EXERCISE 4.19. Let S be a path-connected oriented regular surface. Ifall points of S are umbilical points, prove that S is contained in either a planeor a sphere.HINT: First prove that the principal curvature function k = k1 = k2 isconstant on S. For this, let o : UC R? – VCS be a coordinate chart. Atevery q E U, Nu(q) = dNo(g)(0u(q)) = -k(q) · ou(q). That is, Nu = -k · ouon all of U, and similarly, N, = -k.o, on all of U. Differentiating this firstequation with respect to v and the second with respect to u and subtractingyields kuou = k,őv, which implies that ku = ky = 0; thus, k is constant. Ifk + 0, demonstrate that the point o+N is constant on U, so V is containedin the sphere of radius about this point.%3D%3D

let, σ:U⊂ℝ2→V⊂S be a coordinate chart.N be the normal vector.Since the serface is umbilicale at…

Answered: EXERCISE 4.19. Let S be a…

EXERCISE 4.19. Let S be a path-connected oriented regular surface. If all points of S are umbilical points, prove that S is contained in either a plane or a sphere.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

EXERCISE 4.19. Let S be a path-connected oriented regular surface. If
all points of S are umbilical points, prove that S is contained in either a plane
or a sphere.
HINT: First prove that the principal curvature function k = k1 = k2 is
constant on S. For this, let o : UC R? – VCS be a coordinate chart. At
every q E U, Nu(q) = dNo(g)(0u(q)) = -k(q) · ou(q). That is, Nu = -k · ou
on all of U, and similarly, N, = -k.o, on all of U. Differentiating this first
equation with respect to v and the second with respect to u and subtracting
yields kuou = k,őv, which implies that ku = ky = 0; thus, k is constant. If
k + 0, demonstrate that the point o+N is constant on U, so V is contained
in the sphere of radius about this point.
%3D
%3D

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