  Rotational mixing, the key process in stellar evolution, transports angular
momentum and chemical elements in stellar radiative zones. In the past two
decades, an emphasis has been placed on the turbulent transport induced by the
vertical shear instability. However, instabilities arising from horizontal
shear and the strength of the anisotropic turbulent transport that they may
trigger remain relatively unexplored. This paper investigates the combined
effects of stable stratification, rotation, and thermal diffusion on the
horizontal shear instabilities in the context of stellar radiative zones. The
eigenvalue problem describing linear instabilities of a flow with a
hyperbolic-tangent horizontal shear profile was solved numerically for a wide
range of parameters. As a first step, we consider a polar $f$-plane where the
gravity and rotation vector are aligned. Two types of instabilities are
identified: the inflectional and inertial instabilities. The inflectional
instability that arises from the inflection point is the most unstable when at
a zero vertical wavenumber and a finite wavenumber in the streamwise direction
along the imposed-flow direction. The three-dimensional inflectional
instability is destabilized by stratification, while it is stabilized by
thermal diffusion. The inertial instability is rotationally driven, and a WKBJ
analysis reveals that its growth rate reaches the maximum $\sqrt{f(1-f)}$ in
the inviscid limit as the vertical wavenumber goes to infinity, where $f$ is
the dimensionless Coriolis parameter. The inertial instability for a finite
vertical wavenumber is stabilized as the stratification increases, whereas it
is destabilized by the thermal diffusion. Furthermore, we found a
self-similarity in both the inflectional and inertial instabilities based on
the rescaled parameter $PeN^2$ with the P\'{e}clet number $Pe$ and the
Brunt-V\"{a}is\"{a}l\"{a} frequency $N$.
