4H Dierential Geometry
Sheet 6 Solutions
By denition, the principal curvatures are given by the roots of the quadratic
L
M
det
M
N
E
F
F
G
= 0,
so the Gaussian and mean curvatures are given by the formulae:
LN M 2
,
EG F 2
1 LG 2M F + N E
H =
.
2
EG F 2
K
Mock Exam
Exam Length: 2 hours
Attempt 3 questions out of 4
(1) (a) State, without proof, the formulae for the curvature and torsion of
a regular parametrized curve in R3 (not necessarily parametrized
by arc-length) and prove that they are reparametrizati
4H Dierential Geometry
Sheet 5
2012-13
In these question a , b and c are positive constants, and in the third question a > b .
Qu. 1 Show that the rst and second fundamental forms of the surface
av cos u
u
av sin u
v
bu
are
I =
II =
(a2 v 2 + b2 ) 0
0
a
4E Differential Geometry Sheet 23 201%13
i. The centre of curvature (X, Y) of a point (as, y) on a curve lies at a distance [61 along
the normal to the curve at (ac, y) .
J W .
Showthat
X = mH'lsinH,
Y = yln1c0s0.
Where
sine = -y,
«$2492
cos0 = :6
Find
4H Differential Geometry
Qui The torus is obtained by the rotation of a circle of radius 6 whose center is a distance
a from the z~axis about the zaxis. Show that is is parametrized by the map
Specify the ranges of the parameters.
[You may find the diagra
, th ?, 2012
?a.m. to ?a.m.
EXAMINATION FOR THE DEGREES OF
M.A., B.Sc. AND M.Sci.
HONOURS MATHEMATICS
Dierential Geometry
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display or for graphical
4H Dierential Geometry
Sheet 4
2012-13
Qu. 1 The line through (0, 0, 1) and (x, y, 0) has parametric representation
t (0, 0, 1) + t(x, y, 1) = (tx, ty, 1 t) ,
It meets the sphere S 2 when
(tx)2 + (ty 2 ) + (1 t)2 = 1 ,
or t[t(x2 + y 2 + 1) 2] = 0 . Clearl
4H Dierential Geometry
Sheet 6
2012-13
Qu. 1 Calculate the Gauss and Mean curvatures of the three surfaces dened on example
sheet 5.
Qu. 2 Show that the Enneper surface dened by the parametrization
u u3 /3 + uv 2
u
v v 3 /3 + vu2
v
u2 v 2
is a minimal s
4H Dierential Geometry
Sheet 5 Solutions
Qu. 1 First Fundamental Form:
ru
rv
av sin u
= +av cos u ,
b
+a cos u
= +a sin u .
0
Hence
E = ru .ru = a2 v 2 + b2 ,
F = ru .rv = a2 v( cos u sin u + cos u sin u) = 0 ,
G = rv .rv = a2 .
So I = (a2 v 2 + b2 )du2 +
4H Dierential Geometry
Sheet 9 Solutions
2012-13
Qu. 1
The rotation indices are (a) 3 , (b) 4 , (c) +1 , and (d) 0 .
Qu. 2
The curve is positively oriented. At t = 0 one starts at the point (3, 0) and as t increases
sin t and cos t are positive. Thus the
4H Dierential Geometry
Sheet 7
2012-13
Qu. 1 Show that, for a surface dened by the parametrization
u
,
v
(u, v)
1
1
k u2 + 2 k2 v 2
2 1
that k1 and k2 are the principal curvatures at the point (u, v) = (0, 0) .
Qu. 2 Let (s) be a space curve parametrized
4H Dierential Geometry
Sheet 7 Solutions
2012-13
Qu 1.
First Fundamental Form:
1
ru = 0 ,
k1 u
0
rv = 1
k2 v
so
E = 1 + (k1 u)2 ,
F = k1 k2 uv ,
G = 1 + (k2 v)2 .
The questions says to evaluate curvature at (u, v) = (0, 0) so at this point E = G = 1 , F
Someday, Xth May, 2013
Some time to Another time
EXAMINATION FOR THE DEGREES OF
M.A., B.Sc. AND M.Sci.
4H HONOURS MATHEMATICS
Dierential Geometry: Mock Paper
An electronic calculator may be used provided that it does not have
a facility for either textual
Someday, Xth May, 2013
Some time to Another time
EXAMINATION FOR THE DEGREES OF
M.A., B.Sc. AND M.Sci.
4H HONOURS MATHEMATICS
Dierential Geometry
An electronic calculator may be used provided that it does not have
a facility for either textual storage or
4H Dierential Geometry
Curves in R2
Let t
x(t)
y(t)
= c(t) be a regular parametrized curve.
Arc-length
The arc-length function is given by (where t0 is some arbitrary constant which denes
the point on the curve from which distance is measured):
t
[x( )]2
1
The Sphere
Calculate the Gaussian and Mean curvature of the unit sphere using the parametrization
cos u cos v
u
cos u sin v .
v
sin u
Solution:
Step 1: First Fundamental Form
sin u cos v
ru = sin u sin v
cos u
sin v cos u
rv = cos u cos v
0
So:
E =
4H Dierential Geometry
Sheet 8
2012-13
Qu. 1 Dene functions u etc. by the expansions
uu
ruu = u ru + v rv + aN ,
uu
uu
u
v
ruv = uv ru + uv rv + bN ,
rvv = u ru + v rv + cN
vv
vv
(and hence u = u etc.). Using the expressions obtained in the previous examp