LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION - MATHEMATICS
FIRST SEMESTER – November 2008
AB 29
MT 1807 - DIFFERENTIAL GEOMETRY
Date : 11-11-08
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
Answer ALL the questions
I a) Prove that the curvature is the rate of change of angle of contingency with respect to
arc length.
(or)
b) Show that the necessary and sufficient condition for a curve to be a straight line is that
[5]
k  0 for all points.
c) (1) Find the centre and radius of an osculating circle.
(2) Derive the formula for torsion of a curve in terms of the parameter u.
[8+7]
(or)
d) Derive the Serret-Frenet formulae. Express them in terms of Darboux vector.
[15]
II a) Show that the circle lx  my  nx  0 , x 2  y 2  z 2  2cz has three point contact at the
l 2  m2
bl 2  am2
(or)
b) Derive the necessary and sufficient condition for a space curve to be a helix.
origin with a paraboloid ax 2  by 2  2 z with c 
[5]
c) If two single valued continuous functions k ( s ) and  ( s ) of the real variable s ( 0) are given then
prove that there exists one and only one space curve determined uniquely except for its position in
space, for which s is the arc length, k is the curvature and  is the torsion.
(or)
u
d) Find the intrinsic equation of the curve x  ae cos u, y  aeu sin u, z  beu
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III a) Derive the equation satisfying the principal curvature at a point on the space curve.
(or)
b) Prove that the metric is always positive.
[5]
c) Prove that eg  f 2  0 is a necessary and sufficient condition for a surface to be
developable.
(or)
d) Define developable. Derive polar and rectifying developables associated with a
space curve.
[15]
IV a) State and prove Meusnier Theorem.
(or)
b) Prove that the necessary and sufficient condition for the lines of curvature to be
parametric curves is that f  0 and F  0.
[5]
1
c) (1) Derive the equation satisfying the principal curvature at point on a surface.
(2) How can you find whether the given equation represent a curve or a surface?
(3) Define oblique and normal section.
[9+2+4]
(or)
d) (1) Define geodesic. State the necessary and sufficient condition that the curve
v  c be a geodesic .
(2) Show that the curves u  v  c are geodesics on a surface with metric
[5+10]
(1  u 2 )du 2 - 2uvdudv  (1  v 2 )dv 2 .
V a) Prove that the Gaussian curvature of a space curve is bending invariant.
(or)
b) Show that sphere is the only surface in which all points are umbilics.
c) Derive the partial differential equation of surface theory. Also state Hilbert
Theorem.
(or)
d) State the fundamental theorem of Surface Theory and demonstrate it with an
example.
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2
[5]
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