Orthonormal frame bundle ($O(M,g)$)

For a metric of signature $(r,s)$, the orthonormal frame bundle consists of all linear isometries $u:{\mathbb{R}}^{r,s}\to T_{x}M$. Right composition by $O(r,s)$ makes it a principal bundle. Its standard associated vector bundle is the tangent bundle, via $[u,v]\mapsto u(v)$. A global orthonormal frame is a global section and therefore trivializes this principal bundle. A Lie group with a left-invariant metric has such a frame by left translating an orthonormal basis at the identity.

Represent the Anti-de Sitter spacetime quadric by $x\cdot x=-{\ell }^2$ in ${\mathbb{R}}^{3,2}$. Appending its timelike unit normal $x/\ell$ to an oriented tangent pseudo-orthonormal frame identifies the frame bundle with $SO(3,2)$. The stabilizer of that normal is $SO(3,1)$, so projection realizes $SO(3,2)/SO(3,1)$ as the base of a principal bundle. For the commonly used universal cover of the anti-de Sitter quadric, pull this bundle back along the covering map; the quadric itself has a periodic timelike coordinate.

An oriented pseudo-orthonormal frame in Minkowski spacetime consists of its base point $a$ and a Lorentz matrix $L$ acting on a fixed reference frame. The pairs $(a,L)$ form the Poincaré group, and right multiplication by $(0,h)$ sends the frame to $(a,Lh)$ while fixing its base point. Thus the displayed projection is a principal bundle with group $SO(3,1)$. The full frame bundle uses $O(3,1)$; a specified time orientation restricts to the corresponding connected component.

Represent unit-radius de Sitter spacetime by the spacelike unit hyperboloid in five-dimensional Minkowski spacetime. Appending its position vector as the last column of an oriented pseudo-orthonormal frame produces a matrix in $SO(4,1)$. Projection to that last column identifies $SO(4,1)$ with the oriented orthonormal frame bundle, a principal bundle with group $SO(3,1)$. The full bundle allowing both orientations is $O(4,1)$ with group $O(3,1)$. Requiring a time orientation as well selects the corresponding identity components.

In global coordinates the de Sitter spacetime metric is $-d{\tau }^2+{\cosh }^2\tau g_{S^3}$. A global frame of a vector bundle orthonormal for the round metric $Y_1,Y_2,Y_3$ on $S^3$ gives the displayed global pseudo-orthonormal frame. For example, identify $S^3$ with the unit quaternions and use $qi,qj,qk$. Choosing the spatial orientation and the future direction consistently gives a section of a fiber bundle for the oriented, time-oriented frame bundle. The principal bundle trivialization by a global section then proves that bundle is trivial.