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.[1]
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. 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. [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., vol. 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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