Curvature identities for principle $T^1$-bundles over almost Hermitian manifolds

E. E. Ditkovskaya

Chair of Geometry, Moscow Pedagogical State University, Moscow, Russia

Abstract: We study the equivalence of identities $R_1$, $R_2$, and $R_3$ for an almost Hermitian structure $S$ on the base of a canonical principal $T^1$-bundle and their contact analogs for the induced almost contact metric structure $S^\sharp$ on the total space of this bundle. We prove that the canonical connection of a canonical principal $T^1$-bundle over a Hermitian or a quasi-Kählerian manifold of the class $R_3$ is normal. We also prove that the canonical connection of a canonical principal $T^1$-bundle over a Vaisman–Gray manifold $M$ of the class $R_3$ is normal if and only if the Lie vector of the manifold $M$ belongs to the center of the adjoint $K$-algebra.

Keywords: principal toroidal fiber bundle, almost contact structure, curvature tensor.

UDC: 514.76

Received: 25.07.2008

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2010, 54:7, 49–55

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