Why are liquid surfaces at constant pressure?

[dEdt]

This is probably a stupid question with a simple answer, but please bear with me.

Suppose we have a bucket of water that we begin to spin at a constant angular speed. My textbook asked to find the shape of the surface of water, with the hint that the surface would be a surface of constant pressure. I got the right answer, but I don't fully understand why the surface should all be at the same pressure. I thought it had to do with preventing water from flowing along the surface, but I don't think that's it because different points of the fluid could be at different pressures without fluid flowing between them.

Also, does this (surface liquid makes = surface of constant pressure) hold if viscosity is introduces?

 

[Hassan2]

Surface pressure is equal to the atmospheric pressure which is assumed constant. Isn't it?

 

[dEdt]

Ah of course, that makes sense. Thanks.