Self-duality operators on odd dimensional manifolds
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- by Houhong Fan
- Trans. Amer. Math. Soc. 350 (1998), 3673-3706
- DOI: https://doi.org/10.1090/S0002-9947-98-01954-0
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Abstract:
In this paper we construct a new elliptic operator associated to any nowhere zero vector field on an odd-dimensional manifold and study its index theory. It turns out this operator has several geometric applications to conformal vector fields, self-dual vector fields, locally free $S^{1}$-actions and transversal hypersurfaces of these vector fields in an odd-dimensional manifold. In particular, we reveal a non-stable phenomena about the existence of conformal vector fields and self-dual vector fields in odd dimensions above 3. This is in sharp contrast to the stable phenomena about the existence of nowhere zero vector fields in odd dimensions. Besides these applications, the index formula of this new operator also gives the formulas for the dimensions of self-duality cohomology groups and for the virtual dimensions of the moduli spaces of anti-self-dual connections on 5-cobordisms, which are introduced in author’s previous papers.References
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Bibliographic Information
- Houhong Fan
- Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520
- Email: hhfan@math.yale.edu
- Received by editor(s): August 18, 1995
- Received by editor(s) in revised form: October 6, 1996
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 3673-3706
- MSC (1991): Primary 57R25, 58G10; Secondary 57R80, 57M99, 58F25
- DOI: https://doi.org/10.1090/S0002-9947-98-01954-0
- MathSciNet review: 1422603