Manchester Geometry Seminar 2009/2010

Let M be a compact oriented smooth Riemannian manifold of dimension n without boundary or with boundary and we suppose G is a torus acting by isometries on M. Let X_M be the associated vector field of X on M where X is in the Lie algebra. One defines Witten 's inhomogeneous operator d_XM=d+ι_XM: Ω_G^ev/odd →Ω_G^odd/ev, where Ω_G^ev/odd is the space of invariant forms of even or odd degree. We define the operator δ_XM = (-1)^n(k+1)+1∗d_XM∗.

Then we present the following:

1. The relevant version of the invariant Hodge decomposition theory in terms of the operators d_XM and δ_XM when there is no boundary of M;
2. The relevant version of the invariant Hodge-Morrey-Friedrichs decomposition theory in terms of the operators d_XM and δ_XM when there is a boundary of M;
3. In case there is a boundary of M, we present the relative and absolute X_M-cohomology and X_M-Poincaré-Lefschetz duality.

If I have time I will explain how the X_M-cohomology can help to solve special kinds of differential equations and our decomposition can be used to solve special kinds of the boundary-value problems for invariant differential forms.