A positive cosmological constant as a geometrical artifact

We revisit the classical mechanism to produce $d-$dimensional spacetimes via Kaluza-Klein compactification. We made the assumption that the $\left( d+1\right) -$dimensional bulk geometry $\mathcal{M}$ admits a Homothetic Vector Field (HVF) $\mathbf{H}$ relaxing the existence of a zero i.e. $\mathbf{H}\neq 0$ holds on $\mathcal{M}$ and the homothetic bivector $F_{ab}(\mathbf{H})=0$. Under these circumstances we identify the origin of a positive cosmological constant in the $d-$dimensional spacetime as the homothetic factor representing a geometrical artifact of late time state of the $\left( d+1\right) -$dimensional bulk spacetime.