  Stable fold maps are fundamental tools in a generalization of the theory of
Morse functions on smooth manifolds and its application to studies of
topological properties of smooth manifolds. Round fold maps were introduced as
stable fold maps with singular value sets, defined as the set consisting of all
the singular values, of concentric spheres by the author in 2013; for example,
some special generic maps on spheres are regarded as round fold maps whose
singular value sets are connected. Algebraic invariants such as homology and
homotopy groups of manifolds admitting round fold maps and more precisely, the
homeomorphism and diffeomorphism types of manifolds admitting such maps having
appropriate differential topological structures were studied. Moreover,
explicit round fold maps into the Eucidean space of dimension larger than $1$
are constructed on some fundamental manifolds such as manifolds having the
structures of bundles over the standard sphere of dimension equal to the
Euclidean space whose fibers are closed smooth manifolds and manifolds of
dimension not smaller than twice the dimension of the Euclidean space
represented as the connected sum of manifolds having the structures of bundles
over the standard sphere of dimension equal to the Euclidean space whose fibers
are diffeomorphic to standard spheres. In this paper, we construct new explicit
fold maps on some fundamental manifolds including the manifolds before by using
extended methods of ones used in the constructions before.
