  A geometric generalization of contraction theory called~$k$-contraction was
recently developed using $k$-compound matrices. In this note, we focus on the
relations between $k$-contraction and two other generalized contraction
frameworks: partial contraction (also known as virtual contraction) and
horizontal contraction. We show that in general these three notions of
contraction are different. We here provide new sufficient conditions
guaranteeing that partial contraction implies horizontal contraction, and that
horizontal contraction implies $k$-contraction. We use the Andronov-Hopf
oscillator to demonstrate some of the theoretical results.
