In Riemannian geometry, a Berger sphere, named after Marcel Berger, is a standard 3-sphere with Riemannian metric from a one-parameter family, which can be obtained from the standard metric by shrinking along fibers of a Hopf fibration. It is interesting in that it is one of the simplest examples of Gromov collapse.
More precisely, one first considers the Lie algebra spanned by generators x1, x2, x3 with Lie bracket [xi,xj] = −2εijkxk. This is well known to correspond to the simply connected Lie group S3. Then, taking the product S3×R, extending the Lie bracket so that the generator x4 is left invariant under the operation of the Lie group, and taking the quotient by αx1+βx4, where α2+β2 = 1, we finally obtain the Berger spheres B(β).
There are also higher-dimensional analogues of Berger spheres.