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Oberseminar SoSe 2011 Zusammenfassung des Vortrags von Misha Verbitsky (Moskau, RUS)

An intrinsic volume functional on almost complex 6-manifolds and nearly Kaehler geometry

Let (M,I) be an almost complex 6-manifold. The obstruction to integrability of an almost complex structure (the so-called Nijenhuis tensor) maps a 3-dimensional bundle to a 3-dimensional one. We say that the Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold is called nearly Kaehler if it admits a Hermitian form ω such that ∇(ω) is totally antisymmetric, ∇ being the Levi-Civita connection. I will construct an intrinsic diffeomorphism-invariant functional on the space of almost complex structures on M similar to the Hitchin functional, and compare its extrema with the nearly Kaehler structures.