Answer to Question #82440 in Differential Geometry | Topology for Zaighum abbas

Question

Q no. 3) let
f(t)= { e^-1/t^2 if t not equal to 0
{0. If t=0
You may assume f is C ^infinity with f^(n) =0 for all n. Let a(t) be given by
a(t) = {(t,f(t),0) if t<0
{(0,0,0) if t=0
{(t,0,f(t) if t>0
a) prove that a is regular and c^infinity .
b) show that k=0 at t=0
a consist of curves in two different planes joined together at a point where k=0.
Q no. 4) let a(s) be a unit speed curve
A) prove that the tangent spherical image of alpha (a) is a constant curve iff. a is straight line.
B) prove that the binormal spherical image of a is a constant curve iff. a is a plane curve.
C) prove that the normal spherical image of a is never constant.

Qno. 6 ) let a(s) be a unit speed curve k>0 . Let s star be arc length on the normal spherical image. Prove k= | ds star /ds| iff. a (alpha) is a plane curve.

Answer 1

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