A Cheeger deformation is a one-parameter deformation of a Riemannian metric on a Riemannian manifold carrying a group action by a compact Lie group whose elements act by isometries. It contracts the Riemannian metric in directions tangent to the group orbits and preserves nonnegative sectional curvature when the original metric has it.
More precisely, let 
be a Riemannian manifold on which a compact
Lie group 
acts by isometries, and let 
be a Riemannian metric
on 
invariant under left and right translations. For 
, give 
the product metric 
. The submersion
 |
induces a Riemannian metric 
on 
. The family extends smoothly to 
with 
and shrinks directions tangent to the group
orbits as 
increases. If 
has nonnegative sectional curvature, then
so does 
for 
(Cheeger 1973).
Müter (1987) studied the curvature of this construction in detail and treated 
explicitly as a deformation. Cheeger deformations and their
iterates are widely used to construct Riemannian
metrics with nonnegative or positive sectional
curvature (Ziller 2009).
