  We try to understand the geometric properties of $n$-manifolds ($n\geq 2$)
with geometric structures modeled on $(\bR P^n, \PGL(n+1, \bR))$, i.e.,
$n$-manifolds with projectively flat torsion free affine connections. We define
the notion of $i$-convexity of such manifolds due to Carri\'ere for integers
$i$, $1 \leq i \leq n-1$, which are generalization of convexity. Given a real
projective $n$-manifold $M$, we show that the failure of an $(n-1)$-convexity
of $M$ implies an existence of a certain geometric object, $n$-crescent, in the
completion $\che M$ of the universal cover $\tilde M$ of $M$. We show that this
further implies the existence of a particular type of affine submanifold in $M$
and give a natural decomposition of $M$ into simpler real projective manifolds,
some of which are $(n-1)$-convex and others are affine, more specifically
concave affine. We feel that it is useful to have such decomposition
particularly in dimension three. Our result will later aid us to study the
geometric and topological properties of radiant affine 3-manifolds leading to
their classification. We get a consequence for affine Lie groups.
