Bertrand–Diguet–Puiseux theorem

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

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

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" (PDF), 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