Abstracts of the Arbeitsgruppenseminar

Prof. Bernd Ammann und Mitarbeiter, Zimmer 119

Thomas Tony, Universität Münster

Scalar curvature comparison geometry and the higher mapping degree

April 15, 14:15

Llarull proved in the late '90s that the round n-sphere is area-extremal in the sense that one can not increase the scalar curvature and the metric simultaneously. Goette and Semmelmann generalized Llarulls rigidity statement to certain area-non-increasing spin maps f: M -> N of non-zero A-hat -degree. In this talk, I will prove a generalization in which the topological condition on the A-hat degree is replaced by a less restrictive condition involving the so-called higher mapping degree. For that purpose, I will first present an index formula connecting the higher mapping degree and the Euler characteristic of N with the index of a suitable Dirac operator. Second, I will show that the extremal geometric situation together with the topological assumptions and the index formula give rise to a family of almost constant sections. Third, I will use this family to deduce the extremality and rigidity statement.

Samuel Lockman, KTH Stockholm

Classification of semi-Riemannian Spin_c Manifolds carrying Generalized Killing Spinors

May 6, 16:15, M311

I aim to present the following main result of my thesis: every Lorentzian spin^c manifold carrying a real Killing spinor, that satisfies some rather mild completeness assumption, is isometric to a warped product. With previous results by Bohle, we now have a complete description of all Lorentzian spin manifolds with real Killing spinors, improving the latest results due to Bohle and Leistner. Further, without any completeness assumptions, we apply results by Kühnel and Rademacher, to describe the local geometry of semi-Riemannian spin^c manifolds with generalised Killing spinors, in many cases. In this spirit, inspired by the work of Gutiérrez and Olea, we give conditions on specific neighbourhoods of the manifold which are isometric to a warped product. Also, we give conditions for when a semi-Riemannian spin^c manifold carrying a generalised Killing spinor has a normal semi-Riemannian covering in the form of a warped product. With this, one for example obtains improvements of the classifications made by Große and Nakad of Riemannian spin^c manifolds with imaginary generalized Killing spinors.

Back to the plan of the Arbeitsgruppenseminar

Bernd Ammann, 05.07.2024
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