We study harmonic maps from strictly pseudoconvex CR manifolds into Riemannian manifolds of nonpositive curvature. Some CR analogues of the Corlette and Siu-Sampson formulas are obtained using tools of Spinorial Geometry (Dirac bundles and Dirac operators). As a main application, we obtain results about the curvature of strictly pseudoconvex CR manifolds. In particular, a rigidity theorem for Sasakian manifolds is proved.
We derive Mok-Siu-Yeung type formulas for horizontal maps from compact contact locally sub-symmetric spaces into strictly pseudoconvex CR manifolds and we obtain some rigidity theorems for the horizontal pseudoharmonic maps.
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