A Cheeger-Müter metric is a Riemannian metric on 
obtained by applying a Cheeger deformation
to the standard product metric with respect to
the diagonal group action of the special
orthogonal group 
. The action sends 
to 
for 
. Cheeger (1973) introduced the construction, and
Müter (1987) studied the resulting Riemannian
metrics and their curvature in detail.
A Cheeger-Müter metric has nonnegative sectional curvature. At each generic point, exactly one tangent two-plane has zero sectional
curvature, while all other tangent two-planes have positive sectional
curvature. On the diagonal 
and antidiagonal 
, a one-parameter family of two-planes with
zero sectional curvature remains (Müter
1987, Ziller 2009).
Brendle and Hung (2026) use a Cheeger-Müter metric as the starting point for their announced construction of a Riemannian metric
with positive sectional curvature on 
. Their metric with positive
sectional curvature is a third-order perturbation
of, rather than itself, the Cheeger-Müter metric.
with Positive Sectional Curvature." 19 Aug
2026.