  Within the framework of Landau-de Gennes theory for nematic liquid crystals,
we study the temperature-induced isotropic-nematic phase transition on a
spherical shell. Below a critical temperature, a thin layer of nematic coating
a microscopic spherical particle exhibits non-uniform textures due to the
geometrical frustration. We find the exact value of critical threshold for the
temperature and determine exactly the nematic textures at the transition by
means of a weakly nonlinear analysis. The critical temperature is affected by
the extrinsic curvature of the sphere, and the nematic alignment is consistent
with the Poincar\'e-Hopf index theorem and experimental observations. The
stability analysis of the bifurcate textures at the isotropic-nematic
transition highlight that only the tetrahedral configuration is stable.
