Real hypersurfaces of indefinite Kaehler manifolds

Abstract

We show the existence of (<mml:math alttext="$\varepsilon $" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi> ϵ </mml:mi></mml:math>)-almost contact metric structures and give examples of (<mml:math alttext="$\varepsilon $" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi> ϵ </mml:mi></mml:math>)-Sasakian manifolds. Then we get a classification theorem for real hypersurfaces of indefinite complex space-forms with parallel structure vector field. We prove that (<mml:math alttext="$\varepsilon $" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi> ϵ </mml:mi></mml:math>)-Sasakian real hypersurfaces of a semi-Euclidean space are either open sets of the pseudosphere <mml:math alttext="$S_{2S}^{2n + 1} $" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math> (1) or of the pseudohyperbolic space <mml:math alttext="$H_{2S - 1}^{2n + 1} $" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>S</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math>(1). Finally, we get the causal character of (<mml:math alttext="$\varepsilon $" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi> ϵ </mml:mi></mml:math>) cosymplectic real hypersurfaccs of indefinite complex space-forms.