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Homogeneous geodesics in homogeneous Finsler spaces
In this paper, we study homogeneous geodesics in homogeneous Finsler spaces.
We first give a simple criterion that characterizes geodesic vectors. We show
that the geodesics on a Lie group, relative to a bi-invariant Finsler metric,
are the cosets of the one-parameter subgroups. The existence of infinitely many
homogeneous geodesics on compact semi-simple Lie group is established. We
introduce the notion of naturally reductive homogeneous Finsler space. As a
special case, we study homogeneous geodesics in homogeneous Randers spaces.
Finally, we study some curvature properties of homogeneous geodesics. In
particular, we prove that the S-curvature vanishes along the homogeneous
geodesics
Homogeneous algebras
Various concepts associated with quadratic algebras admit natural
generalizations when the quadratic algebras are replaced by graded algebras
which are finitely generated in degree 1 with homogeneous relations of degree
N. Such algebras are referred to as {\sl homogeneous algebras of degree N}. In
particular it is shown that the Koszul complexes of quadratic algebras
generalize as N-complexes for homogeneous algebras of degree N.Comment: 24 page
Homogeneous and locally homogeneous solutions to symplectic curvature flow
J. Streets and G. Tian recently introduced symplectic curvature flow, a
geometric flow on almost K\"ahler manifolds generalising K\"ahler-Ricci flow.
The present article gives examples of explicit solutions to this flow of
non-K\"ahler structures on several nilmanifolds and on twistor fibrations over
hyperbolic space studied by J. Fine and D. Panov. The latter lead to examples
of non-K\"ahler static solutions of symplectic curvature flow which can be seen
as analogues of K\"ahler-Einstein manifolds in K\"ahler-Ricci flow.Comment: 15 page
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