What is a Cheeger deformation?

Question

I'm currently at a Differential Geometry meeting and there is a mini-course on positively curved Riemannian manifolds. There, we were told that a technique to construct such manifolds is a Cheeger deformation, which (if I understood correctly) is a generalization of a one-parameter family of surfaces of revolution given by $\frac{f}{\lambda f + 1}$, where $f$ is the curve that generates a surface of revolution in Euclidean space and $\lambda$ is a positive parameter that varies over $[0,\infty)$. Can anyone tell me what is the concrete definition of a Cheeger deformation and how are they used to construct manifolds of positive (or non-negative, perhaps; I don't remember) curvature? Thanks a lot.

Answer 1

Cheeger deformations were first introduced in Jeff Cheeger's 1973 paper Some examples of manifolds of nonnegative curvature, and were inspired by the Berger sphere construction. The basic idea is to consider a Lie group acting by isometries on a Riemannian manifold, and then shrink the lengths of vectors tangent to the orbits of the action while keeping those perpendicular to the orbits fixed. Here is a formal description:

---

Let $(M,g_M)$ be a Riemannian manifold with $G$, a closed subgroup of the isometry group of $M$, acting on the left. Choose any bi-invariant metric $g_G$ on $G$, and consider the one-parameter family of product metrics $\{l^2g_G + g_M\}_{l>0}$ on the product manifold $G\times M$. Then $G$ acts on $G \times M$ via the map

$$y\cdot(x,p) = (xy^{-1}, y\cdot p).$$

This action is free, so taking the quotient gives a manifold $(G\times M)/G$, and it is diffeomorphic to $M$. Thus by imposing that the quotient map

$$q_l:(G \times M, l^2g_G + g_M)\to (G\times M)/G\cong M$$

be a Riemannian submersion, we obtain a one-parameter family of metrics $g_l$ on $M$. The family of Riemannian manifolds $\{(M,g_l)\}_{l>0}$ is called a Cheeger deformation of the original manifold $(M,g_M)$.

---

As $l\to \infty$, the Cheeger deformed metrics $g_l$ converge to the original metric $g_M$, and as $l\to 0$, the manifolds $(M,g_l)$ converge in the Gromov-Hausdorff metric to the quotient space $M/G$. Moreover, if the original manifold $(M,g_M)$ is one of nonnegative (resp. positive) sectional curvature, then the Cheeger deformed manifolds $(M,g_l)$ will have nonnegative (resp. positive) sectional curvature for all values of $l$.

The survey by Wolfgang Ziller mentioned by @Dan in the comments is a great reference for general information about these deformations and how they affect curvature. See my blog post for an animation of the Cheeger deformed plane under the circle action given by rotation about the origin.