Homogeneous Einstein Metrics on Spheres and Proiective Spaces.
Homogeneous Einstein metrics on Stiefel manifolds
A Stiefel manifold is the set of orthonormal -frames in , and it is diffeomorphic to the homogeneous space . We study -invariant Einstein metrics on this space. We determine when the standard metric on is Einstein, and we give an explicit solution to the Einstein equation for the space .
Homogeneous Geodesics in 3-dimensional Homogeneous Affine Manifolds
For studying homogeneous geodesics in Riemannian and pseudo-Riemannian geometry (on reductive homogeneous spaces) there is a simple algebraic formula which works, at least potentially, in every given case. In the affine differential geometry, there is not such a universal formula. In the previous work, we proposed a simple method of investigation of homogeneous geodesics in homogeneous affine manifolds in dimension 2. In the present paper, we use this method on certain classes of homogeneous connections...
Homogeneous geodesics in a three-dimensional Lie group
O. Kowalski and J. Szenthe [KS] proved that every homogeneous Riemannian manifold admits at least one homogeneous geodesic, i.eȯne geodesic which is an orbit of a one-parameter group of isometries. In [KNV] the related two problems were studied and a negative answer was given to both ones: (1) Let be a homogeneous Riemannian manifold where is the largest connected group of isometries and . Does always admit more than one homogeneous geodesic? (2) Suppose that admits linearly independent...
Homogeneous geodesics in five-dimensional generalized symmetric spaces.
Homogeneous hessian manifolds
A flat affine manifold is said to Hessian if it is endowed with a Riemannian metric whose local expression has the form where is a -function and is an affine local coordinate system. Let be a Hessian manifold. We show that if is homogeneous, the universal covering manifold of is a convex domain in and admits a uniquely determined fibering, whose base space is a homogeneous convex domain not containing any full straight line, and whose fiber is an affine subspace of .
Homogeneous -symmetric spaces of interior type with simple real fundamental groups and their connection with parabolic spaces.
Homogeneous Kähler manifolds admitting a transitive solvable group of automorphisms
Homogeneous Lorentz manifolds with simple isometry group.
Homogeneous Lorentzian 3-manifolds with a parallel null vector field.
Homogeneous Lorentzian structures on some Gödel-Levichev's spacetimes, and associated reductive decompositions.
Homogeneous parabolic surfaces in .
Homogeneous Poisson structures on loop spaces of symmetric spaces.
Homogeneous quaternionic Kähler structures on Alekseevskian 𝒲-spaces
The homogeneous quaternionic Kähler structures on the Alekseevskian 𝒲-spaces with their natural quaternionic structures, each of these spaces described as a solvable Lie group, and the type of such structures in Fino's classification, are found.
Homogeneous Randers spaces admitting just two homogeneous geodesics
The existence of a homogeneous geodesic in homogeneous Finsler manifolds was investigated and positively answered in previous papers. It is conjectured that this result can be improved, namely that any homogeneous Finsler manifold admits at least two homogenous geodesics. Examples of homogeneous Randers manifolds admitting just two homogeneous geodesics are presented.
Homogeneous Riemannian manifolds with generic Ricci tensor
We describe homogeneous manifolds with generic Ricci tensor. We also prove that if 𝔤 is a 4-dimensional unimodular Lie algebra such that dim[𝔤,𝔤] ≤ 2 then every left-invariant metric on the Lie group G with Lie algebra 𝔤 admits two mutually opposite compatible left-invariant almost Kähler structures.
Homogeneous spaces and isoparametric hypersurfaces.
Homogeneous submanifolds of the hyperbolic space.
Homogeneous surfaces in the equiaffine space R⁴
Homogeneous systems of higher-order ordinary differential equations
The concept of homogeneity, which picks out sprays from the general run of systems of second-order ordinary differential equations in the geometrical theory of such equations, is generalized so as to apply to equations of higher order. Certain properties of the geometric concomitants of a spray are shown to continue to hold for higher-order systems. Third-order equations play a special role, because a strong form of homogeneity may apply to them. The key example of a single third-order equation...