- Or, rather, a generalization of the Gauss curvature at a class of points which, when the Gauss curvature exists, are elliptic.
- It may be of interest to compare Busemann and Feller, Krümmungseigenschaften Konvexer Flächen, Acta Mathematica, vol. 66, 1936, pages 1–47, especially pages 22–23.
- Professor R. E. Gilman has called my attention to a considerable recent literature by Busemann and Feller on the differential geometry of convex surfaces in the large, in which the Gauss curvature (when it exists) is shown to be given by limiting processes involving the area which we denote in section 5 by A. The latest of these publications is Bemerkungen zur Differentialgeometrie der konvexen Flächen, III.
- A Course of Pure Mathematics, seventh edition, Cambridge, 1938.
- It can be shown similarly that Theorem 2.1 remains valid if we replace the condition k <0<h by the weaker condition that (2.2) (&verbar (or mid)h&verbar (or mid) + &verbar (or mid)k&verbar (or mid))/&verbar (or mid)h - k &verbar (or mid) remains bounded.
- Indeed, this particular f(x) remains of the form (3.1) if the condition k<0<h therein is replaced by the condition (2.2).
- These conditions may be regarded as analogs of the conditions for curvature of plane curves given in §1.
- A condition necessary and sufficient that this be the case is easily seen to be the following: fx2(0, 0)fy2(0, 0) > [fxy(0, 0)]2.
- Lest one suppose that the conditions for existence of G-curvature imply that K be of the form (4.3), which is so familiar in the differential geometry of “well-behaved” surfaces, we call attention to the paraboloidal surface (4.4) z = ø(θ)r2/2, which is of the form (4.1) for any continuous, non-vanishing ø(θ) of period π, and for which K = ø(θ).
- It may be of interest to compare this result with the fact that the so-called mean curvature of a surface is twice the mean of K with respect to θ. We have just shown that the Gauss curvature, when positive, is the square of the harmonic mean of K with respect to θ.
- Ibid.
- The condition u→0 means, of course, that the line of intersection of the plane z = mx+ny+q with the xy-plane recedes to infinity.
The American Mathematical Monthly
Volume 47, 1940 - Issue 9