Ιn 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 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
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
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.
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 \( {\displaystyle g^{\mu \nu },} \) which is a rational function of the components \( g_{\mu \nu }. \)
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.
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