Two invariants, called the $c_1$-sectional geometric genus and
the $\co(1)$-sectional geometric genus, are introduced for
generalized polarized manifolds, i.e.,
pairs of a projective manifold and an ample vector bundle on it.
They are generalizations of the $c_1$-sectional genus
and the $\co(1)$-sectional genus introduced by T.~Fujita
for generalized polarized manifolds.
They are also generalizations of the sectional geometric genus
introduced by Y.~Fukuma for polarized manifolds.
We classify generalized polarized manifolds of the smallest
$c_1$ or $\co(1)$-sectional geometric genus
under the spannedness condition for ample vector bundles.