  We study the restriction of the Fourier transform to quadratic surfaces in
vector spaces over finite fields. In two dimensions, we obtain the sharp result
by considering the sums of arbitrary two elements in the subset of quadratic
surfaces on two dimensional vector spaces over finite fields. For higher
dimensions, we estimate the decay of the Fourier transform of the
characteristic functions on quadratic surfaces so that we obtain the
Tomas-Stein exponent. Using incidence theorems, we also study the extension
theorems in the restricted settings to sizes of sets in quadratic surfaces.
Estimates for Gauss and Kloosterman sums and their variants play an important
role.
