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.



What is a Cheeger deformation?
Cheeger deformations were first introduced in Jeff Cheeger's 1973 paper [Some examples of manifolds of nonnegative curvature](http://projecteuclid.org/euclid.jdg/1214431964), 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:




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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 $(M,g\_M)$. 




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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](https://lawrencemouille.wordpress.com/2017/02/27/cheeger-deformations/) for an animation of the Cheeger deformed plane under the circle action given by rotation about the origin.


