On Rigidity and the Albanese Variety for Parallelizable Manifolds

We study the rigidity questions and the Albanese Variety for Complex Parallelizable Manifolds. Both are related to the study of the cohomology group $H^1(X,\mathcal O)$. In particular we show that a compact complex parallelizable manifold is rigid iff $b_1(X)=0$ iff Alb$(X)=\{e\}$ iff $H^1(X,\mathcal O)=0$.