Answer Happy

2. Flat surfaces in R can be bent only along straight lines (see Fig. 6.9). If kı = 0 but k, is never zero, show that the principal curves of k are line seg- ments in Rº. (Hint: With {E} principal and a" = VEE, use Thm. 1.4.) =

ODA A I FIG. 6.9

1.4 Theorem If E1, E2, E3 is an adapted frame field on M CR, and vis tangent to M at p, then V.E: = < 0;(v)E;(p) (15is 3). j=1 The usual interpretation of the connection forms may be read from these equations, and it bears repetition: 00:(v) is the initial rate at which E, rotates toward E; as p moves in the v direction. Since Ez is a unit normal vector field on M, the shape operator of M can be described by connection forms.