On the curvatures of a curve in Riemann space

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Introduction. The curvature and torsion of a curve in ordinary space have three properties which it is the purpose of this paper to attempt to extend to the curvatures of a curve in Riemann space. First, if the curvature vanishes identically the curve is a straight line; if the torsion vanishes identically the curve lies in a plane. Second, the distances of a point of the curve from the tangent line and the osculating plane at a nearby point are given approximately by formulas involving the curvature and torsion. Third, the curvature of a curve at a point is the curvature of its projection on the osculating plane at the point. In extending to Riemann space we take as the Riemannian analogue of the line or plane, a geodesic space generated by geodesics through a point. Such a space possesses the property of the line or plane of being determined by the proper number of directions given at a point, but it will not in general have the three properties given above. On the other hand, if we take as the analogue of line or plane only totally geodesic spaces, then, if such osculating planes exist, the three properties will hold. Curves with a vanishing curvature. Given a curve C: x = x(s), i = 1, n, in a Riemann space Vn with fundamental tensor gij (assumed definite). Following Blaschkel we write the Frenet formulas for the curve. The n associate vectors are given by