A.T. said:
For the analogy I prefer to use the difference in left and right track / wheel distance. Or the inertia of a perfectly balanced turret, as proposed by the OP in post #7.
A gyroscope is tricky within this analogy (2D curved surface embodied in flat 3D), as it tends to keep its orientation within the 3D embedding space, not within the 2D space intrinsic to the curved surface. You might even end up with the gyroscope axis pointing vertically up, and then you cannot detect the turning of the tank. A gyroscope makes sense when you go to higher dimensions, where the 3D space curved.
I mean, it is a two-dimensional gyroscope, i.e., one where the gyroscope is only free to rotate around one axis ...
Ibix said:
The connection coefficients come into the concept of covariant differentiation. Going back to me standing on the plane, as I moved the ##r## and ##\phi## components of my vector changed - that is ##\partial x^a/\partial \tau\neq 0##. But the vector didn't really change (I'm still pointing in the same direction) even if its components did. It would be really useful if we could define a sort of derivative that measures how much a vector really changed (not at all) even when its components do, since all the things like forces would depend on that. That is the covariant derivative, which is basically the familiar partial derivative plus a correction term for how the coordinates changed - i.e. the connection coefficient. Thus if the covariant derivative of a vector is zero as you carry it along a line then it is pointing in the same direction. That is parallel transport.
I'd like to add some additional structure to this: In a Euclidean space - such as the flat plane - it is clear what "the same direction" means between points (precisely because the space is flat). Taking any set of basis vector fields ##\{\vec E_a\}## (let us take ##\{\hat e_r, \hat e_\phi\}## of the polar coordinates for concreteness) that generally depend on the position, a vector field can be written as
$$
\vec v = v^r \hat e_r + v^\phi \hat e_\phi
$$
Along a curve with curve parameter ##s##, we can then consider how ##\vec v## varies along the curve by taking the derivative
$$
\frac{d\vec v}{ds} = \dot v^r \hat e_r + v^r \dot{\hat e}_r + \dot v^\phi \hat e_\phi + v^\phi \dot{\hat e}_\phi
$$
where the dot represents differentiation wrt ##s## and we have simply applied the chain rule. For constant basis vectors, such as the cartesian ones, the derivatives of the basis vectors vanish, but this is not the case in general. Instead, we can expand
$$
\dot{\hat e}_a = \frac{d\hat e_a}{ds} = \sum_b \dot x^b \partial_b \hat e_a
$$
where we know that ##\partial_b \hat e_a## is a vector and may be written as a linear combination of ##\hat e_r## and ##\hat e_\phi## with some coefficients ##\Gamma_{ab}^c##:
$$
\partial_b \hat e_a = \Gamma_{ba}^r \hat e_r + \Gamma_{ba}^\phi \hat e_\phi
$$
The ##\Gamma## coefficients describe precisely how the basis changes when you change position and are the Christoffel symbols, i.e., the connection coefficients of the Levi-Civita connection.
The change in the vector field ##\vec v## along the curve is then described by two separate parts, one describing how the components change along the curve and another that describes how the basis changes:
$$
\frac{d\vec v}{ds} = [\dot v^r \hat e_r + \dot v^\phi \hat e_\phi] + v^r (\dot r [\Gamma_{rr}^r \hat e_r + \Gamma_{rr}^\phi \hat e_\phi]+ \dot\phi [\Gamma_{\phi r}^r \hat e_r + \Gamma_{\phi r}^\phi \hat e_\phi]) + \ v^\phi \dot{\hat e}_\phi (\dot r [\Gamma_{r\phi}^r \hat e_r + \Gamma_{r\phi}^\phi \hat e_\phi]+ \dot\phi [\Gamma_{\phi \phi}^r \hat e_r + \Gamma_{\phi \phi}^\phi \hat e_\phi])
$$
Now that looks like quite a mouthful, but is significantly more compact with Einstein summation notation, where repeated indices are summed over their ranges:
$$
\frac{d\vec v}{ds} = \dot v^a \hat e_a + \dot x^a v^b \Gamma_{ab}^c \hat e_c
$$