Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 57 / 4

A smooth principal bundle with structure Lie group $G$ can be specified by a locally trivial map $\pi :E\to B$ whose fiber coordinate changes have the form $(x,g{)}_i\mapsto (x,h_{ji}(x)g{)}_j$, with the usual transition functions of a principal bundle cocycle on triple overlaps. In each chart define $(x,g)\cdot k=(x,gk)$. On overlaps, $h_{ji}(x)(gk)=(h_{ji}(x)g)k$, so these local right actions agree. They therefore give a global smooth right group action, free and transitive on every fiber. Equivalently, this action together with equivariant local triviality is the definition of a principal bundle.

Use five-dimensional Minkowski spacetime with ambient metric $\eta =\mathrm{diag}(-1,1,1,1,1)$ and realize unit-radius de Sitter spacetime as $\eta (x,x)=1$. Its tangent space is $x^{\perp }$, of Lorentzian signature $(3,1)$. An oriented pseudo-orthonormal frame $(e_0,e_1,e_2,e_3)$ of $x^{\perp }$, together with the last column $x$, is precisely a matrixConversely the last column of such a matrix lies on the hyperboloid, and its other columns give that frame. ThusRight multiplication by $\mathrm{diag}(h,1)$ fixes $x$ and changes the tangent frame by $h$. It is free and transitive on frames over $x$, proving the principal bundle description. Smooth local choices of frames provide its local trivializations. The dimensions are $10=4+6$.

This is the oriented frame bundle of de Sitter spacetime. Allowing orientation reversal gives $O(4,1)$ and structure group $O(3,1)$; imposing a time orientation as well selects $S{O}_0(4,1)$ and $S{O}_0(3,1)$. For radius $a$, take the last column $x/a$.

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.