The almost contact metric structure on a real hypersurface M in complex projective space allows to define on M,  for any nonnull real number k and any operator B,  two tensor fields of type (1,2) denoted by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1761_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$B_F^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>B</mi> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1761_Article_IEq2.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$B_T^{(k)}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>B</mi> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We will classify real hypersurfaces in complex projective space for which the h-operator satisfies that either <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1761_Article_IEq3.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$h_F^{(k)}=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>h</mi> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1761_Article_IEq4.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$h_T^{(k)}=0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>h</mi> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

Commutativity of the h-operator and two other operators on a real hypersurface in complex projective space

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2025年7月16日

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Juan de Dios Pérez,
David Pérez-López

摘要

The almost contact metric structure on a real hypersurface M in complex projective space allows to define on M, for any nonnull real number k and any operator B, two tensor fields of type (1,2) denoted by $B_F^{(k)}$ $B_{F}^{(k)}$ and $B_T^{(k)}.$ $B_{T}^{(k)} .$ We will classify real hypersurfaces in complex projective space for which the h-operator satisfies that either $h_F^{(k)}=0$ $h_{F}^{(k)} = 0$ or $h_T^{(k)}=0.$ $h_{T}^{(k)} = 0 .$