  This thesis applies the Kerr-Schild and the Weyl double copy formalisms to
study various concepts in the physics literature. First we apply both the
Kerr-Schild and the Weyl double copy to solution generating transformations in
General Relativity, where we identify Ehlers transformation as the double copy
of electromagnetic duality transformation. Secondly, as a spin-off of the Weyl
double copy, we use gauge fields defined on curved spacetimes to construct the
Weyl tensor and study a host of solution of Einstein's equations. This study
provides a test of the non-triviality of the double copy formalism.
  The second half of the thesis deals with mathematical concepts of physical
relevance. First we apply the Kerr-Schild double copy to the concept of
holonomy groups of Riemannian manifolds. We find that the single copy of the
Riemannian holonomy operator, which we dub SCH, to be a similar operator
constructed from the single copy gauge-field curvature. This is followed by a
study of this single copy operator on different solutions of Einstein equations
and their respective single copies, where we find that the holonomy and SCH
groups differ for the Taub-NUT metric, while both reducing to $\text{SU}(2)$
for self-dual solutions. Lastly, we apply the Kerr-Schild double copy to the
Ricci flow equation, interpreted as the beta function of the closed string, and
obtain the Yang-Mills flow equation, which is physically interpreted as the
beta function of the open string coupled to a gauge field.
