  The control of the spin degree of freedom is at the heart of spintronics,
which can potentially be achieved by spin-orbit coupling or band topological
effects. In this paper, we explore another potential controlled mechanism under
debate: the spin-deformation coupling (SDC) - the coupling between intrinsic or
extrinsic geometrical deformations and the spin degree of freedom. We focus on
polar-deformed thin films or two-dimensional compounds, where the Rashba
spin-orbit coupling (SOC) is considered as an $SU(2)$ non-Abelian gauge field.
We demonstrate that the dynamics between surface and normal electronic degrees
of freedom can be properly decoupled using the thin-layer approach by
performing a suitable gauge transformation, as introduced in the context of
many-body correlated systems. Our work leads to three significant results: (i)
gauge invariance implies that the spin is uncoupled from the surface's
extrinsic geometry, challenging the common consensus; (ii) the Rashba SOC on a
curved surface can be included as an $SU(2)$ non-Abelian gauge field in
curvilinear coordinates; and (iii) we identify a previously unnoticed scalar
geometrical potential dependent on the Rashba SOC strength. This scalar
potential, independent of spin, represents the residual effect remaining after
decoupling the normal component of the non-Abelian gauge field. The outcomes of
our work open novel pathways for exploring the manipulation of spin degrees of
freedom through the use of the SDC.
