  In this contribution we investigate the Schr\"ordinger equation associated to
the Laplacian on the sphere in the form of sharp Strichartz estimates. We will
provided simple proofs for our main theorems using purely the $L^2\rightarrow
L^p$ spectral estimates for the operator norm of the spectral projections
(associated to the spherical harmonics) proved in [8]. A sharp index of
regularity is established for the initial data in spheres of arbitrary
dimension $d\geq 2$.
