ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53 Veloso, Jose Miguel, Universidade Federal do Para, Brasil Almost pseudo-Hermitian manifolds An almost pseudo-Hermitian manifold M is a contact 2n+1 -manifold defined by a contact form θ and a pseudo-complex structure J on the distribution D = ker θ. We construct a GL(N, C) − structure on M given by the coframes θ, θ^{1}, ..., θ^{2n}, such that J^{∗}θ^{α} = θ^{α+n}, α = 1, ..., n and dθ = \frac{1}{2}g_{αβ}θ^{α} ∧θ^{β} +g_{α\overline{β}}θ^{α} ∧θ\overline{β} + \frac{1}{2}g\overline{αβ}θ\overline{α} ∧θ\overline{β} , On this structure we define a

linear connection given by forms θ_{β}^{α} such that dθ^{α} = θ^{β}∧ θ_{β}^{α}+θ\overline{β} ∧τ α ; \overline{β}+θ∧τ^{α}

, \frac{1}{2}dg_{αβ}−g_{αγ}θ_{β}^{γ}−g_{α\overline{γ}}τ\overline{γ} ; β = A_{αβδ}θ^{δ} , dg_{α\overline{β}}+g_{\overline{β}γ}θ_{α}^{γ}−g_{α\overline{γ}}τ\overline{γ} ; \overline{β}−g_{αγ}τ γ ; \overline{β}+g_{\overline{βγ}}τ\overline{γ} ; α = 0

, g_{αγ}τ^{γ} + g_{α\overline{γ}}τ\overline{γ} = E_{αβ}θ^{β} , A_{αβδ} + A_{βδα} + A_{δαβ} = 0, A_{αβδ} + A_{αδβ} = 0 , E_{αβ} = E_{βα} . This generalizes pseudo-Hermitian manifolds. An example are real hypersurfaces of the tangent manifold of Heinsenberg group.