We show that every closed symplectic four-dimensional manifold (M,ω) admits an almost Kähler metric of negative scalar curvature compatible with ω.

Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and diameter, and almost non-negative scalar curvature. Let r = 1 if n = 2, 3 and r = 2 + 1 if n ≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is ...

We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let X and Y be two smooth vector fields on a two-dimensional manifold M . If X and Y are everywhere linearly independent, then they define a classical Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become l...

on a finsler manifold, we define conformal vector fields and their complete lifts and prove that incertain conditions they are homothetic.