A universal Riemannian foliated space

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• 作者：Jesú ; s A. Á ; lvarez Ló ; pez^a ; ^{jesus.alvarez@usc.es" class="auth_mail" title="E-mail the corresponding author} ; Ramó ; n Barral Lijó ; ^a ; ^{ramon.barral@usc.es" class="auth_mail" title="E-mail the corresponding author} ; Alberto Candel^b ; ^{alberto.candel@csun.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：C&infin ; C&infin ; convergence of Riemannian manifolds ; Locally non-periodic Riemannian manifolds ; Riemannian foliated space
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 February 2016
• 年：2016
• 卷：198
• 期：Complete
• 页码：47-85
• 全文大小：870 K

文摘

It is proved that the isometry classes of pointed connected complete Riemannian n  -manifolds form a Polish space, xl.gif" data-inlimgeid="1-s2.0-S0166864115004964-si1.gif">${\mathcal{M}}_{⁎}^{\infty }(n)$, with the topology described by the xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115004964&_mathId=si2.gif&_user=111111111&_pii=S0166864115004964&_rdoc=1&_issn=01668641&md5=4588b91334d871bb17ecb09a1a2140ff" title="Click to view the MathML source">C^∞$C^{\infty }$ convergence of manifolds. This space has a canonical partition into sets defined by varying the distinguished point into each manifold. The locally non-periodic manifolds define an open dense subspace xl.gif" data-inlimgeid="1-s2.0-S0166864115004964-si3.gif">${\mathcal{M}}_{⁎,\mathrm{lnp}}^{\infty }(n)\subset {\mathcal{M}}_{⁎}^{\infty }(n)$, which becomes a xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115004964&_mathId=si2.gif&_user=111111111&_pii=S0166864115004964&_rdoc=1&_issn=01668641&md5=4588b91334d871bb17ecb09a1a2140ff" title="Click to view the MathML source">C^∞$C^{\infty }$ foliated space with the restriction of the canonical partition. Its leaves without holonomy form the subspace xl.gif" data-inlimgeid="1-s2.0-S0166864115004964-si4.gif">${\mathcal{M}}_{⁎,\mathrm{np}}^{\infty }(n)\subset {\mathcal{M}}_{⁎,\mathrm{lnp}}^{\infty }(n)$ defined by the non-periodic manifolds. Moreover the leaves have a natural Riemannian structure so that xl.gif" data-inlimgeid="1-s2.0-S0166864115004964-si5.gif">${\mathcal{M}}_{⁎,\mathrm{lnp}}^{\infty }(n)$ becomes a Riemannian foliated space, which is universal among all sequential Riemannian foliated spaces satisfying certain property called covering-determination. xl.gif" data-inlimgeid="1-s2.0-S0166864115004964-si5.gif">${\mathcal{M}}_{⁎,\mathrm{lnp}}^{\infty }(n)$ is used to characterize the realization of complete connected Riemannian manifolds as dense leaves of covering-determined compact sequential Riemannian foliated spaces.