a algebraic concept in reimannian geometry?

Question

let $(M,g)$ is a $C^{\infty}$ manifold with product form of $M=R \times \Theta $ the for each element $\varepsilon (r,\theta )\in M$, the tangent space $T_{\varepsilon}M$ is natural isomorphic to $T_{r}R\oplus T_{\theta }\Theta $, the subspaces $T_{r}R$ and$T_{\theta }\Theta $ of $T_{\varepsilon}M$ are orthogonal to $g$. i am pretty naive at riemannian geometry,can somebody please tell what this product means on vector spaces $T_{r}R$ and$T_{\theta }\Theta $.