Bertrand–Diquet–Puiseux theorem

In the mathematical study of the differential geometry of surfaces, the Bertrand–Diquet–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 C.F. Diquet.

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–Diquet–Puiseux theorem asserts that

$K(p) = \lim_{r\to 0^+} 3\frac{2\pi r-C(r)}{\pi r^3} = \lim_{r\to 0^+}12\frac{\pi r^2-A(r)}{\pi r^4}.$

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; Diquet, C.F.; Puiseux, V (1848), "Démonstration d'un théorème de Gauss", Journal de Mathématiques, 13: 80–90
• Spivak, Michael (1999), A comprehensive introduction to differential geometry, Volume II, Publish or Perish Press, ISBN 0-914098-71-3