  We present a new method for the solution of PDEs on manifolds $\mathbb{M}
\subset \mathbb{R}^d$ of co-dimension one using stable scale-free radial basis
function (RBF) interpolation. Our method involves augmenting polyharmonic
spline (PHS) RBFs with polynomials to generate RBF-finite difference (RBF-FD)
formulas. These polynomial basis elements are obtained using the
recently-developed \emph{least orthogonal interpolation} technique (LOI) on
each RBF-FD stencil to obtain \emph{local} restrictions of polynomials in
$\mathbb{R}^3$ to stencils on $\mathbb{M}$. The resulting RBF-LOI method uses
Cartesian coordinates, does not require any intrinsic coordinate systems or
projections of points onto tangent planes, and our tests illustrate robustness
to stagnation errors. We show that our method produces high orders of
convergence for PDEs on the sphere and torus, and present some applications to
reaction-diffusion PDEs motivated by biology.
