Abstract. For every two-dimensional manifold M with locally symmetric linear connection ∇, endowed also with ∇-parallel volume element, we construct a flat connection on some principal fibre bundle P (M, G). Associated with -satisfying some particular conditions -local basis of T M local connection form of such a connection is an R(G)-valued 1-form Ω build from the dual basis ω 1 , ω 2 and from the local connection form ω of ∇. The structural equations of (M, ∇) are equivalent to the condition dΩ − Ω ∧ Ω = 0.This work was intended as an attempt to describe in a unified way the construction of similar 1-forms known for constant Gauss curvature surfaces, in particular of that given by R. Sasaki for pseudospherical surfaces.
We characterize -by describing in local coordinates their sections -the transversal bundles which on the complex hypersurface with type number 1 induce locally symmetric connections. (2000). 53B05, 53C42, 53C56, 32H02.
Mathematics Subject Classifications
In [O2] the Cartan-Norden theorem for real affine immersions was proved without the non-degeneracy assumption. A similar reasoning applies to the case of affine Kähler immersions with an anti-complex shape operator, which allows us to weaken the assumptions of the theorem given in [NP]. We need only require the immersion to have a non-vanishing type number everywhere on M .
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