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Motivated by a general question addressed by Mario Baldassarri in 1956, wediscuss the Pseudo-Abelian Varieties introduced by Roth, and we introduce afirst new notion, of Manifolds Isogenous to a k-Torus Product: the latter havethe last k Chern classes trivial in rational cohomology and vanishing Chernnumbers.  We show that in dimension 2 the latter class is the correct substitute forsome incorrect assertions by Enriques, Dantoni, Roth and Baldassarri: these arethe surfaces with $K_X$ nef and $c_2(X)=0 \in H^4(X, \mathbb{Z})$.  We observe in the last section, using a construction by Chad Schoen, that asimilar picture does not hold in such a simple way in higher dimension.  We discuss then, as a class of solutions to Baldassarri's question,(manifolds isogenous to) projective (respectively: K\"ahler) manifolds whosetangent bundle (resp. cotangent bundle) has a trivial subbundle.  The former class of `partially framed' projective manifolds, that is, whosetangent bundle has a trivial subbundle, consists,  in the case where $K_X$ is nef, of the Pseudo-Abelian varieties of Roth;while the latter class of `partially co-framed' projective manifolds is not yetunderstood; we can however state some new results and formulate open questionsand conjectures.  In the course of the paper we address also the general case of compactcomplex Manifolds, introducing the new notions of suspensions overparallelizable Manifolds, and of twisted hyperelliptic Manifolds, and describethe known results under the K\"ahler assumption.

Manifolds with trivial Chern classes II: Manifolds Isogenous to a Torus Product, coframed Manifolds and a question by Baldassarri