  Path integrals for particles in curved spaces can be used to compute trace
anomalies in quantum field theories, and more generally to study properties of
quantum fields coupled to gravity in first quantization. While their
construction in arbitrary coordinates is well understood, and known to require
the use of a regularization scheme, in this article we take up an old proposal
of constructing the path integral by using Riemann normal coordinates. The
method assumes that curvature effects are taken care of by a scalar effective
potential, so that the particle lagrangian is reduced to that of a linear sigma
model interacting with the effective potential. After fixing the correct
effective potential, we test the construction on spaces of maximal symmetry and
use it to compute heat kernel coefficients and type-A trace anomalies for a
scalar field in arbitrary dimensions up to d=12. The results agree with
expected ones, which are reproduced with great efficiency and extended to
higher orders. We prove explicitly the validity of the simplified path integral
on maximally symmetric spaces. This simplified path integral might be of
further use in worldline applications, though its application on spaces of
arbitrary geometry remains unclear.
