The set of all Abelian simply transitive subgroups of the affine group naturally corresponds to the set of real solutions of a system of algebraic equations. We classify all simply transitive subgroups of the symplectic affine group by constructing a model space for the corresponding variety of solutions. Similarly, we classify the complete global model spaces for flat special Kähler manifolds with a constant cubic form.
In the sub-Riemannian framework, we give geometric necessary and sufficient conditions for the existence of abnormal extremals of the Maximum Principle. We give relations between abnormality, -rigidity and length minimizing. In particular, in the case of three dimensional manifolds we show that, if there exist abnormal extremals, generically, they are locally length minimizing and in the case of four dimensional manifolds we exhibit abnormal extremals which are not -rigid and which can be minimizing...
Summary: In this paper the isometries of the dual space were investigated. The dual structural equations of a Killing tensor of order two were found. The general results are applied to the case of the flat space.
Let us consider a projective manifold and a smooth volume form on . We define the gradient flow associated to the problem of -balanced metrics in the quantum formalism, the -balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the -balancing flow converges towards a natural flow in Kähler geometry, the -Kähler flow. We also prove the long time existence of the -Kähler flow and its convergence towards Yau’s solution to the Calabi conjecture of prescribing the...
We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.