This is a survey on our recent works on bi-harmonic maps on CR-manifolds and foliated Riemannian manifolds, and also a research paper on bi-harmonic maps principal G-bundles. We will show, (1) for a complete strictly pseudoconvex CR manifold , every pseudo bi-harmonic isometric immersion? into a Riemannian manifold of non-positive curvature, with finite energy and finite bienergy, must be pseudo harmonic; (2) for a smooth foliated map of a complete, possibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold, of which transversal sectional curvature is non-positive, we will show that if it is transversally bi-harmonic map with the finite energy and finite bienergy, then it is transversally harmonic; (3) we will claim that the similar result holds for principal G-bundle over a Riemannian manifold of negative Ricci curvature.
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