The fingers are essentially used to strike the keys vertically, not to move the hand horizontally up and down the keyboard (i.e. right and left)*. This function is to be performed by the shoulder, and also (although less frequently) by the pianist’s entire body. Generally speaking, i.e. without getting into the consideration of the differences between fast and slow playing, the mechanism of "gliding" is that the hand (limb) is moved sideways with the shoulder, and the fingers - as if along the way - strike the selected keys.
* However, some must give the first, basic impetus to bounce the Motor Apparatus in the right direction.
The best way to obtain a good “hovering” is to raise the shoulders slightly and to bring one a bit closer to the other. This allows us to integrate the hands (as members) not simply as a mass of inert muscles, but as a compact and elastic block permitting rapid horizontal movements of the motor apparatus (this question is developed in this chapter).
Each excessive relaxing of the shoulder muscles, as well as each use of the finger to make the hand jump, provokes at a quick tempo, a “sticking” motion, i.e. overloading the weak muscles of the fingers which stick the motor apparatus to the keyboard. The result is that the muscles of the arm and forearm contract, which not only slows down the playing, but can also lead to a temporary paralysis of the hand and muscular pain.