The flat set of a Cheeger deformation, and an isolated zero-curvature plane on CP^2 x CP^2

Cheeger deformations are the principal source of nonnegatively curved metrics, and the size of the set of zero-curvature planes they leave behind decides whether the curvature can be pushed all the way to positive. This paper determines that set exactly for Riemannian products carrying a diagonal group action. For a diagonal action of a compact group of rank one, the zero-curvature planes of the Cheeger deformation are precisely the projectivisation of the intersection of the two vertical images, transported by Cheeger's endomorphism, so that their dimension is dim V1 + dim V2 − d − 1, where d is the orbit dimension. This recovers Müter's classical computation for S2×S2, including its exceptional set, and identifies the flat set of S3×S3 as a projective plane. In higher rank the count is governed by Richardson's commuting variety. For CP2×CP2 with the diagonal SU(3) the flat set has dimension 7 − d, and the orbit dimension is 7 away from the codimension-two locus of orthogonal pairs of lines, 6 on it, and 4 on the diagonal: at every point of the open stratum the zero-curvature plane is isolated. The plane is exhibited explicitly over the rationals, and its isolation is the statement that an explicit 12×12 rational Hessian has rank 12, verified by the Lean 4 kernel through a modular inverse certificate. Isolation makes the plane canonical, and the resulting two-plane distribution is integrable and totally geodesic: the open stratum is foliated by flat totally geodesic surfaces tangent to the zero-curvature planes, with the canonical frame at the constant angle arccos(2/3). Both geometric ingredients that the third-order mechanism of Brendle and Hung uses in dimension four are thus present in dimension eight. Complete code reproducing every computation, and the Lean 4 certificate, are included.