I want to map the pdf of the Normal distribution given in Pennec to its tangent space, given mean $\overline x$ and precision matrix $\Gamma$. The Normal law on a manifold $\mathcal M$ is given as $$N_{(\overline{x}, \Gamma)}(y) = k \exp{((-\overrightarrow{\overline{x}y}^T \Gamma \overrightarrow{\overline{x}y}})/2)$$
I have taken the mean to be $0$ and the precision matrix to be the identity matrix. In order to transform the pdf to it's tangent space, I use equation in Sec 3.3 (expression of density in a chart): $$\rho_{x}(y) = p_{\mathrm x}(y) \sqrt{\det (G(y))}$$ where $\rho$ is the density in a chart $x = \pi(x)$, $G$ is the Riemannian metric, and $p$ would be the pdf on the manifold, i.e., $N_{(\overline x, \Gamma)}(y)$.
My confusion comes up here: Do I use $\exp (y)$ (the exponential mapping, not $e$) instead of $y$ in the volume form part of the second equation)? If so, I get the following explicit form of the Normal distribution (given mean $0$, precision matrix as identity) in terms of the usual coordinate chart in $2$ dimensions:
$$f(y) = k.e^{-0.5 ||y||^2} \cdot \frac{4}{(1 - \tanh^2{||y||})^2} $$
and the plot looks as follows (plotted via GeoGebra, taking $k=1$ for simplicity):
And this plot baffles me because I'm not sure what distribution to use for this in $\mathbb R^2$ to perform Rejection Sampling on it.