  The global one-dimensional quantum gravity is the model of quantum gravity
which arises from the global one-dimensionality conjecture within quantum
general relativity, first considered by the author in 2010 and then in 2012. In
this model the global dimension is a determinant of a metric of
three-dimensional space embedded into an enveloping Lorentizan four-dimensional
spacetime. In 2012, it has already been presented by the author that this model
can be extended to any Lorentzian D + 1-dimensional spacetime, where D is a
dimension of space, and resulting in the global one-dimensional model of a
higher dimensional quantum gravity.
  The purely quantum-mechanical part of this model is a minimal effective model
within the quantum geometrodynamics, introduced by J.A. Wheeler and B.S. DeWitt
in the 1960s, but the effective potential is manifestly different from the one
considered by Wheeler & DeWitt. Moreover, in our model the wave functionals
solving the quantum gravity are one-variable smooth functions and, therefore,
the troublesome mathematical technique of the Feynman functional integration
present in the Hawking formulation of quantum gravity, is absent is this model,
what makes it a mathematically consistent theory of quantum gravitation.
  In this paper, we discuss in some detail a certain part of the global
one-dimensional model already proposed in 2010, and then developed in 2012. The
generalized functional expansion of the effective potential and the residual
approximation, which describe the embedded spaces which are maximally symmetric
three-dimensional Einstein's manifolds, whose lead to the Newton-Coulomb type
potential in the quantum gravity model, are considered. Furthermore, scenarios
related to few selected specific forms of the effective potential are suggested
as physically interesting and discussed.
