Riemannian geometry of the space of volume preserving immersions

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• 作者：Martin Bauer^a ; ^b ; ^{bauer.martin@univie.ac.at" class="auth_mail" title="E-mail the corresponding author} ; Peter W. Michor^c ; ^{peter.michor@univie.ac.at" class="auth_mail" title="E-mail the corresponding author} ; Olaf Mü ; ller^d ; ^{Olaf.Mueller@mathematik.uni-regensburg.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：58B20 ; 58D15
• 刊名：Differential Geometry and its Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：49
• 期：Complete
• 页码：23-42
• 全文大小：471 K

文摘

Given a compact manifold M and a Riemannian manifold N   of bounded geometry, we consider the manifold class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516300614&_mathId=si1.gif&_user=111111111&_pii=S0926224516300614&_rdoc=1&_issn=09262245&md5=ad2926f49ec90651abcf06bd690df746" title="Click to view the MathML source">Imm(M,N)class="mathContainer hidden">class="mathCode">$\mathrm{Imm}(M,N)$ of immersions from M to N   and its subset class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516300614&_mathId=si2.gif&_user=111111111&_pii=S0926224516300614&_rdoc=1&_issn=09262245&md5=530035d8915c163d9dae87831803bfde" title="Click to view the MathML source">Imm_μ(M,N)class="mathContainer hidden">class="mathCode">${\mathrm{Imm}}_{\mu }(M,N)$ of those immersions with the property that the volume-form of the pull-back metric equals μ  . We first show that the non-minimal elements of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516300614&_mathId=si2.gif&_user=111111111&_pii=S0926224516300614&_rdoc=1&_issn=09262245&md5=530035d8915c163d9dae87831803bfde" title="Click to view the MathML source">Imm_μ(M,N)class="mathContainer hidden">class="mathCode">${\mathrm{Imm}}_{\mu }(M,N)$ form a splitting submanifold. On this submanifold we consider the Levi-Civita connection for various natural Sobolev metrics, we write down the geodesic equation for which we show local well-posedness in many cases. The question is a natural generalization of the corresponding well-posedness question for the group of volume-preserving diffeomorphisms, which is of importance in fluid mechanics.