<p>In this paper, we study hypersurfaces in a product space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\bar{M}_{\kappa _1}^2\times \bar{M}_{\kappa _2}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mover accent="true"> <mrow> <mi>M</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mrow> <msub> <mi>κ</mi> <mn>1</mn> </msub> </mrow> <mn>2</mn> </msubsup> <mo>×</mo> <msubsup> <mover accent="true"> <mrow> <mi>M</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mrow> <msub> <mi>κ</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> of 2-dimensional space forms for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\kappa _1, \kappa _2\in \{-1,0,1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo>∈</mo> <mrow> <mo stretchy="false">{</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\kappa _1^2+\kappa _2^2\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>κ</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>κ</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We first present all solutions of the Fischer-Marsden equation on hypersurfaces with the product angle function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C^2=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>C</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> as well as on isoparametric hypersurfaces with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|C|<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>C</mi> <mo stretchy="false">|</mo> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Then, we classify the Ricci solitons on Hopf hypersurfaces with the Reeb vector field being the potential vector field.</p>

Fischer-Marsden Equation on Hypersurfaces in the Product Spaces of Space Forms

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发表日期：

2026年3月17日

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Dehe Li,
Xi Zhang

摘要

In this paper, we study hypersurfaces in a product space $\bar{M}_{\kappa _1}^2\times \bar{M}_{\kappa _2}^2$ ${\bar{M}}_{κ_{1}}^{2} \times {\bar{M}}_{κ_{2}}^{2}$ of 2-dimensional space forms for $\kappa _1, \kappa _2\in \{-1,0,1\}$ $κ_{1}, κ_{2} \in {- 1, 0, 1}$ and $\kappa _1^2+\kappa _2^2\ne 0$ $κ_{1}^{2} + κ_{2}^{2} \neq 0$ . We first present all solutions of the Fischer-Marsden equation on hypersurfaces with the product angle function $C^2=1$ $C^{2} = 1$ as well as on isoparametric hypersurfaces with $|C|<1$ $| C | < 1$ . Then, we classify the Ricci solitons on Hopf hypersurfaces with the Reeb vector field being the potential vector field.