The connection defines parallel transport of tensor/spinor fields relative to a given symmetry group G that keeps the global dynamical action of matter fields (classical and quantum including strings and branes) invariant.

The transformations of G need physical definition in terms of measurements at least in sense of gedankenexperiments if not actual technologically achievable measurements.

The most well known anholonomic connection is the vector potential Cartan 1-form A in the G ---> U1 fiber bundle.

The kinetic momentum of a charged particle is

mv = P - (e/c)A

P is the canonical momentum in the Lagrangian formulation

A is the field momentum attached to the charge - needed for local momentum/energy conservation

mv is gauge invariant!

The Abelian gauge transformations are

A --> A' = A + df

f = 0-form

d^2 = 0

When

F = dA =/= 0

A is said to be anholonomic (or non-holonomic)

F = electromagnetic field Cartan 2-form.

In vector calculus F is the 4D curl of A in the Minkowski spacetime of special relativity.

The de Rham integral of A over a closed 1-cycle is detectable as an interferometer fringe shift on charged test particles even if F = 0 on the paths of the particles. This is the Bohm-Aharonov effect.

The generalized Stoke's theorem is:

Integral of the exterior derivative of a p-form over a (p + 1)-co-form (domain of integration) = integral of the p-form over the p-boundary of the (p + 1) co-form (aka "chain" in sense of topological complexes in a discrete version of the manifold like in lattice gauge theory of quantum chromodynamics, Ken Wilson's renormalization group flows on a lattice etc).

Therefore, in the special case of the Bohm-Aharonov effect the enclosed quantized magnetic flux has a nonlocal influence on the quantum phase of the charges in the F = 0 region. Either that, or A is a local observable. More precisely, the equivalence class of A's connected by the df gauge transformation is the local observable.

In the case of the electromagnetic U1 there is no direct way to measure A locally. The situation is not clear to me for SU2 & SU3.

D = d + A/\ = exterior covariant derivative operator

F = DA = dA + A/\A

In the case of gravity there is a local measurement of the connection analogous to A.

In the case of Einstein's 1916 gravity A --> spin-connection A^I^J

F = gravity curvature Cartan 2-form.

For example 1

Including the indices for SU2 & SU3 weak-strong forces

F^a = dA^a + f^abcA^b/\A^c

f^abc are the structure constants of the Lie algebra commutators of the charges Q^a

[Qb,Qc] = f^abcQa

Another example 2 gravity (raise and lower indices I,J,K with the Minkowski metric nIJ)

F^I^J = dA^I^J + A^IK/\A^KJ

A^I^J = -A^J^I = gravity spin connection Cartan 1-form

all of these quantities are GCT LOCAL "GAUGE" INVARIANTS!

The relation to the Levi-Civita connection is indirect and complicated - see Rovelli Ch 2 "Quantum Gravity" (free version online).

I,J,K are indices for the 1905 SR Poincare symmetry Lie group P10 generated by total energy, linear momentum, angular momentum, boosts forming the Lie algebra of "charges."

The spin connection is not a tensor under the Lorentz SO1,3 subgroup of P10.

Example 3 thermodynamics.

The thermodynamic manifold is not in space-time but with respect to a set of intensive-extensive coarse grained conjugate pairs like

T S (temperature - entropy)

u N (chemical potential - particle number)

stress-strain

magnetization-magnetic field (magneto-striction)

the intensive variables are Lagrange multipliers in Legendre transformations etc.

We are given a thermodynamic 0-form E

E is a sum of conjugate pairs.

With 1-form

dE

d^2E = 0 are called the Maxwell relations.

This vanishing curl in the thermodynamic manifold is analogous to zero curvature in gravity and zero electric and magnetic field in Maxwell's theory.

However, new physics will introduce a non-coordinate thermodynamic connection A where

d^2 = 0

but D^2 =/= 0

local experiments measure D not d

i.e.

D = d + A

DdE = d^2A + A/\dE = A/\dE =/= 0

this will look like an apparent violation of the naive Maxwell relations that presuppose a "coordinate" holonomic connection.