We define $\underset\theta\to R$-projective geodesic mappings $(\theta=1,\dots,5)$ of two generalized Riemannian spaces and obtain some invariant geometric objects of these mappings, generalizing Weyl's tensor. Also, we define $\underset\theta\to R$-projectively flat generalized Riemannian spaces $G\overline R_N$ and find necessary conditions for the space $GR_N$ to be $\underset\theta\to R$-projectively flat.