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

Clearly,

so

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

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