  A geometric mechanism that may, in analogy to similar notions in physics, be
considered as "symmetry breaking" in geometry is described, and several
instances of this mechanism in differential geometry are discussed: it is shown
how spontaneous symmetry breaking may occur, and it is discussed how explicit
symmetry breaking may be used to tackle certain geometric problems. A
systematic study of symmetry breaking in geometry is proposed, and some
preliminary thoughts on further research are formulated.
