Hidehiko Yamabe, in his famous paper entitled On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), pp. 21-37, wanted to solve the Poincaré conjecture. 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 , there exists a metric conformal to , such that the corresponding scalar curvature is constant”.
The Yamabe problem was born, since there is a gap in Yamabe’s proof. Now there are many proofs of this statement.
Let us recall the question. Let be a compact -Riemannian manifold of dimension , is its scalar curvature. The problem is:
“Does there exists a metric , conformal to , such that the scalar curvature of the metric is constant?”.
Let us consider the conformal metric with . If and denote the Chrisoffel symbols relating to and , respectively, then
If we consider the conformal deformation in the form (with , ), the scalar curvature satisfies the equation
where . So, Yamabe problem is equivalent to solving the above equation with and the solution must be smooth and strictly positive.
Link to PDF file of the paper Osaka Math. J. 12 (1960), pp. 21-37 can be found here http://projecteuclid.org/euclid.ojm/1200689814