By Nomizu K., Sasaki T.

ISBN-10: 0521441773

ISBN-13: 9780521441773

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**Additional info for Affine differential geometry. Geometry of affine immersions**

**Example text**

But by the Heine-Borel covering theorem the bounded closed interval s0 < s < sx can be covered by choosing a finite number of the open neighborhoods within each of which 6(s) is continuous. It is then clear that the definition of 6 in each of the neighborhoods can be chosen so that 6 is continuous throughout the interval. Fig. 6 The circular image. 11) itself cannot be so generalized. The unit tangent vectors X(s) are carried to the fixed point 0, so that their end points trace out an arc of the unit circle centered at 0 (cf.

Note also that a dot was used to denote differentiation with respect to this special parameter—a notation which will be used frequently in Chap. II P L A N E CURVES 20 this book from now on in dealing with curves for which the arc length is the parameter. 8). 8. Arc Length as an Invariant Since the length of a given arc of a curve is defined with the aid of vectors and an invariant operation on them, it is clear that it is a number that has a value independent of the choice of a coordinate system.

24) thus become v l = a 2V2> anc ^ v 2 = "-«2V1' It is known that fj = KV2 [cf. 17)]. 27) v2 = -KV1 *, in which the *'s have been put in purely to emphasize the skew-symmetric character of the right-hand sides. These differential equations are called the equations of Frenet for the special case of plane curves. Sec. 14 EVOLUTE AND INVOLUTE OF A PLANE CURVE 29 These equations form a system of linear homogeneous differential equations for the determination of the vectors v^«), v2(«). They have a uniquely determined solution for these vectors as soon as the initial values Vj(0), v2(0) are prescribed, provided that K(S) is any arbitrarily given continuous function of s (see Appendix B, for example).

### Affine differential geometry. Geometry of affine immersions by Nomizu K., Sasaki T.

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