To any finite collection of smooth real vector fields Xj in R we associate a metric in the phase space T ∗Rn. The relation between the asymptotic behavior of this metric and hypoellipticity of ∑ X j , in the smooth, real analytic, and Gevrey categories, is explored. To Professor P. Lelong, on the occasion of his 85th birthday.

in this paper, we study projective randers change and c-conformal change of p-reduciblemetrics. then we show that every p-reducible generalized landsberg metric of dimension n  2 must be alandsberg metric. this implies that on randers manifolds the notions of generalized landsberg metric andberwald metric are equivalent.

The invention of the battery can be attributed to Alessandro Volta (1745-1827) of Como, Italy, who in 1800 described an assembly consisting of plates of two different metals, such as Zn and Cu, placed alternately in a stack-like fashion separated by paper soaked in an aqueous solution, such as brine or vinegar. As discovered by Volta, this contraption was capable of producing an electrical shoc...