We are interested in the optimal constant problem for the critical Sobolev embedding of the space Hk(M) into Lp*(M), where k is a positive integer, (M,g) is a closed Riemannian manifold of dimension n>2k, and where p*= 2n/(n-2k) is the critical Sobolev exponent. We show that the optimal constant in front of the leading-order term of the Sobolev norm can be taken to be the same as in the Euclidean setting. This result extends the cases k=1 showed by Hebey-Vaugon in 1996, and k=2 showed by Hebey in 2003. For k≥3, we propose a new strategy of proof that works independently of the order of the problem: We study the pointwise behavior of an asymptotically singular sequence of positive solutions of minimal energy to the critical polyharmonic equation (∆g+α)ku=up*-1 in M, as α→∞. We develop a new analytical formalism that provides a precise description of the solutions near the blow-up point, allowing to overcome the difficulties that come with the higher-order setting and the diverging coefficients in the equation. Based on a recent joint work with F. Robert, we will show that the geometry of the manifold plays a crucial role in the validity of the optimal inequality, depending on the nature of the lower-order remainder term.