Download e-book for kindle: A First Course in Differential Geometry by Vaisman

By Vaisman

ISBN-10: 0824770633

ISBN-13: 9780824770631

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R s, 6 , are substituted in the expressions (73) where, for the sake of brevity, = 1,2,8 = l, 2, we have put CURVES IN SPACE 28 The and coefficients of U, F, Win (73) are of the same form as the expressions (71) for a, I, X ; ft m, p, ; 7, n, v. Moreover, the are equivalent to (72). Hence these equations of condition (59) coefficients are the direction-cosines of three fixed directions in another. * These results may be stated thus : be If the general solution of equation (65) the curve whose radii of respectively is given first and second curvature are p and T "by must be remarked that the new axes of coordinates are not necessarily real, so that when it is important to know whether the It will be advisable to consider the general formulas example of this will be given later.

6. satisfy equations similar to (85), They namely are the direction-cosines of the If a, 6, c same direction, with respect to the mov- = ing trihedral at J/, we must have ax a cos co+c sin o>, a sin a> c cos o>, for all possible cases, 11 5, tfj = = as + When enumerated above. these values are sub- stituted in the above equations, we a sin f cos < ds sin o> cosw &> :] 1 k [sin cos co k o> 1 p fsin , ^ . T T t T Sin . , , -j- (e o, Pl sin co + a cos da> A = 0. G)J Since these equations must be true for every fixed line, the coefficients of a, 6, c in each of these equations must be zero.

S f denote coordinates referring to the axes at 77, M those with reference to the axes at Since the rate of change of a (see fig. 4). f , and zero is and :=! at J/, the cosine of the angle between the f- and f'-axes is 1 to within terms of higher than the first order in As. Likewise the cosine of the angle between the - and Tj'-axes cosines of the angles between all the axes, and the results is As/p. We calculate the may be tabulated as follows: f (81) FIG, 4 Let at P be M are f , a point whose coordinates with respect to the trihedral 77, f.

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A First Course in Differential Geometry by Vaisman

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