Say we have a free quantum mechanical particle constrained to move on the surface of S^6; x1^2 + x2^2 + x3^2 + x4^2 + x5^2 + x6^2 + x7^2 = R^2 Can we add a simple potential such that for low energies the particle is "constrained" to move in some sub space of S^6, say S^3, but for higher energies the particle has "full access" of the space S^6. What might be such a simple potential? Does this potential have a topology? Is their a group represented by the quantum mechanics of a free particle constrained to move on the surface S^3? Thanks for your help.