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We further develop on the study of the conditions for the existence of locally stable non-supersymmetric vacua with vanishing cosmological constant in supergravity models involving only chiral super fields. Starting from the two necessary conditions for flatness and stability derived in a previous paper (which involve the Kahler metric and its Riemann tensor contracted with the supersymmetry breaking auxiliary fields) we show that the implications of these constraints can be worked out exactly not only for factorizable scalar manifolds, but also for symmetric coset manifolds. In both cases, the conditions imply a strong restriction on the Kahler geometry and constrain the vector of auxiliary fields de ning the Goldstino direction to lie in a certain cone. We then apply these results to the various homogeneous coset manifolds spanned by the moduli and untwisted matter fields arising in string compactications, and discuss their implications. Finally, we also discuss what can be said for completely arbitrary scalar manifolds, and derive in this more general case some explicit but weaker restrictions on the Kahler geometry.