Differential Gemometry

Cerrado Publicado Nov 2, 2014 Pagado a la entrega
Cerrado Pagado a la entrega

question 17, page 55 Shifrin's book

Below is the question : 17

How is the Frenet frame along an asymptotic curve related to the geometry of the surface?

Suppose K.P / < 0. If C is an asymptotic curve with .P / ¤ 0, prove that its torsion satisfies

p

j.P /j D

K.P /. (Hint: If we choose an orthonormal basis fU; Vg for T P .M / with U tangent

to C , what is the matrix for S P ? See the Remark on p. 47.)

Note : please check the file also in order to solve the question .

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