I am having a hard time figuring out how to formalize this problem, so I am asking for references that I can follow.
Let's say that $X$ is a random variable defined on the probability space $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d),\mu)$ and let $M$ be a $r<d$ dimensional connected manifold embedded in $\mathbb{R}^d$. I have that $\mathbb{P} (X \in M) = 1$. Is there a way to find a sequence of gaussian distributions indexed by $n$, $$ X_n \sim N(\mu_n, \Sigma_n),$$ where each diagonal element of $\Sigma_n$ is strictly positive, such that $$ X_n \xrightarrow{D} X $$
As an example that I have in mind if $M$ is just a point $m \in \mathbb{R}^d$, then we can let $$ X_n \sim N\left(m, \frac{1}{n} I_d\right), $$ Where $I_d$ is the identity matrix. I do not know how to think about this problem for the general case and I would be very glad if you can point out textbooks or articles.