  In the description of the extrinsic geometry of the string world sheet
regarded as a conformal immersion of a 2-d surface in $R^3$, it was previously
shown that, restricting to surfaces with $h\surd{g}\ =\ 1$, where $h$ is the
mean scalar curvature and $g$ is the determinant of the induced metric on the
surface, leads to Virasaro symmetry. An explicit form of the effective action
on such surfaces is constructed in this article which is the extrinsic
curvature analog of the WZNW action. This action turns out to be the gauge
invariant combination of the actions encountered in 2-d intrinsic gravity
theory in light-cone gauge and the geometric action appearing in the
quantization of the Virasaro group. This action, besides exhibiting Virasaro
symmetry in $z$-sector, has $SL(2,C)$ conserved currents in the
$\bar{z}$-sector. This allows us to quantize this theory in the
$\bar{z}$-sector along the lines of the WZNW model. The quantum theory on
$h\surd{g}\ =\ 1$ surfaces in $ R^3$ is shown to be in the same universality
class as the intrinsic 2-d gravity theory.
