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A surgery formula for the smooth Yamabe invariant

where [g0] denotes the set of all metrics conformal to g and of volume 1. The solution of the Yamabe problem (see e.g. [10] for a good overview) states that this infimum is always attained. The scalar curvature of the mimimzing metric satisfies Scalg ≡ Y (M, [g0]). It is not difficult to show that Y (M, [g0]) ≤ Y (S) = n(n− 1)ωn where S denotes the sphere with the standard metric, and where ωn = Vol(S). The smooth Yamabe invariant, also called Yamabe number or Schoen’s σinvariant is then defined as