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Along a smooth curve $\gamma :I\to M$, a horizontal lift of a linear connection on a vector bundle has fibre equation $v^{\prime }=-\Gamma (\dot{\gamma })v$. On each compact time interval the coefficient matrix is bounded, and the Gronwall inequality prevents finite-time blowup. A finite cover by vector bundle trivializations and ODE uniqueness patch local solutions. Exhausting the open interval gives a unique lift on all of $I$. Its endpoint map is linear, and reversal of the path gives its inverse; no completeness of the base manifold is needed.

A horizontal lift is determined by the horizontal distribution of a principal connection or the corresponding connection on a vector bundle. A horizontal lift of a base curve is a curve in the bundle projecting to it and tangent to the horizontal distribution. In a local frame its fiber coordinate solves a linear ordinary differential equation.

For a principal bundle $\pi :E\to B$, the vertical space is $V_e=\ker d{\pi }_e$. A principal connection specifies smooth complements $T_{e}E=H_e\oplus V_e$ with $(R_g{)}_{\ast }{H}_e=H_{eg}$. Equivalently its Lie-algebra-valued connection one-form $\mathcal{A}$ satisfies $\mathcal{A}({\xi }_E)=\xi$ and $R_g^{\ast }\mathcal{A}={\mathrm{Ad}}_{g^{-1}}\mathcal{A}$. 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 $\hat{\Omega }=\dot{R}{R}^{-1}$ encode the spatial angular velocity. The instantaneous contact velocity isThe 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 areTranslations 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 isHere $\Omega$ 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 liftThis is a kinematic connection; inertia and gravity determine which of the allowed paths is dynamically realized.

For a sphere of radius $a$, $r=-an$, 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 $\dot{R}=\hat{\Omega }R$. 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.

Let $E\to M$ have rank $r$. A vector bundle covariant derivative is a real-linear operatorPairing its one-form slot with a vector field $X$ gives ${\nabla }_{X}s$, with ${\nabla }_{fX}s=f{\nabla }_{X}s$ and ${\nabla }_X(fs)=X(f)s+f{\nabla }_{X}s$.

In the horizontal description, a connection on a vector bundle $A$ is a smooth splitting $T_{a}E=V_{a}E\oplus H_{a}E$, where $V_{a}E=\ker (d\pi {)}_a$, and the horizontal choice is compatible with fibre addition and scalar multiplication. This linearity is essential; an arbitrary nonlinear horizontal splitting is not a vector-bundle connection. Identify a vertical vector at $a\in E_x$ with the element of $E_x$ obtained by differentiating the curve $a+tv$. Let $K_A:T_{a}E\to E_x$ be vertical projection followed by this identification. The induced vector bundle covariant derivative isIn a local frame of a vector bundle $e=(e_1,\dots ,e_r)$, a linear horizontal splitting has the formfor a matrix-valued connection one-form $\Gamma$. Thus for $s=ev$, ${\nabla }^{A}s=e(dv+\Gamma v)$, which directly verifies the Leibniz rule.

Conversely, the Leibniz rule makes $\nabla$ local: if a section vanishes near a point, multiply it by a smooth cutoff function supported in that neighborhood and equal to one near the point to see that its derivative vanishes there too. Local sections can therefore be differentiated by any extension agreeing near the point. Given $\nabla$, define its local connection one-form by $\nabla e_b={\sum }_a{e}_a{\Gamma }_b^a$. The Leibniz rule gives $\nabla (ev)=e(dv+\Gamma v)$. Under $e^{\prime }=eg$, $v^{\prime }=g^{-1}v$, one hasThe tangent fibre coordinate transforms by $w^{\prime }=g^{-1}w-g^{-1}(dg)(\xi )g^{-1}v$, so $w=-\Gamma (\xi )v$ transforms precisely to $w^{\prime }=-{\Gamma }^{\prime }(\xi )v^{\prime }$. Consequently these local horizontal subspaces of a vector bundle connection glue to a smooth linear splitting inducing the given $\nabla$. Its local formula also shows uniqueness. This proves the horizontal connection associated to a covariant derivative construction.

A horizontal lift $\tilde{\gamma }$ of $\gamma$ is a smooth curve in $E$ with $\pi \circ \tilde{\gamma }=\gamma$ and $\dot{\tilde{\gamma }}(t)\in H_{\tilde{\gamma }(t)}E$. In a local frame write $\tilde{\gamma }(t)=e(\gamma (t))v(t)$. Horizontality is exactly the linear ordinary differential equationWe use existence and uniqueness for an ODE with smooth coefficients, and the continuation theorem that a solution extends across a finite interval when it remains bounded inside the coordinate domain. For a linear system $\dot{v}=-C(t)v$ with continuous coefficients bounded by $L$ on a compact time interval, the Gronwall inequality bounds $|v(t)|$ by its initial norm times $e^{L|t-t_0|}$, preventing finite-time fibre escape. Cover a compact subinterval of $I$ by finitely many bundle charts and partition it so each piece lies in one chart. Solve successively, changing fibre frames at the partition points. Coordinate invariance of the horizontal equation and uniqueness make the pieces agree. Exhausting $I$ by compact intervals containing $0$ produces a unique horizontal lift on all of $I$ for every prescribed $a_0$. This is global continuation of linear parallel transport.

The lift equation is linear in its initial value. Hence the endpoint map $P_{\gamma }:E_{\gamma (0)}\to E_{\gamma (1)}$ is linear. Solving the same equation backwards, or transporting along $t\mapsto \gamma (1-t)$, gives its inverse, soThis is parallel transport. For two parallel lifts $s_1(t),s_2(t)$, the assumed metric connection identity extends along the embedded curve and becomesHere $D_t$ is the covariant derivative along a curve. The permitted extension of sections and the tangent vector along the embedded curve justifies applying the bundle identity; equivalently it follows in each local frame. Thereforeso parallel transport is an isometry. This is the principle that parallel transport preserves a fibre metric.

A horizontal lift of $\gamma$ from $p$ is a curve $\tilde{\gamma }$ in $E$ projecting to $\gamma$, starting at $p$, and tangent to the horizontal distribution of the connection. In a local frame write $\tilde{\gamma }(t)=(\gamma (t),v(t))$. Horizontality is the linear ordinary differential equationwhose initial-value theorem gives local existence and uniqueness; successive trivializations continue the lift.

A geodesic satisfies ${\nabla }_{\dot{\gamma }}\dot{\gamma }=0$, andfor the geodesic with initial velocity $v$. Since $d({\exp }_p{)}_0=1$, the inverse function theorem makes ${\exp }_p$ a diffeomorphism near zero; its inverse gives normal coordinates. A geodesic sphere is ${\exp }_p(\{v:|v|=r\})$ inside such a normal neighborhood.

The Gauss lemma statesFor the variation ${\gamma }_s(t)={\exp }_p(t(v+sw))$, let $J={\partial }_s{\gamma }_s{|}_{s=0}$. The coordinate vector fields commute, so metric compatibility and constant geodesic speed giveSince $J(0)=0$, evaluation at $t=1$ proves the formula. In particular radial and spherical directions are orthogonal.