The variational formulas of the volume function in the equiaffine geometry

Abstract

We obtain the first and second variational formulas for the volume function over the space of all equiaffine immersions. By using the first variational formula, we define some kinds of volume functions over the space of all non-degenerate immersions and obtain the Euler-Lagrange equation for those volume functions. Also, in the case where the ambient space is flat, we obtain the second variational formulas for special ones of those volume functions.