Solutions to Final Exam Differential Geometry II June 18, 2010
Let Mbe a surface in R3with shape operator S.
Question 1 (5 points) Prove that if Mis given in implicit form M:g(x, y, z) = c,
where c∈Ris a constant, then Mis closed in R3(that is,
its complement R3\Mis an open set in R3).
Answer: This follows from the continuity of g.
Question 2 (5 points) Let vbe any nonzero tangent vector to M. Prove that the
normal curvature k(v)of Min the direction of vis equal to
S(v)•v/v •v.
Answer: Follows from the linearity of Sand •.
Question 3 (5 points) Let x:D→Mbe a patch in M. Prove that p=x(u, v)is an
umbilic point in Mif and only if there exists k∈Rsuch that
L(u, v) = kE(u, v),M(u, v) = kF (u, v ),and N(u, v) = kG(u, v),
where E=xu•xu, F =xu•xv, G =xv•xv,and
L=S(xu)•xu,M=S(xu)•xv,N=S(xv)•xv.
Answer: If pis umbilic, then there exists k∈Rsuch that
S(w) = k·wfor all w∈Tp(M),and the three equalities
follow immediately.
Conversely, if w=a·xu(u, v) + b·xv(u, v) is any unit tangent
vector to Mat p, then a direct calculation shows that the
above three equalities and the linearity of Sand •imply
k(w) = S(w)•w=k, hence pis umbilic.
