  The theory described by the sum of the Einstein-Hilbert action and the action
of conformal scalar field possesses the duality symmetry which includes some
special conformal transformation of the metric, and also inversion of scalar
field and of the gravitational constant. In the present paper the conformal
duality is generalized for arbitrary space-time dimension $n \neq 2$ and for
the general sigma-model type conformal scalar theory. We also consider to apply
the conformal duality for the investigation of quantum gravity in the strong
curvature regime. The trace of the first coefficient of the Schwinger-DeWitt
expansion is derived and it's dependence on the gauge fixing condition is
considered. After that we discuss the way to extract the gauge-fixing
independent result and also it's possible physical applications.
