What does holonomy measure? I have difficulty understanding conceptually what holonomy measures.
it can return a phase shift of the vector transported parallel along the connection. If there is no phase shift, it means that the connection is flat, and if there is phase shift, then it should indicate that the space is curved.
But I have found examples where a flat space can have non-trivial holonomy, for example a cone has non-trivial holonomy (see On a flat surface, can a holonomy can be nontrivial around certain curves).
So my question: what information does holonomy give us? anything about the curvature?
 A: The holonomy group of a Riemannian manifold $(M,g)$ tells you (at least) five things about $(M,g)$:

*

*Topological information.  That is, the holonomy group of $(M,g)$ encodes information about the fundamental group $\pi_1(M)$.  More precisely, there is a surjective group homomorphism $\pi_1(M) \to \text{Hol}(g)/\text{Hol}^0(g)$, where $\text{Hol}(g)$ is the (full) holonomy group and $\text{Hol}^0(g)$ is the reduced holonomy group.  This is what you are seeing in the cone example.


*Curvature. The holonomy group constrains the possible values of the Riemann curvature tensor.  Roughly speaking, the smaller the holonomy group, the flatter your manifold.  Conversely, by the Ambrose-Singer Theorem, the Riemann curvature determines the holonomy group, demonstrating the slogan "holonomy is curvature." Interestingly, if the Riemannian holonomy group $\text{Hol}_x(g)$ lies in certain special subgroups of $\text{SO}(T_xM)$, then $g$ is an Einstein metric,
meaning that the Ricci curvature $\text{Ric}(g)$ satisfies
$\text{Ric}(g) = cg$ for some constant $c \in \mathbb{R}$.


*Parallel tensor/spinor fields. The holonomy group completely determines the existence (or non-existence) of parallel vector fields, parallel differential forms, parallel spinor fields, etc.  This is sometimes known as the holonomy principle.


*Product structure. The holonomy group of $g$ can be used to determine whether or not $g$ is (locally or globally) a product metric: i.e., whether or not $g = g_1 \times g_2$ for some Riemannian metrics $g_1, g_2$.  This is the de Rham Decomposition Theorem.  There is both a local version of the theorem and a global one.


*Extra compatible geometric structure. Is your Riemannian manifold $(M,g)$ secretly a Kahler manifold?  In other words, does there exist an integrable complex structure $J$ on $M$, compatible with $g$, so that $(M,g,J)$ is a Kahler manifold?  This is completely determined by the holonomy group.  More generally, the presence of extra geometric structure on $(M,g)$ that is both "compatible with $g$" and "flat to first order" is determined by the holonomy group.
