 Solution

A principal bundle with structural Lie group $G$ is locally a product $U\times G$, with changes of trivialization of the form $(x,g)\mapsto (x,t(x)g)$. Define a right action locally by $(x,g)\cdot h=(x,gh)$. It is independent of trivialization because $t(x)(gh)=(t(x)g)h$. It is smooth and obeys the group-action law. Right multiplication in $G$ is free and transitive, so the global action is free and each fiber is a single orbit. Equivalently these properties and equivariant local triviality can be included in the definition of a principal bundle.

For a representation $\rho :G\to GL(V)$, form the associated vector bundleProjection sends $[p,v]$ to the base point of $p$. A local section $s(x)$ gives coordinates $[s(x),v]$ and hence a local product $U\times V$; on overlaps the principal transition functions act through $\rho$. Because these maps are linear in $v$, the fibers have well-defined vector-space structures.

For an $n$-dimensional manifold, its frame bundle has points $u:{\mathbb{R}}^n\to T_{x}M$ that are linear isomorphisms. Right action is composition $u\cdot A=u\circ A$ for $A\in GL(n,\mathbb{R})$. Coordinate frames provide local trivializations. Its associated bundle for the defining representation is the tangent bundle, by the well-defined map $[u,v]\mapsto u(v)$. A metric restricts the frames to linear isometries and reduces the group to $O(r,s)$; choosing orientation gives the corresponding special-orthogonal reduction.

In Minkowski spacetime choose a reference oriented pseudo-orthonormal frame at the origin. Every such frame is specified by its base point $a\in {\mathbb{R}}^{3,1}$ and a Lorentz transformation $L\in SO(3,1)$. The Poincaré group has multiplicationand its right Lorentz action is $(a,L)\cdot h=(a,Lh)$. Henceis the oriented orthonormal frame bundle of Minkowski spacetime. The same argument with $O(3,1)$ gives the full unoriented bundle. If time orientation is fixed, use the orthochronous components throughout; the use of $SO$ in the printed quotient is an orientation reduction.

For four-dimensional de Sitter spacetime, use the spacelike hyperboloid $x\cdot x={\ell }^2$ in ${\mathbb{R}}^{4,1}$. An element of $SO(4,1)$ maps a fixed unit spacelike vector to $x/\ell$, and its other columns give an oriented pseudo-orthonormal tangent frame. The stabilizer of that unit vector is $SO(3,1)$. This identifies the oriented frame bundle of de Sitter spacetime and its projection:The right stabilizer action changes the tangent frame without changing $x$.

For Anti-de Sitter spacetime, take $x\cdot x=-{\ell }^2$ in ${\mathbb{R}}^{3,2}$. A unit timelike normal has orthogonal complement of signature $(3,1)$, so its stabilizer is again $SO(3,1)$. Appending this normal to a tangent frame realizes the oriented orthonormal frame bundle of anti-de Sitter spacetime asConnected components or orientation/time-orientation restrictions must be chosen consistently. This quotient is the anti-de Sitter quadric, which has a periodic timelike coordinate. If “anti-de Sitter spacetime” denotes its universal cover, pull the frame bundle back to that cover, rather than identify the covered spacetime with the unmodified quotient.