By B. A. Dubrovin, A. T. Fomenko, S. P. Novikov (auth.)
Read Online or Download Modern Geometry — Methods and Applications: Part I. The Geometry of Surfaces, Transformation Groups, and Fields PDF
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Extra info for Modern Geometry — Methods and Applications: Part I. The Geometry of Surfaces, Transformation Groups, and Fields
Distances at perihelion). This law can be regarded as a consequence oft he fact that the behaviour of such a mechanical system, consisting of a particle moving in a Newtonian attractive field of potential
Xi - xi) 2 . (10) The space-time separation of two distinct events P 1 , P 2 can be positive, negative, or even zero (see §6). Rf ,2 , namely by pseudo-spherical co-ordinates, which are defined as follows. 2 . ·< p < 00 , x1 = p sinh x cos cp, 0 ::; X < oo, x2 = p sinh x sin 0 ::; cp < 2n. cp, (II) Hence (xo)2 _ (x 1 )2 _ (x2)2 = pz ~ O. e. in the interior of the cone (x 0 ) 2 = (x 1) 2 + (x 2) 2 (Figure 4). All points of this region except those on the x 0 -axis are ordinary points of the pseudo-spherical co-ordinate system.
Secondly, the number x(l) (which may be positive, negative or 0) is called the torsion of the space curve (at the value I of the natural parameter). From the third ofthe equations (22) we see that its magnitude IxI = Idbjdll. If the curve is planar (lies in some plane), then b is a constant vector, so that dbjdl = 0, and therefore x = 0. 3. We begin by proving the third formula dbjdl = xn. 2 that b= [tl, n] + [v, n]. (23) 46 l. Geometry in Regions of a Space. ln. Hence li = o:v + (Jb for some numbers o:, (J.
Modern Geometry — Methods and Applications: Part I. The Geometry of Surfaces, Transformation Groups, and Fields by B. A. Dubrovin, A. T. Fomenko, S. P. Novikov (auth.)