<p>In this paper, we mainly focus on space-like PMCV surfaces in Robertson–Walker spaces. First, we derive certain geometrical properties of biconservative surfaces in the Robertson–Walker space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2395_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$L^n_1(f, c)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mn>1</mn> <mi>n</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of arbitrary dimension. Then, we get complete local classifications of PMCV surfaces in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2395_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$L^4_1(f,0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mn>1</mn> <mn>4</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2395_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$L^5_1(f,0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mn>1</mn> <mn>5</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2395_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$L^5_1(1,\pm 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mn>1</mn> <mn>5</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Finally, we prove that a space-like PMCV biconservative surface in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2395_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">$L^n_1(f,0),\ n\ge 6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mn>1</mn> <mi>n</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mi>n</mi> <mo>≥</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> lies on a totally geodesic submanifold with dimension either 4 or 5.</p>

Biconservative Surfaces in Robertson–Walker Spaces

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2025年3月29日

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Nurettin Cenk Turgay,
Rüya Yeǧin Şen

摘要

In this paper, we mainly focus on space-like PMCV surfaces in Robertson–Walker spaces. First, we derive certain geometrical properties of biconservative surfaces in the Robertson–Walker space $L^n_1(f, c)$ $L_{1}^{n} (f, c)$ of arbitrary dimension. Then, we get complete local classifications of PMCV surfaces in $L^4_1(f,0)$ $L_{1}^{4} (f, 0)$ , $L^5_1(f,0)$ $L_{1}^{5} (f, 0)$ and $L^5_1(1,\pm 1)$ $L_{1}^{5} (1, \pm 1)$ . Finally, we prove that a space-like PMCV biconservative surface in $L^n_1(f,0),\ n\ge 6$ $L_{1}^{n} (f, 0), n \geq 6$ lies on a totally geodesic submanifold with dimension either 4 or 5.