You must be thinking of Lang's conjecture which predicts that a smooth projective variety is (Brody) hyperbolic if and only if all of its irreducible subvarieties are of general type.

This is still not known in general but there are many special cases that are known. A good example is McQuillan's theorem - a smooth surface of general type which satisfies $c_{1}^{2} > c_{2}$ and does not contain any rational or elliptic curves is Brody hyperbolic.