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Differential Geometry Homework

  1. Feb 26, 2007 #1
    (1) The gradient of a differential function f: S --> R is a differentiable map grad f: S --> R^3 which assigns to each point p in S a vector grad f(p) in the tangent space of p s.t.

    <grad f(p), v> at p = dfp(v) for every v in the tangent space Tp(S)

    (a) If E, F, G are the coefficients of the first fundamental form in a parameterization X: U a subset o R^2 --> S, then grad f on X(U) is given by

    grad f = [(df/du *G - df/dv *F)/(EG - F^2)]*dX/du + [(df/dv *E - df/du *F)/(EG - F^2)]*dX/dv

    - Oh man, i'm really lost on this one. My book didn't really cover this too much, and I'm not sure what all the notation means.

    (b) If you let p in S be fixed and v vary in the unit circle, ie |v| = 1 in Tp(S), then dfp(v) is maximum iff v = grad f/|grad f|

    (c) if grad f is nonzero at all points of the level curve C = {q in S | f(q) = constant}, then C is a regular curve on S and grad f is normal to C at all pojnts of C.

    Notation: S is a regular surface, ie. a manifold. X is a mapping from the uv plane to the surface S such that it is differentiable and a homeomorphism. Tp(S) is the tangent plane of p for a given p in S.

    (2) Show that at a hyperbolic point (a point who Gaussian curvature is less than zero), the principal directions bisect the asymptotic directions.

    Pf.

    I have that the asymptotes are indeed the asymptotic directions and via the Dupin indicatrix, i see that principal directions e1 and e2 which are eigenvectors for our curvatures k1 and k2 which determine K do indeed bisect the asymptotes, but I don't have no godly reason why they do.

    (3) Let C be a subset of a regular surface S with Gaussian curvature K > 0. Show that the curvature k of C at p satisfies:

    k is greater than or equal to min (|k1|, |k2|)
    where k1 and k2 are principal curvatures of S at p.

    Pf.

    Again, kinda stuck like the previous problem

    (4) Show that the mean curvature H at p in S is given by:

    H = 1/pi * int from 0 to pi (kn(theta)) d(theta) where kn(theta) is the normal curvature at p along a direction making an angle theta with a fixed direction.

    - What exactly is normal curvature? My book uses the term at will and I don't understand what it means in this context?

    (5) Show that the sum of normal curvatures for any pair of orthogonal directions, at a point p in S, is constant.

    - ????????????????

    (6) Prove that the absolute value of the torsion T at a point of an asymptotic curve, whose curvature is nowhere zero, is given by:

    |T| = sqrt(-K) where K is the Gaussian curvature of the surface at a given point.

    Sorry for not showing any work I've done, but I'm really really stumped on these problems.
     
  2. jcsd
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