Manifolds in quantum chemistry
The use of the word "manifold" in most of the quantum chemistry literature is purely as a synonym for multiple or different. Usually it appears when the number may be very large. Nothing more complicated than that.

For a simpler example than coupled cluster, consider configuration interaction with only single excitations (CIS). One of the key quantities is the set of single excitation amplitudes, $t_{ia}$ or $t_{i}^{a}$ or $k_{ai}\hat{a}^{\dagger}\hat{i}$ depending on your preference for notation. $i$ is an occupied MO index, and $a$ is a virtual MO index. We might say that the set $\{t\}$ forms a manifold, and the number quickly becomes large as you increase the basis set size: there will be $N_{ov} = N_{\text{occ}} \times N_{\text{virt}}$ single excitation amplitudes.