The first initial boundary value problem for Hessian equations of parabolic type on Riemannian manifolds

In this paper, we are concerned with the first initial boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. As usual, the establishment of the a priori C^2 estimates is our main part. Based on these estimates, the existence of classical solutions is proved under conditions which are nearly optimal.