  A geometric torsion dynamics leading to an effective curvature in a second
order formalism on a D4-brane is revisited with a renewed interest. We obtain
two effective $AdS_4$ brane geometries on a vacuum created pair of $(D{\bar
D})_3$-brane. One of them is shown to describe an AdS Schwarzschild spinning
black hole and the other is shown to describe a spinning black hole bound
state. It is argued that a D-instanton in a vacuum created anti D3-brane within
a pair may describe a quintessence. It may seen to incorporate a varying vacuum
energy density in a brane universe. We consider the effective curvature scalar
on $S^1\times S^1$ to analyze torsion-less geometries on a vacuum created pair
of $(D{\bar D})_2$-brane. The emergent $AdS_3$ brane is shown to describe a
Schwarzschild and a Reissner-Nordstrom (RN) geometries in presence of extra
dimension(s).
