TITLE: geometric interpretation of "Euclidean domain"
QUESTION [5 upvotes]: I cite from Wikipedia: 
    Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields
All of these properties have a well-known geometric interpretation, except perhaps "(norm) Euclidean domain". Do you know one?

REPLY [3 votes]: Although this isn't "if and only if", one property of Euclidean domains $R$ which is not true of all PID's is that the special Whitehead group $SK_1(R)$ is trivial.
This is Theorem 2.3.2 of Rosenberg's book "Algebraic K-theory and its applications".
I leave it to others to decide whether this property is geometric.