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 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 [3] is the only scalar invariant (or absolute invariant) linear in the second derivatives of the metric tensor .

See also

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 is linear in the second derivatives of the metric tensor , quadratic in the first derivatives and contains the inverse matrix which is a rational function of the components .

References

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