= Oriented frame bundle of de Sitter spacetime
{title2=$SO(4,1)\longrightarrow \mathrm{dS}_4$}
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.
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