The tightness of certain almost complex submanifolds

Author:
Cristián U. Sánchez

Journal:
Proc. Amer. Math. Soc. **110** (1990), 807-811

MSC:
Primary 53C40; Secondary 53C15, 53C30, 57T15

MathSciNet review:
1025282

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Abstract: This paper contains a proof of the following fact. If a map from a connected Riemannian manifold with an almost complex structure into a euclidean space has the following properties: (a) is nondegenerate and (b) for almost all height functions, the linear Hessian of the composition with commutes with the almost complex structure, then is tight. This gives some information about the homology groups of the manifold. This result yields a new proof of a well-known theorem of R. Bott.

**[1]**Armand Borel,*Kählerian coset spaces of semisimple Lie groups*, Proc. Nat. Acad. Sci. U. S. A.**40**(1954), 1147–1151. MR**0077878****[2]**Raoul Bott,*An application of the Morse theory to the topology of Lie-groups*, Bull. Soc. Math. France**84**(1956), 251–281. MR**0087035****[3]**Theodore Frankel,*Fixed points and torsion on Kähler manifolds*, Ann. of Math. (2)**70**(1959), 1–8. MR**0131883****[4]**Shoshichi Kobayashi,*Imbeddings of homogeneous spaces with minimum total curvature*, Tôhoku Math. J. (2)**19**(1967), 63–70. MR**0212726****[5]**Nicolaas H. Kuiper,*Minimal total absolute curvature for immersions*, Invent. Math.**10**(1970), 209–238. MR**0267597****[6]**-,*Tight imbedding and maps. Submanifolds of geometric class three in*, Chern Symposium (Berkeley 1979), Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 97-145.**[7]**Hsien-Chung Wang,*Closed manifolds with homogeneous complex structure*, Amer. J. Math.**76**(1954), 1–32. MR**0066011**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1990-1025282-0

Keywords:
Almost complex structure,
nondegenerate function,
tight function,
Hessian

Article copyright:
© Copyright 1990
American Mathematical Society