Locally Conformal Kähler Geometry

1 L.c.K. Manifolds.- 2 Principally Important Properties.- 2.1 Vaisman’s conjectures.- 2.2 Reducible manifolds.- 2.3 Curvature properties.- 2.4 Blow-up.- 2.5 An adapted cohomology.- 3 Examples.- 3.1 Hopf manifolds.- 3.2 The Inoue surfaces.- 3.3 A generalization of Thurston’s manifold.- 3.4 A four-dimensional solvmanifold.- 3.5 SU(2) x S1.- 3.6 Noncompact examples.- 3.7 Brieskorn & Van de Ven’s manifolds.- 4 Generalized Hopf manifolds.- 5 Distributions on a g.H. manifold.- 6 Structure theorems.- 6.1 Regular Vaisman manifolds.- 6.2 L.c.K.0 manifolds.- 6.3 A spectral characterization.- 6.4 k-Vaisman manifolds.- 7 Harmonic and holomorphic forms.- 7.1 Harmonic forms.- 7.2 Holomorphic vector fields.- 8 Hermitian surfaces.- 9 Holomorphic maps.- 9.1 General properties.- 9.2 Pseudoharmonic maps.- 9.3 A Schwarz lemma.- 10 L.c.K. submersions.- 10.1 Submersions from CH?n.- 10.2 L.c.K. submersions.- 10.3 Compact total space.- 10.4 Total space a g.H. manifold.- 11 L.c. hyperKähler manifolds.- 12 Submanifolds.- 12.1 Fundamental tensors.- 12.2 Complex and CR submanifolds.- 12.3 Anti-invariant submanifolds.- 12.4 Examples.- 12.5 Distributions on submanifolds.- 12.6 Totally umbilical submanifolds.- 13 Extrinsic spheres.- 13.1 Curvature-invariant submanifolds.- 13.2 Extrinsic and standard spheres.- 13.3 Complete intersections.- 13.4 Yano’s integral formula.- 14 Real hypersurfaces.- 14.1 Principal curvatures.- 14.2 Quasi-Einstein hypersurfaces.- 14.3 Homogeneous hypersurfaces.- 14.4 Type numbers.- 14.5 L. c. cosymplectic metrics.- 15 Complex submanifolds.- 15.1 Quasi-Einstein submanifolds.- 15.2 The normal bundle.- 15.3 L.c.K. and Kähler submanifolds.- 15.4 A Frankel type theorem.- 15.5 Planar geodesic immersions.- 16 Integral formulae.- 16.1 Hopf fibrations.- 16.2 The horizontallifting technique.- 16.3 The main result.- 17 Miscellanea.- 17.1 Parallel IInd fundamental form.- 17.2 Stability.- 17.3 f-Structures.- 17.4 Parallel f-structure P.- 17.5 Sectional curvature.- 17.6 L. c. cosymplectic structures.- 17.7 Chen’s class.- 17.8 Geodesic symmetries.- 17.9 Submersed CR submanifolds.- A Boothby-Wang fibrations.- B Riemannian submersions.