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Gaussian Curvature, Normal Curvature, and the Shape Operator

  1. Mar 30, 2010 #1
    1. The problem statement, all variables and given/known data
    Let [tex]u_1, u_2[/tex] be orthonormal tangent vectors at a point p of M. What geometric information can be deduced from each of the following conditions on S at p?

    a) [tex]S(u_1) \bullet u_2 = 0[/tex]

    b) [tex]S(u_1) + S(u_2) = 0[/tex]

    c) [tex]S(u_1) \times S(u_2) = 0[/tex]

    d) [tex]S(u_1) \bullet S(u_2) = 0[/tex]


    2. Relevant equations
    If v and w are linearly independent tangent vectors at a point p of M, then [tex]S(v) \times S(w) = K(p)v \times w[/tex], where [tex]K= det S[/tex].



    3. The attempt at a solution

    a) Since [tex]u_1, u_2[/tex] form a basis to [tex]T_p (M)[/tex], we can write [tex]S(u_1)=au_1 + bu_2[/tex]. Then [tex](au_1 + bu_2) \bullet u_2 = 0[/tex]. Since the dot product is linear we can write [tex]au_1 \bullet u_2 + bu_2 \bullet u_2 = 0 = bu_2 \bullet u_2 = 0 = b[/tex]. Hence [tex]S(u_1)=au_1[/tex], so the shape operator is just scalar multiplication. Does this mean M is a sphere?

    c) By the given formula, we know that [tex]K(p) = 0[/tex] since [tex]u_1 \times u_2 \neq 0[/tex]. But when [tex]K(p)=0[/tex], there are two cases, depending on the principal curvature, which I don't have any information about.

    I don't have any information on b) or d). Thanks for any input.
     
  2. jcsd
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