The Riemann Geometry of Space and Gravitational Waves With The Spin $s=1$
It is taken into account that not the Ricci tensor ${R_{il}}$ (Einstein equation), but the Riemann tensor ${R_{iklm}}$ provides the most general description of the space geometry. If ${R_{il}=0}$ (the space empty with matter, but it can be occupied by gravitational waves) then ${R_{iklm}={C_{iklm}}} $ . The tensor ${C_{iklm}}$ is the Weyl tensor, which disappears by conversion:${R_{il}={g^{km}}{R_{iklm}}}$ and we lose all information about the space structure, which is described by ${C_{iklm}}$ .The symmetry of ${R_{il}}$ provides the existents of gravitational waves with the spin s=2. We show that ${C_{iklm}}$ describes gravitational waves with s=1. Such gravitational waves can be created in inhomogeneous media, where the selected directions are determined by derivates of the energy-momentum tensor ${T^m}_{i,k}$ of matter. It is taken into account that gravitation is described not only by the metric tensor $g^{ik} = (1/2)(γ^i γ^k + γ^k γ^i)$, but also by the anti-symmetric tensor $σ^{ik} = (i/2)({γ^i}{γ^k} - {γ^k}{γ^i})$, where ${γ^k}(x)$ are Clifford matrices. We show that the tensor ${K_{ik}} = (1/4){σ^{lm}} {R_{iklm}}$, is the anty-simmetric analog to the Ricci tensor. The ${K_{lm}}$ describes various kinds of space metrics, not described by the Ricci tensor. It includes the Weyl tensor ${C_{iklm}}$, because $σ^{lm} {C_{lmik}}$ is not zero.