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

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 $g^{\mu \nu },$ 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.

[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 $R$ 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 $g_{\mu \nu }$ .

[1]

[2]

[3]

Standard version of the theorem

See also

Notes

References