= Solution
For a <principal bundle> $\pi:E\to B$, the vertical space is $V_e=\ker d\pi_e$. A <principal connection> specifies smooth complements $T_eE=H_e\oplus V_e$ with $(R_g)_*H_e=H_{eg}$. Equivalently its Lie-algebra-valued connection one-form $\mathcal A$ satisfies $\mathcal A(\xi_E)=\xi$ and $R_g^*\mathcal A=\operatorname{Ad}_{g^{-1}}\mathcal A$. Every base vector has a unique <horizontal lift> because $d\pi:H_e\to T_{\pi(e)}B$ is an isomorphism. A base curve lifts from a chosen initial fiber point by solving $\mathcal A(\dot e)=0$; this defines <parallel transport>. Closed curves can give nontrivial <holonomy>, and failure of horizontal fields to close under brackets is measured by the <curvature of a principal connection>.
For a smooth strictly convex <rigid body>, assume contact with the plane is unique and varies smoothly with orientation. Write a configuration as $(R,c_{\parallel})\in SO(3)\times\mathbb R^2$. Contact determines the reference point's height $h(R)$, so the full reference position is $c=(c_{\parallel},h(R))$. Let $r(R)$ be the spatial vector from that point to the contacting material point and let $\widehat\Omega=\dot R R^{-1}$ encode the spatial <angular velocity>. The instantaneous contact velocity is
$$
v_{\mathrm{contact}}=\dot c+\Omega\times r(R).
$$
The no-slip condition is its vanishing. Its vertical component follows already from differentiating contact: the normal component of the changing contact point along the body surface is zero, giving $\dot h=-(\Omega\times r)_z$. The two remaining equations are
$$
\dot c_{\parallel}=-(\Omega\times r(R))_{\parallel}.
$$
Translations of $c_{\parallel}$ define a free right $\mathbb R^2$ action, with quotient $SO(3)$. The <translational connection for a convex body rolling on a plane> is
$$
\boxed{\mathcal A=dc_{\parallel}+(\Omega\times r(R))_{\parallel}.}
$$
Here $\Omega$ denotes the vector-valued right <Maurer-Cartan form> on $SO(3)$. This one-form is invariant under translations and returns a translation vector on a vertical translation, so it satisfies both <principal connection> axioms. A prescribed orientation path has the unique rolling <horizontal lift>
$$
c_{\parallel}(t)=c_{\parallel}(0)-\int_0^t(\Omega(s)\times r(R(s)))_{\parallel}\,ds.
$$
This is a kinematic connection; inertia and gravity determine which of the allowed paths is dynamically realized.
For a sphere of radius $a$, $r=-a n$, where $n$ is the upward unit normal. The same no-slip condition gives $\dot c_{\parallel}=a\Omega\times n$. It leaves the component $\Omega\cdot n$ arbitrary. If one additionally forbids twisting about $n$, then $\Omega=n\times\dot c_{\parallel}/a$ and a plane path uniquely lifts to an orientation path by $\dot R=\widehat\Omega R$. This is the familiar $SO(3)$ <principal connection> over the plane; its horizontal lifts have noncommuting rotation generators and hence nonzero <curvature>. No-slip alone does not supply that latter two-dimensional horizontal distribution. Corners or nonsmooth changes of contact require piecewise treatment rather than the smooth bundle construction.
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