Yamabe wanted to solve the Poincare conjecture (see 9.14). For this he thought, as a first step, to exhibit a metric with constant scalar curvature. He considered conformal metrics (the simplest change of metric is a conformal one), and gave a proof of the following statement “On a compact Riemannian manifold (M, g), there exists a metric g′ conformal to g, such that the corresponding scalar curvature R′ is constant”. The Yamabe problem was born, since there is a gap in Yamabe’s proof. Now there are many proofs of this statement. We will consider some of them, but if the reader wants to see one proof, he has to read only sections 5.11, 5.21, 5.29 and 5.30.