Affine curvature of plane geodesic lines on affine hypersurfaces

Abstract

We establish a necessary and sufficient condition for a geodesic line on a nondegenerate hypersurface to be a plane curve. We deduce a formula for the affine curvature of a plane geodesic line on the affine hypersurface in terms of the affine fundamental form and the shape operator. We present the definition of transverse curvature and determine some of its elementary properties.