The Ricci flow on generalized Wallach spaces

We consider the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as special planar dynamical systems. All non symmetric generalized Wallach spaces can be naturally parametrized by three positive numbers $a_1, a_2, a_3$. Our interest is to determine the type of singularity of all singular points of the normalized Ricci flow on all such spaces. Our main result gives a qualitative answer for almost all points $(a_1,a_2,a_3)$ in the cube $(0,1/2]\times (0,1/2]\times (0,1/2]$.Comment: 18 pages, 3 figures, comments are welcom

On the characteristic connection of gwistor space

We give a brief presentation of gwistor space, which is a new concept from G_2 geometry. Then we compute the characteristic torsion T^c of the gwistor space of an oriented Riemannian 4-manifold with constant sectional curvature k and deduce the condition under which T^c is \nabla^c-parallel; this allows for the classification of the G_2 structure with torsion and the characteristic holonomy according to known references. The case with the Einstein base manifold is envisaged.Comment: Many changes since first version, including title; Central European Journal of Mathematics, 201

Geodesic flows on Riemannian g.o. spaces

We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.Comment: 12 pages, minor corrections, final versio

Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces (M=G∕H,g) whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (Sp(n)∕Sp(n1)×⋯×Sp(ns),g), with 0<n1+⋯+ns≤n. Such spaces include spheres, quaternionic Stiefel manifolds, Grassmann manifolds and quaternionic flag manifolds. The present work is a contribution to the study of g.o. spaces (G∕H,g) with H semisimple. © 2021 Elsevier B.V

Two-step homogeneous geodesics in pseudo-Riemannian manifolds

Given a homogeneous pseudo-Riemannian space (G/H,⟨,⟩), a geodesic γ: I→ G/ H is said to be two-step homogeneous if it admits a parametrization t= ϕ(s) (s affine parameter) and vectors X, Y in the Lie algebra g, such that γ(t) = exp (tX) exp (tY) · o, for all t∈ ϕ(I). As such, two-step homogeneous geodesics are a natural generalization of homogeneous geodesics (i.e., geodesics which are orbits of a one-parameter group of isometries). We obtain characterizations of two-step homogeneous geodesics, both for reductive homogeneous spaces and in the general case, and undertake the study of two-step g.o. spaces, that is, homogeneous pseudo-Riemannian manifolds all of whose geodesics are two-step homogeneous. We also completely determine the left-invariant metrics ⟨,⟩ on the unimodular Lie group SL(2 , R) such that (SL(2,R),⟨,⟩) is a two-step g.o. space

A review of compact geodesic orbit manifolds and the g.o. condition for SU(5)/ S(U(2) × U(2))

Geodesic orbit manifolds (or g.o. manifolds) are those Riemannian manifolds (M, g) whose geodesics are integral curves of Killing vector fields. Equivalently, there exists a Lie group G of isometries of (M, g) such that any geodesic γ has the simple form γ(t) = etX · p, where e denotes the exponential map on G. The classification of g.o. manifolds is a longstanding problem in Riemannian geometry. In this brief survey, we present some recent results and open questions on the subject focusing on the compact case. In addition we study the geodesic orbit condition for the space SU(5)/ S(U(2) × U(2)) © Balkan Society of Geometers, Geometry Balkan Press 202

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces (M= G/ H, g) whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (G/H, g), such that G is one of the compact classical Lie groups SO(n), U (n) , and H is a diagonally embedded product H1× ⋯ × Hs, where Hj is of the same type as G. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces (G/H, g) with H semisimple. © 2021, The Author(s), under exclusive licence to Springer Nature B.V