Nonlinear generalized sections of vector bundles

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• 作者：E.A. Nigsch ^{eduard.nigsch@univie.ac.at" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlinear generalized functions ; Diffeomorphism invariance ; Colombeau algebra ; Singular pseudo-Riemannian geometry ; Covariant derivative
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 August 2016
• 年：2016
• 卷：440
• 期：1
• 页码：183-219
• 全文大小：765 K

文摘

We present an extension of J.F. Colombeau's theory of nonlinear generalized functions to spaces of generalized sections of vector bundles. Our construction builds on classical functional analytic notions, which is the key to having a canonical geometric embedding of vector bundle valued distributions into spaces of generalized sections. This permits to have tensor products, invariance under diffeomorphisms, covariant derivatives and the sheaf property. While retaining as much compatibility to L. Schwartz' theory of distributions as possible, our theory provides the basis for a rigorous and general treatment of singular pseudo-Riemannian geometry in the setting of Colombeau nonlinear generalized functions.