The index of a holomorphic mapping and the index theorem

Ishihara, Tôru

doi:10.1090/S0002-9939-1977-0494249-7

The index of a holomorphic mapping and the index theorem
HTML articles powered by AMS MathViewer

by Tôru Ishihara

Proc. Amer. Math. Soc. 66 (1977), 169-174

DOI: https://doi.org/10.1090/S0002-9939-1977-0494249-7

PDF | Request permission

Abstract:

The index theorem for a harmonic mapping of riemannian manifolds is given. Let $f:M \to N$ be a holomorphic mapping of Kaehler manifolds. Then it is shown that the index of f is zero and that a Jacobi field along f is a holomorphic section of the bundle ${f^ \ast }T(N)$ induced by f.

References

Samuel I. Goldberg and Toru Ishihara, Harmonic quasiconformal mappings of Riemannian manifolds, Bull. Amer. Math. Soc. 80 (1974), 562–566. MR 341335, DOI 10.1090/S0002-9904-1974-13499-3
S. I. Goldberg, T. Ishihara, and N. C. Petridis, Mappings of bounded dilatation of Riemannian manifolds, J. Differential Geometry 10 (1975), no. 4, 619–630. MR 390964
André Lichnerowicz, Applications harmoniques et variétés Kähleriennes, Rend. Sem. Mat. Fis. Milano 39 (1969), 186–195 (French, with English summary). MR 270305, DOI 10.1007/BF02924136
E. Mazet, La formule de la variation seconde de l’énergie au voisinage d’une application harmonique, J. Differential Geometry 8 (1973), 279–296 (French). MR 336767
James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
S. Smale, On the Morse index theorem, J. Math. Mech. 14 (1965), 1049–1055. MR 0182027, DOI 10.1111/j.1467-9876.1965.tb00656.x
R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229–236. MR 375386, DOI 10.1090/S0002-9939-1975-0375386-2