We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for the limiting case of the estimates. MSC(2000): 53C25, 53C27, 58B20

We consider the existence and regularity of weakly polyharmonic almost complex structures on a compact Hermitian manifold $$M^{2m}$$ . Such objects satisfy elliptic system $$[\varDelta ^m J, J]=0$$ weakly. prove general theorem for semilinear systems in critical dimensions (with growth nonlinearities), which includes dimension four six.