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Intrinsic harmonicily of Morse functions

Published online by Cambridge University Press:  26 February 2010

Patrizio Frosini  and
Claudia Landi
Patrizio Frosini
Affiliation:
Università di Bologna, Dipartimento di Matematica, P.zza Porta San Donato, 5, I-40127 Bologna, Italia, E-mail: frosini@dm.unibo.it
Claudia Landi
Affiliation:
Università di Modena e Reggio Emilia, DISMI, Viale Allegri, 12, I-42100 Reggio Emilia, Italia, E-mail: clandi@unimore.it
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Abstract

Consider a real valued Morse function f on a C2 closed connected n-dimensional manifold M. It is proved that a suitable Riemannian metric exists on M, such that f is harmonic outside the set of critical points of f of index 0 and n. The proof is based on a result of Calabi [1], providing a criterion for a closed one-form on a closed connected manifold to be harmonic with respect to some Riemannian metric.

Type
Research Article
Information
Mathematika , Volume 50 , Issue 1-2 , December 2003 , pp. 167 - 170
Copyright
Copyright © University College London 2003

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References

1.Calabi, E., An intrinsic characterization of harmonic one-forms. Global Analysis. Papers in Honor of K. Kodaira, (Spencer, D. C. and Iyanaga, S., eds.) (1969) 101–117.CrossRefGoogle Scholar
2.Farber, M., Katz, G., and Levine, J.. Morse theory of harmonic forms. Topology 37 (1998). 469–483.CrossRefGoogle Scholar
3.Munkres, J. R., Elementary Differential Topology, Annals of Mathematical Studies, n. 54 (Princeton University Press, 1963).CrossRefGoogle Scholar
4.Pctcrsen, P., Riemannian Geometry (Springer, New York-Heidelberg Berlin. 1998).Google Scholar