On invariant projectively flat affine connections

Constructing Complete Projectively Flat Connections

Theorem 1. Let T 2 be the two dimensional torus. Then for any positive integer m there is a complete torsion free projectively flat connection, ∇, on T 2 such that for any point p ∈ T 2 there is a point q ∈ T 2 with the property that any broken ∇-geodesic between p and q has at least m breaks. Moreover if T 2 is viewed as a Lie group in the usual manner, this connection is invariant under trans...

Hermitian–Einstein connections on principal bundles over flat affine manifolds

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G–bundle EG over M admits a Hermitian–Einstein structure if and only if EG is po...

On a class of locally projectively flat Finsler metrics

‎In this paper we study Finsler metrics with orthogonal invariance‎. ‎We‎ ‎find a partial differential equation equivalent to these metrics being locally projectively flat‎. ‎Some applications are given‎. ‎In particular‎, ‎we give an explicit construction of a new locally projectively flat Finsler metric of vanishing flag curvature which differs from the Finsler metric given by Berwald in 1929.

74 the Deformation Spaces of Projectively Flat

Projectively Flat Finsler Metrics of Constant Curvature

It is the Hilbert’s Fourth Problem to characterize the (not-necessarilyreversible) distance functions on a bounded convex domain in R such that straight lines are shortest paths. Distance functions induced by a Finsler metric are regarded as smooth ones. Finsler metrics with straight geodesics said to be projective. It is known that the flag curvature of any projective Finsler metric is a scala...