Vermeil's theorem

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.[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 ${\displaystyle R}$[3] 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. 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. 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.
3. 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.