Integral geometry for differential forms (Q1374934)

The authors study certain problems of integral geometry for differential forms connected with inverse problems for differential hyperbolic second-order equations, the theory of representations, finding integral invariants, etc. There are given two existence theorems for solutions \((u,F)\) of the equation \[ \xi^iu_{x^i}-\Gamma^i_{jk} \xi^i \xi^k u_{\xi^i} =a_{i_1\dots i_m} \xi^{i_1} \cdots\xi^{im} \] where \(\Gamma^i_{ij}\) are the Christoffel symbols of the Riemannian metric. It is used a method of \textit{A. Kh. Amirov} [Sib. Math. J. 34, 199-209 (1993; Zbl 0844.53048)].