  The electronic spectrum on the spherical surface of a topological insulator
reflects an active property of the helical surface state that stems from a
constraint on its spin on a curved surface. The induced effective vector
potential (spin connection) can be interpreted as an effective vector potential
associated with a fictitious magnetic monopole induced at the center of the
sphere. The strength of the induced magnetic monopole is found to be g=2pi,
-2pi, being the smallest finite (absolute) value compatible with the Dirac
quantization condition. We have established an explicit correspondence between
the bulk Hamiltonian and the effective Dirac operator on the curved spherical
surface. An explicit construction of the surface spinor wave functions implies
a rich spin texture possibly realized on the surface of topological insulator
nanoparticles. The electronic spectrum inferred by the obtained effective
surface Dirac theory, confirmed also by the bulk tight-binding calculation,
suggests a specific photo absorption/emission spectrum of such nanoparticles.
