The authors consider a compact Riemannian manifold M with boundary. They suppose that M is dispersive, i.e. each geodesic starting from any point of M intersects \(\partial M\). For any symmetric tensor field f of order m on M and any geodesic \(\gamma\) there is defined the integral \[ If(\gamma)=\int^{+\infty}_{-\infty}f_{i_ 1...i_ m}(\gamma (t)){\dot \gamma}^{i_ 1}(t)...{\dot \gamma}^{i_ m}(t)dt. \] The question is: To what extent is the field f uniquely determined by \(\gamma\) \(\mapsto If(\gamma)?\)
After the proof of the theorem on the solenoidal decomposition \(f=\tilde f+dv\), \(\delta\) \(\tilde f=0\), where d is the symmetric part of the covariant derivative and \(\delta\) denotes divergence, the authors prove the following main theorem of the paper: Let M be compact without geodesics of infinite length and with boundary being strictly convex, and let the sectional curvature be non-positive. Then for f having locally square integrable generalized derivative we have the estimate \(\| u\|^ 2_ 0\leq C(m \| f\|_ 1 \| If\|_ 0+\| If\|^ 2_ 1)\) with the constant C independent of f, where \(\| \cdot \|_ 0\) and \(\| \cdot \|_ 1\) denote the norms in appropriate Hilbert spaces. From the theorem it follows that for any Riemannian manifold of an arbitrary dimension, dispersive and having non- positive sectional curvature, f is determined by If(\(\cdot)\) up to dv, where v is a symmetric (m-1)-field on M.