In playing scales and arpeggios, the thumb should never be turned under the hand.
I disagree, and I think most people on this forum who observe themselves playing piano, will find many instances where the thumb comfortably, naturally, and practically bends underneath the hand. After all, that is what it is positioned to do.
It is true that in a lot of the repertoire from Liszt onwards, perhaps even most of the repertoire, depends on playing with the hand as if it were one unit. Connection between the fingers is less important, and often impossible, as maintaining a consistent sound through moving the hand all over the place.
Playing without bending the thumb underneath the hand means playing everything by lifting and moving the hand around the keyboard. But let's not get too dogmatic. I can think of a hundred instances in polyphonic music, when the thumb must be bent under, in order to maintain multiple lines. Don't get too carried away with one technique, and imagine that it must be the answer to every problem.
Every pianist should know how to connect music whether they must lift their hand off the keyboard, or whether they choose to keep it on.
Walter Ramsey