The concept of the Riemann-Cartan manifold was introduced by E.~Cartan. The Riemann-Cartan manifold is a triple $(M,g,\bar\nabla)$, where $(M,g)$ is a Riemann $n$-dimensional $(n\geq2)$ manifold with linear connection $\bar\nabla$ having nonzero torsion $\bar S$ such that $\bar\nabla g=0$. In our paper, we have considered scalar and total scalar curvatures of the Riemann-Cartan manifold $(M,g,\bar\nabla)$ and proved some formulas connecting these curvatures with scalar and total scalar curvatures of the Riemannian $(M,g)$. In particular we have analyzed these formulas for the case of Weitzenbök manifolds. And in an inference we have proved some vanishing theorems.