In this paper we establish the best constant $\widetilde
A_{opt}(\overline{M})$ for the Trace Nash inequality on a $n-$dimensional
compact Riemannian manifold in the presence of symmetries, which is an
improvement over the classical case due to the symmetries which arise and
reflect the geometry of manifold. This is particularly true when the data of
the problem is invariant under the action of an arbitrary compact subgroup $G$
of the isometry group $Is(M,g)$, where all the orbits have infinite cardinal.