I don't know -- In riding, you find men in the areas where there's a lot of adrenaline. There are more of them in jumping, and in reining, for example. You don't find a lot of them in dressage, which is detail-oriented and requires obsessive-compulsive personality qualities. But in diving, I find for example that DIR is a lot like dressage -- Perfectionist, detail-oriented, requiring a lot of relatively unexciting practice -- and yet, at least where I am, the DIR divers are almost entirely men.
BTW, Drbill, after having raised animals for the last 15 years, I am completely convinced that there ARE genetic differences in personality between males and females. You can see it in foals within days of birth. The colts very quickly start to explore their environment and are much more independent of the mares. Fillies stay near mom, and are much more subdued in their play. You can't convince me that, just because we have a bigger frontal cortex, we are immune from those kinds of influences.