Vermeil's theorem

In differential geometry, Vermeil's theorem essentially states that that the scalar curvature is the only (non-trivial) absolute invariant among those of prescribed type suitable for Einstein’s theory.^[1] The theorem was proved by the German mathematician Hermann Vermeil in 1917.^[2]

Standard version of the theorem

The theorem states that the Ricci scalar $R$^[3] is the only scalar invariant (or absolute invariant) linear in the second derivatives of the metric tensor $g_{\mu\nu}$.

See also

• Scalar curvature
• Differential invariant
• Einstein–Hilbert action

Notes

1. Kosmann-Schwarzbach, Y. (2011), The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century: Invariance and Conservation Laws in the 20th Century, New York Dordrecht Heidelberg London: Springer, p. 71, doi:10.1007/978-0-387-87868-3, ISBN 978-0-387-87867-6
2. H. Vermeil (1917). "Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen". Mathematisch physikalische Klasse 21: 334–344. doi:10.1007/BF01457097.
3. Let us recall that Ricci scalar $R$ is linear in the second derivatives of the metric tensor $g_{\mu\nu}$, quadratic in the first derivatives and contains the inverse matrix $g^{\mu\nu},$ which is a rational function of the components $g_{\mu\nu}$.

References

• H. Vermeil (1917). "Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen". Mathematisch physikalische Klasse 21: 334–344. doi:10.1007/BF01457097.