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Working on the four-sphere S4 , a flat four-torus, S x S , or a compact hyperbolic space, with a metric which is an arbitrary positive function times the standard one, we give explicit formulas for the functional determinants of the conformai Laplacian (Yamabe operator) and the square of the Dirac operator, and discuss qualitative features of the resulting variational problems. Our analysis actually applies in the conformai class of any Riemannian, locally symmetric, Einstein metric on adoi:10.1090/s0002-9939-1991-1050018-8 fatcat:dhpdxs4z2fdrdpl5xh7jmmcudm