  In this paper, we aim to obtain some results under the condition that the
dual of a module has finite Gorenstein dimension. In this direction we derive
results involving vanishing of Ext as well as the freeness or totally
reflexivity of modules. For instance, we provide a generalization of a
celebrated theorem by Evans and Griffith, obtain criteria for the totally
reflexivity of modules over Cohen-Macaulay rings as well as of locally totally
modules reflexive on the punctured spectrum, and recover a result by Araya.
Moreover, we prove that the Auslander-Reiten conjecture holds true for all
finitely generated modules $M$ over a Noetherian local ring $R$ such that
$\operatorname{G-dim}_R(\operatorname{Hom}_R(M,R))<\infty$ and
$\operatorname{pd}_R(\operatorname{Hom}_R(M,M))<\infty$. Additionally, we
derive Gorenstein criteria under the condition that the dual of certain modules
is of finite Gorenstein dimension. Furthermore, we explore some applications in
the theory of the modules of K\"ahler differentials of order $n\geq 1$,
specifically concerning the $k$-torsionfreeness of these modules and the
Herzog-Vasconcelos's conjecture.
