Jaw Rigging Issue

The jaw of my seal works well with the mouth, but the neck and body moves along with it. I don't mind the neck moving because I want to have a shadow underneath the jaw.

I am also confused by the triangle geometry. I have experience with 3D modeling and it looks like there is not separate geometry for every part. Instead it looks like the puppet is just one set of modelling geometry. Is that the way it's supposed to be?

This is my first puppet from scratch.