Let M be a real hypersurface in complex projective space equipped with both the Levi-Civita and kth generalized Tanaka-Webster connections, for any nonnull real number k. For any operator B on M we can define two tensor fields of type (1,2) on M, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="22_2025_757_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle B_F^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="22_2025_757_Article_IEq2.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle B_T^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </math></EquationSource> </InlineEquation>, related to both connections. In the particular case of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="22_2025_757_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle B=L$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi>B</mi> <mo>=</mo> <mi>L</mi> </mrow> </mstyle> </math></EquationSource> </InlineEquation>, the structure Lie operator on M, we study purity and hybridness of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="22_2025_757_Article_IEq4.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle L_F^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="22_2025_757_Article_IEq5.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle L_T^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </math></EquationSource> </InlineEquation> with respect to the structure operator <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="22_2025_757_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle \phi $</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </math></EquationSource> </InlineEquation>.

New conditions on some derivatives of the structure Lie operator on a real hypersurface in complex projective space

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2025年6月29日

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Juan de Dios Pérez,
David Pérez-López

摘要

Let M be a real hypersurface in complex projective space equipped with both the Levi-Civita and kth generalized Tanaka-Webster connections, for any nonnull real number k. For any operator B on M we can define two tensor fields of type (1,2) on M, $\displaystyle B_F^{(k)}$ $B_{F}^{(k)}$ and $\displaystyle B_T^{(k)}$ $B_{T}^{(k)}$ , related to both connections. In the particular case of $\displaystyle B=L$ $B = L$ , the structure Lie operator on M, we study purity and hybridness of $\displaystyle L_F^{(k)}$ $L_{F}^{(k)}$ and $\displaystyle L_T^{(k)}$ $L_{T}^{(k)}$ with respect to the structure operator $\displaystyle \phi $ $ϕ$ .