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P. 313
Complex Analysis and Differential Geometry
Notes 24.7 Self Assessment
3
1. Let X : U 3 , and X : U be parametric surfaces. We say that X is reparametrization
of X if X X , where : U U is a ...................
2. The tangent space T X of the parametric surface ................... at u U is the 2-dimensional
u
linear subspace of spanned by the two vectors X and X . 1
3
1
2
3. A ................... Y is tangent to X if Y (u) T X for all u U. A vector field Y is normal to X
u
if Y(u) T X for all u U.
u
4. The ................... is invariant under orientation-preserving reparametrization.
5. Let X : U be a parametric surface, and let N : U be its Gauss map. Let X X
3
2
be an orientation-preserving ................... of X. Then the Gauss map of X is N .
24.8 Review Questions
1. Let X : U 3 and X : U be two parametric surfaces. The angle between them is
3
the angle between their unit normals: cos N N. Let be a regular curve which lies on
both X and X, and suppose that the angle between X and X is constant along . Show that
is a line of curvature of X if and only if it is a line of curvature of X.
2. Let X : U be a parametric surface, and let be an asymptotic line with curvature k
3
0, and torsion . Show that K
3. Denote by SO(n) the set of orthogonal n × n matrices, and by D(n) the set of n × n diagonal
k
matrices. Let A :(a,b) S n n be a C function, and suppose that A maps into the set of
matrices with distinct eigenvalues. Show that there exist C functions Q: (a, b) SO(n) and
k
: (a, b) D(n) such that Q AQ = . Conclude the matrix function A has C eigenvector
k
1
fields e , ,e :(a,b) n ,Ae j e . j Give a counter-example to show that this last
n
j
1
conclusion can fail the eigenvalues of A are allowed to coincide.
4. Let M n×n be the space of all n × n matrices, and let B: (a, b) M n×n be continuously
differentiable. Prove that:
detB tr B * B ,
where B* is the matrix of co-factors of B.
5. Two harmonic surfaces X,Y : U are called conjugate, if they satisfy the Cauchy-
3
Riemann Equations:
X = Y , X = Y ,
u
u
v
v
where (u, v) denote the coordinates in U. Prove that if X is conformal then Y is also
conformal. Let X and Y be conformal conjugate minimal surfaces. Prove that for any t:
Z = X cos t + Y sin t
is also a minimal surface. Show that all the surfaces Z above have the same first fundamental
form.
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