Berger's sphere

In the mathematical field of Riemannian geometry, the Berger spheres form a special class of examples of Riemannian manifolds diffeomorphic to the 3-sphere. They are named for Marcel Berger who introduced them in 1962.

Geometry of the Berger spheres

The Lie group SU(2) is diffeomorphic to the 3-sphere. Its Lie algebra is a three-dimensional real vector space spanned by

$u_1=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right),u_2=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),u_3=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),$

which are complex multiples of the Pauli matrices. It is direct to check that the commutators are given by [u₁, u₂] = 2u₃ and [u₁, u₃] = −2u₂ and [u₂, u₃] = 2u₁. Any positive-definite inner product on the Lie algebra determines a left-invariant Riemannian metric on the Lie group. A Berger sphere is a metric so obtained by making the inner product on the Lie algebra have matrix

$\left(\begin{array}{ccc}t& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

relative to the basis u₁, u₂, u₃. Here Template:Mvar is a positive number to be freely chosen; each choice produces a different Berger sphere. If it were chosen negative, a Lorentzian metric would instead be produced. Using the Koszul formula it is direct to compute the Levi-Civita connection:

$\begin{array}{rlrl}{\mathrm{\nabla }}_{u_1}{u}_2& =(2-t)u_3& {\mathrm{\nabla }}_{u_2}{u}_1& =-t{u}_3\\ {\mathrm{\nabla }}_{u_2}{u}_3& =u_1& {\mathrm{\nabla }}_{u_3}{u}_2& =-u_1\\ {\mathrm{\nabla }}_{u_3}{u}_1& =t{u}_2& {\mathrm{\nabla }}_{u_1}{u}_3& =(t-2)u_2.\end{array}$

The curvature operator has eigenvalues t, t, 4 − 3t. The left-invariant Berger metric is also right-invariant if and only if t = 1.Template:Sfnm

The left-invariant vector field on SU(2) corresponding to u₁ (or to any other particular element of the Lie algebra) is tangent to the circular fibers of a Hopf fibration SU(2) → S².Template:Sfnm As such, the Berger metrics can also be constructed via the Hopf fibration, by scaling the tangent directions to the fibers. Unlike the above construction, which is based on a Lie group structure on the 3-sphere, this version of the construction can be extended to the more general Hopf fibrations S^{2n + 1} → CPⁿ of odd-dimensional spheres over the complex projective spaces, using the Fubini–Study metric.

Significance

A well-known inequality of Wilhelm Klingenberg says that for any smooth Riemannian metric on a closed orientable manifold of even dimension, if the sectional curvature is positive then the injectivity radius is greater than or equal to π⁄K^1/2, where Template:Mvar is the maximum of the sectional curvature. The Berger spheres show that this does not hold if the assumption of even-dimensionality is removed.Template:Sfnm

Likewise, another estimate of Klingenberg says that for any smooth Riemannian metric on a closed simply-connected manifold, if the sectional curvatures are all in the interval Template:Closed-open, then the injectivity radius is greater than π⁄2. The Berger spheres show that the assumption on sectional curvature cannot be removed.Template:Sfnm

Any compact Riemannian manifold can be scaled to produce a metric of small volume, diameter, and injectivity radius but large curvature. The Berger spheres illustrate the alternative phenomena of small volume and injectivity radius but without small diameter or large curvature. They show that the 3-sphere is a collapsing manifold: it admits a sequence of Riemannian metrics with uniformly bounded curvature but injectivity radius converging to zero. This sequence of Riemannian manifolds converges in the Gromov–Hausdorff metric to a two-dimensional sphere of constant curvature 4.Template:Sfnm

Generalizations

Berger–Cheeger perturbations

The Hopf fibration S³ → S² is a principle bundle with structure group U(1). Furthermore, relative to the standard Riemannian metric on S³, the unit-length vector field along the fibers of the bundle form a Killing vector field. This is to say that U(1) acts by isometries.Template:Sfnm

In greater generality, consider a Lie group Template:Mvar acting by isometries on a Riemannian manifold (M, g). In this generality (unlike for the specific case of the Hopf fibration), different orbits of the group action might have different dimensionality. For this reason, scaling the tangent directions to the group orbits by constant factors, as for the Berger spheres, would produce discontinuities in the metric. The Berger–Cheeger perturbations modify the scaling to address this, in the following way.Template:Sfnm

Given a right-invariant Riemannian metric Template:Mvar on Template:Mvar, the product manifold G × M can be given the Riemannian metric th ⊕ g. The left action of Template:Mvar on this product by x⋅(y, m) = (y x⁻¹, xm) acts freely by isometries, and so there is a naturally induced Riemannian metric on the quotient space, which is naturally diffeomorphic to Template:Mvar.Template:Sfnm

Canonical variation of a Riemannian submersion

The Hopf fibration S³ → S² is a Riemannian submersion relative to the standard Riemannian metrics on S³ and S². For any Riemannian submersion f: M → B, the canonical variation scales the vertical part of the metric by a constant factor. The Berger spheres are thus the total space of the canonical variation of the Hopf fibration. Some of the geometry of the Berger spheres generalizes to this setting. For instance, if a Riemannian submersion has totally geodesic fibers then the canonical variation also has totally geodesic fibers.Template:Sfnm

References

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