Berger's sphere

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 [x_i,x_j] = −2ε_ijkx_k. This is well known to correspond to the simply connected Lie group S³. Denote by ω₁, ω₂, ω₃ the left invariant 1-forms on S³ which equal the dual covectors to x₁, x₂, x₃. Then the standard metric on S³ is ω₁²+ω₂²+ω₃². The Berger metric is βω₁²+ω₂²+ω₃², for any constant β>0.

There are also higher-dimensional analogues of Berger spheres.