An affine invariant characterization of flat gravity curves (Q1034940)

In the case of a locally strongly convex surface \(x(M)\subset{\mathbb R}^{3}\), \textit{W. Blaschke} proved that the affine normal of \(x(M)\) at a point \(p\in M\) coincides with the tangent to the so called \textit{gravity} \textit{curve} at \(p\) [Vorlesungen über Differentialgeometrie, und geometrische Grundlagen von Einsteins Relativitätstheorie. II. Berlin: J. Springer, IX u. 259 S. gr. \(8^{\circ}\) (1923). (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellung mit besonderer Berücksichtigung der Anwendungsgebiete. Band VII.) (1923; JFM 49.0499.01)]. The gravity curve is defined as follows: Consider intersections of the surface \(x(M)\) with planes parallel to the tangent plane at \(p\); then the centers of gravity of these intersections form a differentiable curve. The result of Blaschke was generalized to locally convex hypersurfaces in \({\mathbb R}^{n+1}\) by \textit{K. Leichtweiß} [Arch. Math. 53, No.~6, 613--621 (1989; Zbl 0661.53005)]. The proofs of Blaschke and Leichtweiß use Taylor expansions of \(x(M)\) up to order three. In the present paper the author calculates two further coefficients in order to link second order properties of the gravity curve to affine invariants of \(x(M)\). Denoting the gravity curve to be \textit{flat} at \(p\) iff the Euclidean curvature vanishes (which is indeed affinely invariant) the main result is: Let \(x:M^n \rightarrow {\mathbb R}^{n+1}\) be a locally convex affine hypersurface. Then the gravity curve is flat at \(p\in M\) if and only if \(4(n-1)dJ| _p+3(n+2)dH| _p=0\), where \(J\) and \(H\) denote the affine Pick invariant and the affine mean curvature respectively.