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. 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.