Department:
Student Number:
Name:
Final-term Examination
Di erential Geometry 1
2019. 6. 20
1. (a) ( 10 pts ) Prove that curves ; : I ! R2 are
congruent if ~ = ~ and they have the same
speed.
1
Check:
(b) (p10 pts ) Show that the space curves (t) =
( 2t; t2 ; 0) and (t) = ( t; t; t2 ) are congruent.
Find an isometry that carries to .
Final-term Examination
Di erential Geometry 1
2019. 6. 20
2. ( 20 pts ) Consider the unit sphere
X = f(x; y; z) 2 R jx
3
2
+ y 2 + z 2 = 1g
and the hyperboloid
H = f(x; y; z ) 2 R3 jx2 + y2 z 2 = 1g:
P
Denote by N = (0; 0; 1) and S = (0; 0; 1) the
north and south poles of , respectively, and let
M =
fN; S g. Let F : M ! H be de ned as
follows:
If p 2 M , draw the line l orthogonally out from the z
axis through p 2 M . Then the perpendicular from p
to the z axis meet at q. (i.e, l is the half line starting
at q and containing p.) Let F (p) be the point at
which this line l rst meets H . Then F (p) = l \ H .
Prove that F is a mapping.
p
[Hint: sinh 1 s = ln(s + s2 + 1),
p
cosh 1 s = ln(s + s2 1)]
P
.
2
Department:
Student Number:
Name:
Final-term Examination
Di erential Geometry 1
2019. 6. 20
3. ( 10 pts ) Give the de nition of the exterior derivative
of di erential 1-forms on a surface M in R3 and show
that it is well-de ned.
3
Check:
4. ( 10 pts ) Let F : M ! N be a mapping of surfaces ,
and let and be forms on N .
Prove that F ( ^ ) = F ^ F .
Final-term Examination
Di erential Geometry 1
4
2019. 6. 20
5. ( 10 pts ) State and prove Stokes' theorem.
6. ( 10 pts ) Prove that the punctured plane R2
not simply connected.
f0g is
Department:
Student Number:
Name:
Final-term Examination
Di erential Geometry 1
2019. 6. 20
7. ( 10 pts ) Prove that a closed 1-form on a simply
connected surface is an exact form.
5
Check:
8. (a) ( 10pts) A surface M in R3 is orientable if and
only if there exists a unit normal vector eld on
M.
Final-term Examination
Di erential Geometry 1
6
2019. 6. 20
(b) ( 10 pts ) Let f be a di erentiable real valued
function and c a scalar. Prove that a surface M
de ned implicitly by f is orientable.
9. ( 10 pts ) For any sets A and B the Cartesian product
A B consists of all ordered pair (a; b) with a 2 A
and b 2 B .
If x : D ! M and y : E ! N are patches in surfaces
M and N in R3 , de ne x y : D E ! M N by
(x y)(u; v; u0 ; v 0 ) = (x(u; v ); y(u0 ; v 0 )):
Show that x y is an abstract patch and that the
collection P of all such patches makes M N a 4dimensional manifold.
(M N is called the Cartesian product of M and N .)