement is in this edited version of a published paper; instead $(G,W)$ determines under good conditions a holonomy topological groupoid $H(G,W)$ which maps down to $G$ and is universal for maps of topological groupoids into $(G,W)$.

The situation is different for monodromy, which relates to the old "monodromy principle", that a local morphism should be extendible to a universal cover. See papers [88,89] in my publication list, in which a "universal cover" is replaced by a "monodromy groupoid" of a topological groupoid. Sorry to be so brief on this.

Both the monodromy and holonomy are morphisms from the fundamental group to another group. In case of the monodromy of the universal cover $p:\tilde{X}\to X$ is a morphism from $\pi_1(X,x_0)$ to the group of bijections of the fiber $p^{-1}(x_0)$. The holonomy of a flat connection connection on a vector bundle $E\to X$ over a smooth manifold $X$ is a morphism $\pi_1(X,x_0)\to GL(E_{x_0})$. There is also a notion of monodromy of locally constant sheaves. When asking about monodromy you should specify the monodromy of what object are you interested in.