Abstract
Yamabe wanted to solve the Poincaré 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.
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© 1998 Springer-Verlag Berlin Heidelberg
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Aubin, T. (1998). The Yamabe Problem. In: Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-13006-3_5
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DOI: https://doi.org/10.1007/978-3-662-13006-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08236-8
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