Bertrand–Diguet–Puiseux theorem
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Short description: Gives the Gaussian curvature of a surface from the length of a geodesic circle or its area
In the mathematical study of the differential geometry of surfaces, the Bertrand–Diguet–Puiseux theorem expresses the Gaussian curvature of a surface in terms of the circumference of a geodesic circle, or the area of a geodesic disc. The theorem is named for Joseph Bertrand, Victor Puiseux, and Charles François Diguet.
Let p be a point on a smooth surface M. The geodesic circle of radius r centered at p is the set of all points whose geodesic distance from p is equal to r. Let C(r) denote the circumference of this circle, and A(r) denote the area of the disc contained within the circle. The Bertrand–Diguet–Puiseux theorem asserts that
- [math]\displaystyle{ K(p) = \lim_{r\to 0^+} 3\frac{2\pi r-C(r)}{\pi r^3} = \lim_{r\to 0^+} 12\frac{\pi r^2-A(r)}{\pi r^4}. }[/math]
The theorem is closely related to the Gauss–Bonnet theorem.
References
- Berger, Marcel (2004), A Panoramic View of Riemannian Geometry, Springer-Verlag, ISBN 3-540-65317-1
- Bertrand, J; Diguet, C.F.; Puiseux, V (1848), "Démonstration d'un théorème de Gauss", Journal de Mathématiques 13: 80–90, http://sites.mathdoc.fr/JMPA/PDF/JMPA_1848_1_13_A11_0.pdf
- Spivak, Michael (1999), A comprehensive introduction to differential geometry, Volume II, Publish or Perish Press, ISBN 0-914098-71-3
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