Affine Killing complete and geodesically complete homogeneous affine surfaces

Abstract

An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting of homogeneous affine surfaces.2010 Mathematics Subject Classification. 53C21.

An almost Zoll affine surface

“…Proof. We follow the discussion in Gilkey et al [9]. A direct computation shows that {e cx 2 cos(x 2 ), e cx 2 sin(x 2 ), x 1 } ⊂ Q(M(c)).…”

Affine Killing vector fields on homogeneous surfaces with torsion

“…The structure group for the set of Type B models is not the full general linear group but rather the ax + b group acting by the shear (x 1 , x 2 ) → (x 1 , bx 1 + ax 2 ); again we say two Type B models are linearly equivalent if they are in the same orbit of the induced linear action on R 8 . The work of [31,32] mentioned previously does not provide a full classification of all the Type B models without torsion up to linear equivalence. It does suffice, for our purposes, in that it does classify the torsion free Type B models with dim{K} > 2 by providing models N j i ( ; 0) where is an auxiliary parameter present in some instances.…”

“…Brozos-Vázquez et al [31] and Gilkey and Valle-Regueiro [32] have classified, up to linear equivalence, all the Type A and Type B models without torsion where dim{K} > 2. Given an arbitrary model T M of Type A or Type B with torsion, we pass to the associated torsion free model 0 M and write down a basis for K( 0 M).…”

“…The original analysis of Brozos-Vázquez et al [31] ignored the flat geometries as being uninteresting as they are in the torsion free setting. But once torsion is added, it is necessary to include these geometries as the flat geometries give rise to non-trivial geometries with torsion and for this the additional analysis of Gilkey and Valle-Regueiro [32] is required.…”

“…We say that two Type A models are linearly equivalent if they lie in the same orbit of this representation. The works [31,32] mentioned previously classify all Type A torsion free models up to linear equivalence. We restrict this classification to those torsion free models where dim{K} > 2 to obtain models M j i ( ; 0) where there is an auxiliary parameter in certain examples.…”