Problems in Dierential Geometry (3)
1. Compute the curvature of r(t) =
t sinh t cosh t, 2 cosh t
2. In the catenary r(t) = (t, cosh t) prove that the curvature of the curve and arclength
function s satisfy
1
=
1 + s2
where the arclength s is measured with
Problems in Dierential Geometry (1)
1. Sketch roughly the following curves :
(A) r(t) = t, 5 + sin t
(B) r(t) = t, cosh t
Catenary
(C) r(t) = sin (3t) cos t, sin (3t) sin t
Three petal rose
(D) r(t) = 5 cos t, 2 sin t, t
(E) r(t) = (5 + sin t) cos t, (5 +
Problems in Dierential Geometry (8)
1. Prove that the mean curature and the Gaussian curvature of the surface
q(u, v) = (u, v, uv)
are respectively
H=
uv
1
and K =
.
2 )3/2
2 + v 2 )2
(1 + + v
(1 + u
u2
2. Let b > 0. Prove that the surface
q : (b, ) R R3
Problems in Dierential Geometry (11)
1. Prove that on the right helicoid
q(u, ) = (u cos , u sin , )
a geodesic is of the form r = q with
(t) = (u(t), (t)
where
d2 u
d
u( )2 = 0
2
dt
dt
du d
d2
+ 2u
= 0
2
dt
dt dt
Deduce that u when expressed as a funct
Problems in Dierential Geometry (4)
I. Let a, b > 0 be constants. Evaluate the curvature and torsion of the curve
r(t) = a(t sin t) , a(1 cos t) , bt
.
2. Let a, b, c > 0 be constants. Write down the equations of the osculating, normal and
rectifying plan
Problems in Dierential Geometry (7)
1. Let a, b, c R be constants. Compute the rst and the second fundamental forms of
the following surfaces.
(A) q(u, v) = a sin u cos v, b sin u sin v, c cos u
Elipsoid
(B) q(u, v) = au cos v, bu sin v, u2
Elliptic parab
Problems in Dierential Geometry (2)
1. Prove that following curves are parametrised by arclength :
(A) r(t) =
12 cos t
5 cos t 28
,
sin t ,
13
193
13
(B) r(t) =
(1 + t)3/2 (1 t)3/2 t
,
,
3
3
2
2. Prove that following curves are of constant speed :
t2
a
Problems in Dierential Geometry (5)
1 (A) Given a curve r : J R3 with unit tangent eld T, torsion and nowhere
vanishing curvature , prove that the following are equivalent :
(i) There exists a direction with which the tangents to the curve make a constant
Problems in Dierential Geometry (10)
1. Let = (0, ) (0, 2) . Write down the set of equations which describe the parallel
translation of a tangent vector on the surface q : R3 dened by
q(, ) = [sin cos , sin sin , cos ]
(A) along the curve r = q where : (/
Problems in Dierential Geometry (9)
1. Prove that in a surface with F = 0,
1 1
K=
2 EG v
1 E
EG v
+
u
1 G
EG u
.
A surface with this property is said to be a a surface with orthogonal parametrisation.
2. Let q : R3 be a surface of unit normal eld n with
Problems in Dierential Geometry (6)
1. Write down the equation of the surface obtained by rotating the curve y = x2 + 1
about the x-axis.
2. Write down the equation of the surface obtained by rotating the curve y = x + 4 + sin x
about the line y = x.
3. W