LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – NOV 2006
AA 21
MT 1807 - DIFFERENTIAL GEOMETRY
Date & Time : 02-11-2006/1.00-4.00 Dept. No.
Max. : 100 Marks
Answer ALL the questions
I a) Obtain the equation of tangent at any point on the circular helix.
(or)
b) Show that the necessary and sufficient condition for a curve to be a plane curve
is  r , r , r  = 0.
[5]
c) Derive the equation of the osculating plane at a point on the curve of intersection of
two surfaces f ( x, y, z )  0  g ( x, y, z ) in terms of the parameter u.
[15]
(or)
d) Derive the Serret-Frenet formulae and deduce them in terms of Darboux vector.
II a) Define involute and find the curvature of it.
(or)
b) Prove that a curve is of constant slope if and only if the ratio of curvature to torsion
is constant .
[5]
c) State and prove the fundamental theorem for space curve.
[15]
(or)
d) Find the intrinsic equations of the curve given by x  aeu cos u, y  aeu sin u , z  beu
III a) What is metric? Prove that the first fundamental form is invariant under the
transformation of parameters.
(or)
b) Derive the condition for a proper transformation from regular point.
[5]
c) Show that a necessary and sufficient condition for a surface to be developable is
that the Gaussian curvature is zero.
[15]
(or)
d) Define envelope and developable surface. Derive rectifying developable associated
with a space curve.
IV a) State and prove Meusnier Theorem.
(or)
b) Prove that the necessary and sufficient condition that the lines of curvature may be
parametric curve is that f  0 and F  0.
[5]
c) Prove that on the general surface, a necessary and sufficient condition that the curve
v  c be a geodesic is EE2  FE1  2 EF1  0 for all values of the parameter u . [15]
(or)
d) Find the principal curvature and principal direction at any point on a surface
x  a(u  v), y  a(u  v), z  uv
V a) Derive Weingarten equation.
[5]
(or)
b) Prove that in a region R of a surface of a constant positive Gaussian curvature
without umbilics, the principal curvature takes the extreme values at the boundaries.
c) Derive Gauss equation.
[15]
(or)
d) State the fundamental theorem of Surface Theory and illustrate with an example
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