On products of harmonic forms

AbstractThe purpose for this short note is to describe the space of harmonic spinors on hyperbolicn-spaceHn. This is a natural continuation of the study of harmonic functions onHnin [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms onHnwere considered. The connection between invariant differential equations on a Riemannian semisimple symmetric spaceG/Kand homological aspects of the representation theory ofG, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.

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Border Reduction in Existence Problems of Harmonic Forms

Nagoya Mathematical Journal ◽

10.1017/s0027763000024235 ◽

1967 ◽

Vol 29 ◽

pp. 137-143 ◽

Cited By ~ 1

Author(s):

Mitsuru Nakai ◽

Leo Sario

Keyword(s):

Complete Solution ◽

Harmonic Forms ◽

Harmonic Field ◽

Regular Region ◽

Special Case

Given an arbitrary Riemannian n-space V let σ be a harmonic field in the complement V-V0 of a regular region V0. The problem of constructing in v a harmonic field ρ with the property was given a complete solution in [2]. The corresponding problem for harmonic forms σ, ρ remains open iri the general case. In the special case of locally flat spaces the construction can be carried out by replacing by the point norm [3].

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TRANSVERSAL HARMONIC THEORY FOR TRANSVERSALLY SYMPLECTIC FLOWS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788708000190 ◽

2008 ◽

Vol 84 (2) ◽

pp. 233-245 ◽

Cited By ~ 2

Author(s):

HONG KYUNG PAK

Keyword(s):

Compact Manifold ◽

Hodge Structure ◽

Contact Manifold ◽

Symplectic Manifolds ◽

Poisson Manifold ◽

Harmonic Forms ◽

Cosymplectic Manifolds ◽

Almost Cosymplectic Manifold ◽

Differential Complex ◽

Hard Lefschetz Theorem

AbstractWe develop the transversal harmonic theory for a transversally symplectic flow on a manifold and establish the transversal hard Lefschetz theorem. Our main results extend the cases for a contact manifold (H. Kitahara and H. K. Pak, ‘A note on harmonic forms on a compact manifold’, Kyungpook Math. J.43 (2003), 1–10) and for an almost cosymplectic manifold (R. Ibanez, ‘Harmonic cohomology classes of almost cosymplectic manifolds’, Michigan Math. J.44 (1997), 183–199). For the point foliation these are the results obtained by Brylinski (‘A differential complex for Poisson manifold’, J. Differential Geom.28 (1988), 93–114), Haller (‘Harmonic cohomology of symplectic manifolds’, Adv. Math.180 (2003), 87–103), Mathieu (‘Harmonic cohomology classes of symplectic manifolds’, Comment. Math. Helv.70 (1995), 1–9) and Yan (‘Hodge structure on symplectic manifolds’, Adv. Math.120 (1996), 143–154).

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