2015/03/02 by Andreas Arvanitoyeorgos, Nikolaos Panagiotis Souris
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Combinatorics #Exponential function #Geodesic #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Invariant (physics) #Isotropy #Mathematical analysis #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum mechanics #math.DG #msc:53C25 #msc:53C30
paper · pdf · doi:10.1007/s00022-015-0268-0
published as J. Geometry 106 (3) (2015) 583-603 · Journal of Geometry (2015)
openalex publication_date 2015/03/02 · arxiv created 2015/03/14 · arxiv updated 2015/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces M=G/K whose isotropy representation decomposes into a direct sum of three submodules \frakm=\frakm1⊕\frakm2⊕\frakm3, satisfying the relations [\frakmi,\frakmi]⊂ \frakk. Assuming that the submodules \frakmi are pairwise non isomorphic, we study geodesics on such spaces of the form γ(t)=exp (tX)exp (tY)exp (tZ)⋅ o, where X∈\frm1, Y∈\frm2, Z∈\frm3 (o=eK), with respect to a G-invariant metric. Our investigation imposes certain restrictions on the G-invariant metric, so the geodesics turn out to be orbits of two exponential terms. We give a point of view using Riemannian submersions. As an application, we describe geodesics in generalized flag manifolds with three isotropy summands and with second Betti number b2(M)=2, and in the Stiefel manifolds SO(n+2)/S(n). We relate our results to geodesic orbit spaces (g.o. spaces).