TITLE: For what metrics are circles solutions of the isoperimetric problem?
QUESTION [11 upvotes]: A classical result is that solutions of the isoperimetric problem on the plane, the sphere, and the hyperbolic plane are circles. Are there any other Riemannian metrics on these spaces that share this property or does this characterize metrics of constant curvature ?
I should add that by circles I mean really standard circles, not geodesic circles for the new metric.

REPLY [2 votes]: It is rather a comment and a question to the author than the  answer (but it is too long for the comment window). 
What is the   (solutions of the) isoperimetric   problem? If it is   a closed curve of a  fixed length such that it  bounds the region of a maximal volume, or  a shortest closed curve that bounds the region of certain  fixed  volume, I believe that  the solutions of isoperimetric problem are very rare. 
In order to confirm my suggestion, I would like to mention that for  metrics of constant positive curvature  the length of the circle bounding the ball of volume 1 is less than that of  for metrics of constant negative curvature.  This suggests that for the sphere  of revolution  such that the maximums of the curvature are  the poles 
the isoperimentric circles (of sufficiently small length) 
are circles (= boundary of the balls) centered at the poles. Then, in a certain coordinate system they are usual circles.   
(To see that the solutions of the isoperimetric problem are rare you could also combine two statements from the answer of Robert Bryant: curves of constant  geodesic curvature  are usually  not closed and isoperimetric curves must have constant geodesic curvature.)