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We give a negative answer to a question posed by Severi in 1951, whether theAbelian Varieties are the only projective manifolds with trivial Chern classes.  By Yau' s celebrated result, compact K\"ahler manifolds with trivial Chernclasses must be flat, that is, they belong to the class of HyperellipticManifolds (quotients $T/G$ of a complex torus $T$ by the free action of afinite group $G$).  We exhibit simple examples of projective Hyperelliptic Manifolds which arenot Abelian varieties and whose Chern classes are zero not only in integralcohomology, but also in the Chow ring.  We prove moreover that the Bagnera-de Franchis manifolds (quotients $T/G$ asabove but where the group $ G$ is cyclic) have topologically trivial tangentbundle.  Our results naturally lead to the question of classifying all compactK\"ahler manifolds with topologically trivial tangent bundle, and all thecounterexamples to Severi's question.

Manifolds with trivial Chern classes I: Hyperelliptic Manifolds and a question by Severi