2019/09/30 by Tillmann Jentsch, Gregor Weingart
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebra over a field #Algebraic geometry #Algebraic number #Complex space #Differential geometry #Differential operator #Generalization #Heisenberg group #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Reductive group #math.DG #msc:53C21 #msc:53C25 #msc:53C30
paper · pdf · doi:10.1007/s10455-020-09740-7
published as Ann Glob Anal Geom (2020) · 46 pages. Some minor corrections
openalex created_date 2019/09/19 · openalex publication_date 2020/10/29 · arxiv created 2020/11/09 · arxiv updated 2020/11/10 · openalex updated_date 2026/08/05
Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that are closer to symmetric spaces than others. On the one hand, there is the series of Hopf fibrations over complex space forms, including the Heisenberg groups with their metrics of type H. On the other hand, there exist certain naturally reductive spaces in dimensions six and seven whose torsion forms have a distinguished algebraic property. All these spaces generalize geometric or algebraic properties of 3--dimensional naturally reductive spaces and have the following point in common: along every geodesic the Jacobi operator satisfies an ordinary differential equation with constant coefficients which can be chosen independently of the given geodesic.