2015/12/31 by Rudolf Zeidler
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Canonical bundle #Cobordism #Codimension #Curvature #Geometric Analysis and Curvature Flows #Geometry #Holonomy #Manifold (fluid mechanics) #Mathematical analysis #Mathematical physics #Mathematics #Prescribed scalar curvature problem #Pure mathematics #Scalar (mathematics) #Scalar curvature #Sectional curvature #Submanifold #math.GT #math.KT #msc:46L80 #msc:53C23 #msc:58J22
paper · pdf · doi:10.2140/agt.2017.17.3081
published as Algebr. Geom. Topol. 17 (2017), 3081-3094 · 12 pages; v2: Changed title, minor revision following the referee's suggestions. To appear in Algebr. Geom. Topol
arxiv created 2017/07/09 · openalex created_date 2017/07/21 · openalex publication_date 2017/09/19 · arxiv updated 2017/09/25 · openalex updated_date 2026/08/05
We exhibit geometric situations where higher indices of the spinor Dirac operator on a spin manifold [math] are obstructions to positive scalar curvature on an ambient manifold [math] that contains [math] as a submanifold. In the main result of this note, we show that the Rosenberg index of [math] is an obstruction to positive scalar curvature on [math] if [math] is a fiber bundle of spin manifolds with [math] aspherical and [math] of finite asymptotic dimension. The proof is based on a new variant of the multipartitioned manifold index theorem which might be of independent interest. Moreover, we present an analogous statement for codimension-one submanifolds. We also discuss some elementary obstructions using the [math] -genus of certain submanifolds.