Canonical almost Hermitian structures of normal bundles and applications to Kähler forms

Po-Hsun Hsieh

Vol. 184 (1998), No. 2, 257–277

DOI: 10.2140/pjm.1998.184.257

Abstract

For any (real) submanifold L of an almost Hermitian manifold (M,J,g,ω) (ω = g(J⋅,⋅)), there is a canonical almost Hermitian structure (Ĵ,ĝ,ω) (ω = ĝ(Ĵ⋅,⋅)) on (the total space of) the normal bundle L^⊥. We have three main topics: (i) We investigate conditions under which (L^⊥,Ĵ,ĝ) is Kähler or almost Kähler. (ii) If ω is a symplectic form, then ω is called the canonical symplectic form of L^⊥. We investigate conditions for two such canonical symplectic forms to be isomorphic. (iii) If (M,J,g) is Kähler, we investigate conditions under which ω and ω are isomorphic: We obtain a single theorem which synthesizes, generalizes, and improves two of McDuff’s theorems on Kähler forms of Kähler manifolds with certain nonpositive curvature.

Milestones

Received: 15 November 1996

Revised: 12 May 1997

Published: 1 June 1998