The paper represents the lectures given by the author at the 16th Winter School on Geometry and Physics, Srni, Czech Republic, January 13-20, 1996. He develops in an elegant manner the theory of conformal covariants and the theory of functional determinant which is canonically associated to an elliptic operator on a compact pseudo-Riemannian manifold. The presentation is excellently realized with a lot of details, examples and open problems.

Let $(M^n,g)$ be a closed Riemannian manifold and $g_E$ the Euclidean metric. We show that for $m>1$, $\left(M^n\times {𝐑}^m,(g+g_E)\right)$ is not conformal to a positive Einstein manifold. Moreover, $\left(M^n\times {𝐑}^m,(g+g_E)\right)$ is not conformal to a Riemannian manifold of positive Ricci curvature, through a radial, integrable, smooth function, $\varphi :{𝐑}^{𝐦}\to {𝐑}^{+}$, for $m>1$. These results are motivated by some recent questions on Yamabe constants.