We proved that there does not exist a proper <mml:math alttext="$\rm CR$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>CR</mml:mtext></mml:mrow></mml:math>-hypersurface of <mml:math alttext="$S^6 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>S</mml:mi><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:math> with parallel second fundamental form. As a result of this we showed that <mml:math alttext="$S^6 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>S</mml:mi><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:math> does not admit a proper <mml:math alttext="$\rm CR$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>CR</mml:mtext></mml:mrow></mml:math>-totally umbilical hypersurface. We also proved that an Einstein proper <mml:math alttext="$\rm CR$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>CR</mml:mtext></mml:mrow></mml:math>-hypersurface of <mml:math alttext="$S^6 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>S</mml:mi><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:math> is an extrinsic sphere.

<mml:math alttext="$\rm CR$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>CR</mml:mtext></mml:mrow></mml:math>-hypersurfaces of the six-dimensional sphere


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International Journal of Mathematics and Mathematical Sciences

CR-hypersurfaces of the six-dimensional sphere

Abstract

Copyright