  In this paper we establish the best constant $\widetilde A_{opt}(\bar{M})$
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.
