The Yamabe Invariant in Two Dimensions
In the case that, (so that M is a closed surface) the Einstein–Hilbert functional is given by
where is the Gauss curvature of g. However, by the Gauss–Bonnet theorem, the integral of the Gauss curvature is given by, where is the Euler characteristic of M. In particular, this number does not depend on the choice of metric. Therefore, for surfaces, we conclude that
For example, the 2-sphere has Yamabe invariant equal to, and the 2-torus has Yamabe invariant equal to zero.
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