Manifolds in quantum chemistry Sometimes the concept "manifold" is used in quantum chemistry, for instance,
in Calculation of size‐intensive transition moments from the coupled cluster singles and doubles linear response function:


  where $\tau_{2s}$ and $\tau_{2t}$ denote the singlet-singlet and triplet-triplet spin coupled double excitation manifolds where spin couplings initially are carried out on the occupied-occupied and unoccupied-unoccupied orbital indices. 


However, in general, manifold refers to a geometric object that is locally Euclidian. This is not essential to the coupled cluster method. 

My question is, what is the motivation to introduce manifolds in the coupled cluster method?
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.