By Bernard Diu
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Extra resources for La mathématique du physicien
Since, moreover, T is a continuous mapping, this proves that it is topological. Let us now show that Tis a geodesic mapping. For this it is enough to demonstrate that the inverse map r- 1 is geodesic. Let us take any plane in £ 0 • It is given by the equation (ay) +b = 0, where a is a vector, and b is a scalar. Substituting Tx into this equationinplaceofy, we obtain the equation for its image in R under r- 1 • The equation (a, Tx) +b = 0 becomes linear with respect to x after multiplication by eo (x + Ax), and, consequently, is the equation of plane R.
Let us join the 38 II. CONVEX BODIES AND CONVEX SURFACES ends of the curve 1 by a simple geodesic open-polygon ak be the angles between consecutive links in the polygon, and a and ~ be angles formed by the initial and terminal link with 1 , all angles being taken from the side of the region bounded by the curves 1 and r. The right swerve of the curve is given by r in the right-hand semi-neighborhood. Let The left swerve is defined similarly. The swerve has the following properties. If the point C divides the curvey into 11 and12.
Then we obtain: And this is just the system from which we find the coefficients of the corresponding derivation formulae for Euclidean surfaces. We can draw a similar conclusion with regard to the coefficients and B! Bi . , = r! , = B2Xu + B2Xv, Xu,,= where the coefficients r andB are expressed in terms of E, F, G, L, M and N in exactly the same way as the corresponding coefficients of the derivation formulae for Euclidean space surfaces. The conditions which the coefficients of the first and second fundamental forms have to satisfy may be obtained in the same way as for Euclidean space surfaces.