Mathematics – Differential Geometry
Scientific paper
2009-09-10
Mathematics
Differential Geometry
51 pages, 6 figures
Scientific paper
On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and boundary of the manifold. The principal angles between these interior subspaces are all acute and are called Poincare duality angles. This paper determines the Poincare duality angles of a collection of interesting manifolds with boundary derived from complex projective spaces and from Grassmannians, providing evidence that the Poincare duality angles measure, in some sense, how "close" a manifold is to being closed. This paper also elucidates a connection between the Poincare duality angles and the Dirichlet-to-Neumann operator for differential forms, which generalizes the classical Dirichlet-to-Neumann map arising in the problem of Electrical Impedance Tomography. Specifically, the Poincare duality angles are essentially the eigenvalues of a related operator, the Hilbert transform for differential forms. This connection is then exploited to partially resolve a question of Belishev and Sharafutdinov about whether the Dirichlet-to-Neumann map determines the cup product structure on a manifold with boundary.
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