the unitary group ;
the special orthogonal group ;
For general abstract properties usually the first characterization is the most important one. Notably it implies that the circle group fits into a short exact sequence
the “real exponential exact sequence”.
(On the other hand, the last characterization is usually preferred when one wants to be concrete.)
is the compact real form of the multiplicative group over the complex numbers, see at form of an algebraic group – Circle group and multiplicative group.