  Let $Y$ be an algebraic manifold of dimension 3 with $H^i(Y, \Omega^j_Y)=0$
for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$. Let $X$ be a smooth
completion of $Y$ such that the boundary $X-Y$ is the support of an effective
divisor $D$ on $X$ with simple normal crossings. We prove that the
$D$-dimension of $X$ cannot be 2, i.e., either any two nonconstant regular
functions are algebraically dependent or there are three algebraically
independent nonconstant regular functions on $Y$. Secondly, if the
$D$-dimension of $X$ is greater than 1, then the associated scheme of $Y$ is
isomorphic to Spec$\Gamma(Y, {\mathcal{O}}_Y)$. Furthermore, we prove that an
algebraic manifold $Y$ of any dimension $d\geq 1$ is affine if and only if
$H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and it is regularly separable,
i.e., for any two distinct points $y_1$, $y_2$ on $Y$, there is a regular
function $f$ on $Y$ such that $f(y_1)\neq f(y_2)$.
