  Let M be a compact connected oriented Riemannian manifold. The purpose of
this paper is to investigate the long time behavior of a degenerate stochastic
differential equation on the state space $M\times \mathbb{R}^{n}$; which is
obtained via a natural change of variable from a self-repelling diffusion
taking the form $$dX_{t}= \sigma dB_{t}(X_t) -\int_{0}^{t}\nabla
V_{X_s}(X_{t})dsdt,\qquad X_{0}=x$$ where $\{B_t\}$ is a Brownian vector field
on $M$, $\sigma >0$ and $V_x(y) = V(x,y)$ is a diagonal Mercer kernel.
  We prove that the induced semi-group enjoys the strong Feller property and
has a unique invariant probability $\mu$ given as the product of the normalized
Riemannian measure on M and a Gaussian measure on $\mathbb{R}^{n}$. We then
prove an exponential decay to this invariant probability in $L^{2}(\mu)$ and in
total variation.
