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A local characterization of affine holomorphic immersions with an anti-complex and ∇-parallel shape operator

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Robaszewska M.

Annales Polonici Mathematici

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2002

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tom 78

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nr 1

59-84

EN

We study the complex hypersurfaces $f:M^{(n)} → ℂ^{n+1}$ which together with their transversal bundles have the property that around any point of M there exists a local section of the transversal bundle inducing a ∇-parallel anti-complex shape operator S. We give a class of examples of such hypersurfaces with an arbitrary rank of S from 1 to [n/2] and show that every such hypersurface with positive type number and S ≠ 0 is locally of this kind, modulo an affine isomorphism of $ℂ^{n+1}$.

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Affine analogues of the Sasaki-Shchepetilov connection

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Robaszewska M.

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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2016

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tom 15

EN

For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM ⊕ (ℝ × M) and two on TM ⊕ (ℝ2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝN, after suitable local affine embedding of (M,∇) into ℝN.

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On the Cartan-Norden theorem for affine Kähler immersions

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Robaszewska M.

Annales Polonici Mathematici

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2000

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tom 75

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nr 1

69-77

EN

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.

4

On some flat connection associated with locally symmetric surface

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Robaszewska M.

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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2014

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tom 13

EN

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 TM local connection form of such a connection is an R(G)-valued 1-form build from the dual basis ω1, ω2 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.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

4 Robaszewska M.

1 2016

1 2014

1 2002

1 2000

A local characterization of affine holomorphic immersions with an anti-complex and ∇-parallel shape operator

Affine analogues of the Sasaki-Shchepetilov connection

On the Cartan-Norden theorem for affine Kähler immersions

On some flat connection associated with locally symmetric surface