  After establishing the positivity constraint and spin content of the theory
for gravitons interacting with a necessarily, and \textit{a priori},
\textit{non}-conserved external energy-momentum tensor, the expectation value
formalism of the theory is developed at \textit{finite} temperature in the
functional \textit{differential} treatment of quantum field theory. The
necessity of having, \textit{a priori}, a non-conserved external
energy-momentum tensor is an obvious technical requirement so that its
respective ten components may be varied \textit{independently} in order to
generate expectation values and non-linearities in the theory. The covariance
of the \textit{induced} Riemann curvature tensor, in the initial vacuum, is
established even for the quantization in a gauge corresponding only to two
physical states of the gravitons as established above. As an application, the
\textit{induced} correction to the metric and the underlying geometry is
investigated due to a closed string arising from the Nambu action as a solution
of a circularly oscillating string as, perhaps, the simplest generalization of
a limiting point-like object. Finally it is discussed on why the geometry of
spacetime may, in general, depend on temperature due to radiative corrections
and its physical significance is emphasized.
