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In differential geometry, Vermeil's theorem essentially states that the scalar curvature is the only (non-trivial) absolute invariant among those of prescribed type suitable for Albert Einstein’s theory of General Relativity. The theorem was proved by the German mathematician Hermann Vermeil in 1917.

Standard version of the theorem

The theorem states that the Ricci scalar ${\displaystyle R}$ is the only scalar invariant (or absolute invariant) linear in the second derivatives of the metric tensor ${\displaystyle g_{\mu \nu }}$.

See also

• Scalar curvature
• Differential invariant
• Einstein–Hilbert action
• Lovelock's theorem

Notes

1. Let us recall that Ricci scalar ${\displaystyle R}$ is linear in the second derivatives of the metric tensor ${\displaystyle g_{\mu \nu }}$, quadratic in the first derivatives and contains the inverse matrix ${\displaystyle g^{\mu \nu },}$ which is a rational function of the components ${\displaystyle g_{\mu \nu }}$.

References

• Vermeil, H. (1917). "Notiz über das mittlere Krümmungsmaß einer n-fach ausgedehnten Riemann'schen Mannigfaltigkeit". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse. 21: 334–344.
• Weyl, Hermann (1922). Space, time, matter. Translated by Brose, Henry L. Courier Corporation. ISBN 0-486-60267-2. JFM 48.1059.12.

• Theorems in differential geometry
• Invariant theory