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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?

  • In measures "how many ways" there exist to go away from a point and come back to it. On the cone, the holonomy measures how many loops you have made around the vertex of the cone before coming back. – Didier Jun 17 '21 at 08:52
  • Thank you for your answer. So it doesn't say anything about the curvature, am I right? –  Jun 17 '21 at 08:54
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    It does say something about curvature, because how you measure the difference between two ways for coming back to your point is parallel transportation, which is closely related to curvature. It does not just take account how many loops you have made but also how much you have to bend along them. – Didier Jun 17 '21 at 08:55
  • The holonomy group is almost completely equivalent to the curvature. But the holonomy group is more complicated than that, as it also contains topological information. It's the topology of the cone (not the curvature of the cone) that's at play in your link. Still, a large part of why differential geometers care about holonomy is because of how deeply it relates to curvature. – Jesse Madnick Jun 17 '21 at 08:57
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    The choice of a "cone" as an example is somewhat deceptive, because the "cone point" is not a smooth point. Nonetheless, from a more advanced point of view one can view the cone point as an atom of positive curvature which the holonomy has indeed detected. – Lee Mosher Jun 17 '21 at 16:07
  • If a closed curve bounds a region $R$ in your surface, then the holonomy around the curve is given by $\iint_R K,dA$. – Ted Shifrin Jun 18 '21 at 03:37

2 Answers2

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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.

Jesse Madnick
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As a modest addendum by a naïve physicist to the previous detailed answer, my understanding is that parallel transport on flat but topologically non trivial surfaces will generate a fixed and quantized phase: for example $n$ times around a Möbius strip will pick up a phase of $n$ $\pi$; similarly an integer multiple of some constant if one goes around the apex of a cone (zero, on the other hand, if the apex is not inside the loop). Phases picked up on curved surfaces (e.g. the sphere), instead, depend on the path. (Of course I have no idea how general this is or what possible exceptions are.)

Vincenzo Fiorentini
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  • I like the general idea but the specific difficulty with your Möbius band example is that for odd values of $n$ the holonomy is a reflection, not a rotation. – Lee Mosher Nov 27 '24 at 15:25
  • Thanks for the comment. This attracted my attention because of examples of phase behavior in spinor rotations or in loops around singular band structure loci in solids (e.g. graphene); I am just now starting to study topological effects in materials. So I guess in this case one (or 3, 5, ...) loop(s) around the strip is equivalent to a local reflection, and a further loop gives another reflection, "unlooping" the vector back to its original self at $n=2$ (4, 6...). The proper nomenclature is way over my head, I guess. – Vincenzo Fiorentini Nov 28 '24 at 19:14