Gaussian distribution propoprties

Good day,

I've a question regarding Gaussian distribution properties over lattices :

Let $\mathcal{L}$ := $ \mathcal{L}(\,b_{1}$,..., $b_{m})$ be a lattice over $\mathbb{R}^{n}$, and $W$ = span($b_{1}$,..,$b_{m}$)$^{\perp}$. define $\pi_{W}$ to be the orthogonal projection onto $W$.

If i sample a vector b from a Gaussian distribution of support $\mathcal{L}$, standard deviation parameter $s$ and center parameter $c \in$ span($\mathcal{L}$).

Can i pretend that $\pi_{W}(\,b)\,$ can be sampled from a Gaussian distribution of support $\pi_{W}(\,\mathcal{L})\,$, standard deviation parameter $s$ and center parameter $\pi_{W}(\,c)\,$?

Thanks

As Daniel's comment mentions, the answer to your question appears to be "no, for trivial reasons" (plausibly due to some typo in your question).

Still, any question of this form has a relatively straightforward answer --- namely to look into Improved Discrete Gaussian and Subgaussian Analysis for Lattice Cryptography. In particular, section 3 discusses how Gaussians on lattices transform under linear transformations (including projections), and seems to be what you are interested in.