We investigate the structure of complete Riemannian or Kaehler manifolds
that admit a weighted Poincare inequality and whose Ricci curvature tensor
is bounded from below in terms of the weight function. This subject has
been intensively studied recently by professors P. Li and J. Wang. We will
recall some of their fundamental results and discuss new ideas on the
problem.

In 1986 A. Ancona showed, using the Koebe one-quarter Theorem, that for a simply-connected planar domain the constant in the Hardy inequality with the distance to the boundary is greater than or equal to 1/16. We consider classes of domains for which there is a stronger version of the Koebe Theorem. This implies better estimates for the constant appearing in the Hardy inequality.

We will talk about transversality and intersection of submanifolds. Then we will define intersection forms of 4-manifolds and 6-manifolds and give some examples.