Differential geometry of hydrodynamic Vlasov equations

Recurrent metrics in the geometry of second order differential equations

Given a pair (semispray $S$, metric $g$) on a tangent bundle, the family of nonlinear connections $N$ such that $g$ is recurrent with respect to $(S, N)$ with a fixed recurrent factor is determined by using the Obata tensors. In particular, we obtain a characterization for a pair $(N, g)$ to be recurrent as well as for the triple $(S, stackrel{c}{N}, g)$ where $stackrel{c}{N}$ is the canonical ...

Differential equations and integral geometry

be the operator of mean value over a radius r sphere centered at y ∈ R. The integral transform I is clearly injective. Let C be a compact hypersurface in R isotopic to a sphere. Theorem 1.1 Let f(x) be a smooth function vanishing near C. Then one can recover f from its mean values along the spheres tangent to C, and the inversion is given by an explicit formula. In fact we will show that this t...

recurrent metrics in the geometry of second order differential equations

given a pair (semispray $s$, metric $g$) on a tangent bundle, the family of nonlinear connections $n$ such that $g$ is recurrent with respect to $(s, n)$ with a fixed recurrent factor is determined by using the obata tensors. in particular, we obtain a characterization for a pair $(n, g)$ to be recurrent as well as for the triple $(s, stackrel{c}{n}, g)$ where $stackrel{c}{n}$ is the canonical ...

Integral geometry and differential equations

Differential Geometry of Strongly Integrable Systems of Hydrodynamic Type

Here the matrix (gij) (assumed nondegenerate) defines a pseudo-Riemannian metric (with upper indices) of zero curvature on the u-space, Fjk i = Fjki(u) being the corresponding Levi-Civita connection. Thus, the integrability condition can be formulated in terms of the differential geometry of SHT. For such integrable systems S. P. Tsarev [3] found a generalization (for N _> 3) of the hodograph m...