What are the types of plant geotropisms? Why do the stem and the root present opposite geotropisms?

The parts of geotropisms are the positive geotropism, that in which the plant grows in favor of the gravitational force, as for example in roots, and the negative geotropism, that against the gravitational force, for example, in the stem.

Root geotropism and stem geotropism are opposed due to dissimilar sensitivities to auxin concentration in these structures. The following experiment can demonstrate the phenomenon: Stem and root are placed in a horizontal position (parallel to the ground) and naturally auxins concentrate with their bottom part. Under this condition it is observed that the stem grows upwards and the root grows downwards. This occurs because in the stem the high auxin concentration in the bottom makes this side grow (longitudinally) more and the structures arcs upwards. In the root the high auxin concentration in the bottom inhibits the growth of this side and the upper side grows more making the root to curve downwards.