Yamabe invariant

In mathematics, in the field of differential geometry, the Yamabe invariant (also referred to as the sigma constant) is a real number invariant associated to a smooth manifold that is preserved under diffeomorphisms. It was first written down independently by O. Kobayashi and R. Schoen and takes its name from H. Yamabe.

Let ${\displaystyle M}$ be a compact smooth manifold (without boundary) of dimension ${\displaystyle n\geq 2}$. The normalized Einstein–Hilbert functional ${\displaystyle {\mathcal {E}}}$ assigns to each Riemannian metric ${\displaystyle g}$ on ${\displaystyle M}$ a real number as follows:

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