  A new variable in the Riemannian geometry is introduced by the tetrad and the
Ricci's coefficients of rotation, the characters of curve of the Riemannian
geometry are determined completely by the new variable; for general relativity,
all the Einstein-Hilbert action, the Einstein equation in general relativity
and the Dirac equation in curved spacetime can be expressed by the new
variable, and, further, as well the action of the theory on the interaction of
gravitational, electromagnetic and spinor field (TGESF). All the characters of
transformations of the new variable, the Einstein-Hilbert action and the action
of TGESF under the general coordinate transformations and the local Lorentz
transformations are discussed, respectively. After presenting the method of
introduction of gravitational field in terms of the principle of gauge
invariance based on the Dirac equation, and the ten constraint conditions for
the tetrad are given, the vacuum-vacuum transition amplitude with the
Faddeev-Popov ghost and the terms of external sources of the pure gravitational
field is presented; finally, as well that of TGESF.
