In contrast to classical chemical reaction kinetics, for diffusion limited chemical reactions the anisotropy of the geometry has far reaching effects. We use tubular two and three-dimensional spaces to illustrate and discuss the dimensional crossover in A + B --> 0 reactions due to dimensional compactification. We find that the crossover time t(c) = W-alpha scales as alpha = beta/(a - b), where a, b, and beta are given by the earlier and the late time inverse density scaling of rho(-1) similar to t(a) and rho(-1) similar to t(b)W(beta), respectively. We also obtain a critical width W-c below (above) which the chemical reaction progresses without (with) traversing a two or three-dimensional Ovchinnikov-Zeldovich (OZ) reaction regime. As a result we find that there exist different hierarchies of dimensionally forced crossovers, depending on the initial conditions and geometric restrictions. Kinetic phase diagrams are employed, and exponents are given for various Euclidean and fractal compactified geometries, for the A + B and A + A elementary reactions. Monte Carlo simulations illustrate some of the kinetic hierarchies.