Given matrices <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3030_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma _{1},\cdots ,\Gamma _{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi mathvariant="normal">Γ</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and g (g being symmetric), we give necessary and sufficient conditions for the existence of a vector field <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3030_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> satisfying <Equation ID="Equ18"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3030_Article_Equ18.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="138" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \nabla _{\Gamma }\psi +\left( \nabla _{\Gamma }\psi \right) ^{t}=g \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mi mathvariant="normal">Γ</mi> </msub> <mi>ψ</mi> <mo>+</mo> <msup> <mfenced close=")" open="("> <msub> <mi mathvariant="normal">∇</mi> <mi mathvariant="normal">Γ</mi> </msub> <mi>ψ</mi> </mfenced> <mi>t</mi> </msup> <mo>=</mo> <mi>g</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3030_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\nabla _{\Gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">∇</mi> <mi mathvariant="normal">Γ</mi> </msub> </math></EquationSource> </InlineEquation> denotes the covariant derivative. We then give several applications of our result.

The symmetrized covariant derivative

期刊论文
发表日期：

2025年5月28日

查看全文

S. Bandyopadhyay,
B. Dacorogna

摘要

Given matrices $\Gamma _{1},\cdots ,\Gamma _{n}$ $Γ_{1}, \dots, Γ_{n}$ and g (g being symmetric), we give necessary and sufficient conditions for the existence of a vector field $\psi $ $ψ$ satisfying $\begin{aligned} \nabla _{\Gamma }\psi +\left( \nabla _{\Gamma }\psi \right) ^{t}=g \end{aligned}$ $\begin{matrix} \nabla_{Γ} ψ + {(\nabla_{Γ}, ψ)}^{t} = g \end{matrix}$ where $\nabla _{\Gamma }$ $\nabla_{Γ}$ denotes the covariant derivative. We then give several applications of our result.