  A closed symmetric differential of the 1st kind is a differential that
locally is the product of closed holomorphic 1-forms. We show that closed
symmetric 2-differentials of the 1st kind on a projective manifold $X$ come
from maps of $X$ to cyclic or dihedral quotients of Abelian varieties and that
their presence implies that the fundamental group of $X$ (case of rank 2) or of
the complement $X\setminus E$ of a divisor $E$ with negative properties (case
of rank 1) contains subgroup of finite index with infinite abelianization.
Other results include: i) the identification of the differential operator
characterizing closed symmetric 2-differentials on surfaces (which provides in
this case a connection to flat Riemannian metrics) and ii) projective manifolds
$X$ having symmetric 2-differentials $w$ that are the product of two closed
meromorphic 1-forms are irregular, in fact if $w$ is not of the 1st kind (which
can happen), then $X$ has a fibration $f:X \to C$ over a curve of genus $\ge
1$.
