  In this paper we consider new geometric flow equations, called D-flow, which
describe the variation of space-time geometries under the change of the number
of dimensions. The D-flow is originating from the non-trivial dependence of the
volume of space-time manifolds on the number of space-time dimensions and it is
driven by certain curvature invariants. We will work out specific examples of
D-flow equations and their solutions for the case of D-dimensional spheres. The
discussion of the paper is motivated from recent swampland considerations,
where the number $D$ of space-time dimensions is treated as a new swampland
parameter.
