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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.[1]

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).^[2]

There are also higher-dimensional analogues of Berger spheres.
References

Greene, Robert E. (1997), "A genealogy of noncompact manifolds of nonnegative curvature: history and logic", Comparison geometry (Berkeley, CA, 1993–94), Math. Sci. Res. Inst. Publ. 30, Cambridge: Cambridge Univ. Press, pp. 99–134, MR 1452869. See in particular p. 122.
Cheeger, Jeff; Ebin, David G. (2008), Comparison theorems in Riemannian geometry, Providence, RI: AMS Chelsea Publishing, p. 70, ISBN 978-0-8218-4417-5, MR 2394158.

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