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<p>A characterization of immersions of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\({\mathbb {R}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\({\mathbb {R}}^{n+m}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> in Monge form making constant Jordan angles with a fixed subspace is shown. The characterization states that the immersion <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\({\textbf {x}}(u) = \left( u,f_1(u),f_2(u),\dots ,f_m(u)\right) ,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfenced close=")" open="("> <mi>u</mi> <mo>,</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>f</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> has constant Jordan angles if and only if for any <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(k=1,2,\dots , n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, <Equation ID="Equ87"> <MediaObject ID="MO1"> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40687_2025_555_Equ87_HTML.png" Format="PNG" Height="112" Rendition="HTML" Resolution="300" Type="Linedraw" Width="899" /> </MediaObject> </Equation>Explicit examples of this result are worked out.</p>

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Graph immersions of \({\mathbb {R}}^n\) in \({\mathbb {R}}^{n+m}\) making constant Jordan angles with a fixed subspace

  • 期刊论文

  • 发表日期：

    2025年9月12日

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  • J. Monterde

摘要

A characterization of immersions of \({\mathbb {R}}^n\) R n in \({\mathbb {R}}^{n+m}\) R n + m in Monge form making constant Jordan angles with a fixed subspace is shown. The characterization states that the immersion \({\textbf {x}}(u) = \left( u,f_1(u),f_2(u),\dots ,f_m(u)\right) ,\) x ( u ) = u , f 1 ( u ) , f 2 ( u ) , ⋯ , f m ( u ) , has constant Jordan angles if and only if for any \(k=1,2,\dots , n\) k = 1 , 2 , ⋯ , n , Explicit examples of this result are worked out.