We study the statistical mechanics and the equilibrium dynamics of a system of classical Heisenberg spins with frustrated interactions on a d -dimensional simple hypercubic lattice, in the limit of infinite dimensionality d -> infinity . In the analysis we consider a class of models in which the matrix of exchange constants is a linear combination of powers of the adjacency matrix. This choice leads to a special property: the Fourier transform of the exchange coupling J ( k ) presents a ( d - 1) -dimensional surface of degenerate maxima in momentum space. Using the cavity method, we find that the statistical mechanics of the system presents for d -> infinity a paramagnetic solution which remains locally stable at all temperatures down to T = 0. To investigate whether the system undergoes a glass transition we study its dynamical properties assuming a purely dissipative Langevin equation, and mapping the system to an effective single -spin problem subject to a colored Gaussian noise. The conditions under which a glass transition occurs are discussed including the possibility of a local anisotropy and a simple type of anisotropic exchange. The general results are applied explicitly to a simple model, equivalent to the isotropic Heisenberg antiferromagnet on the d -dimensional face -centered -cubic lattice with first- and secondnearest -neighbor interactions tuned to the point J 1 = 2 J 2 . In this model, we find a dynamical glass transition at a temperature T g separating a high -temperature liquid phase and a low -temperature vitrified phase. At the dynamical transition, the Edwards -Anderson order parameter presents a jump demonstrating a first -order phase transition.