X_j,[\mathbf X_i,\mathbf X^j]]$ taking the role of a noncommutative Laplacian). The string state $|\psi {\rangle \hspace{-0.2em} \rangle}$ obeys to a Schrödinger equation with quantum Hamiltonian $H^{eff}(t) = \sigma^i \otimes(X_i(t)-x_i(t))$ [@Berenstein; @Badyn].\
The initial conditions concerning $X_i(t=0)$ fix an initial topology for the noncommutative D2-brane. Usually, the works concerning such a model use a fuzzy sphere at $t=0$. In order to test the independance of our analysis from the initial topology we consider also a fuzzy ellipsoid, a fuzzy cylinder, a fuzzy torus and two noncompact noncommutative manifolds, a noncommutative plane and a fuzzy hyperboloid. The characteristics of these models are presented table \[NCmanifold\].

+-----------------------+-----------------------+-----------------------+
| [*manifold*]{}        | [*algebra*]{}         | [*coordinates*]{}     |
+:======================+:======================+:======================+
| Fuzzy sphere          | $j$-representation of | $X_i = \frac{1}{j} J_ |
|                       | $\mathfrak{su}(2)$    | i$                    |
|                       | ($N=2j+1$)\           |                       |
|                       | $[J_i,J_j] = \imath \ |                       |
|                       | epsilon_{ijk} J_k$    |                       |
+-----------------------+-----------------------+-----------------------+
| Fuzzy ellipsoid       |                       | $X_1=\frac{2}{j} J_1$ |
|                       |                       | ,                     |
|                       |                       | $X_2=\frac{2}{j} J_2$ |
|                       |                       | ,                     |
|                       |                       | $X_3=\frac{1}{j} J_3$ |
+-----------------------+-----------------------+-----------------------+
| Fuzzy cylinder        | $j$-representation of | $X_1 = \frac{1}{j} \f |
|                       | $\mathfrak{iso}(2)$   | rac{V+V^\dagger}{2}$, |
|                       