  We have recently studied a simplified version of the path integral for a
particle on a sphere, and more generally on maximally symmetric spaces, and
proved that Riemann normal coordinates allow the use of a quadratic kinetic
term in the particle action. The emerging linear sigma model contains a scalar
effective potential that reproduces the effects of the curvature. We present
here further details on the construction, and extend its perturbative
evaluation to orders high enough to read off the type-A trace anomalies of a
conformal scalar in dimensions d = 14 and d = 16.
