LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
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M.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – APRIL 2008
MT 1807 – DIFFERENTIAL GEOMETRY
Date : 05-05-08 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
Answer all the questions
I a) Obtain the equation of the tangent at any point on the circular helix.
(or)
- b) Prove that the curvature is the rate of change of angle of contingency with respect to
arc length. [5]
- c) Derive the formula for torsion of a curve in terms of the parameter u and hence
calculate the torsion and curvature of the curve.
(or)
- d) Derive the Serret-Frenet formulae and deduce them in terms of Darboux vector.[15]
II a) If the curve has three point contact with origin
withthen prove that .
(or)
- b) Prove that the necessary and sufficient condition that a space curve may be helix is
that the ratio of its curvature to torsion is always a constant. [5]
- c) Define evolute and involute. Also find their equations.
(or)
- d) State and prove the fundamental theorem of space curves. [15]
III a) Derive the equation satisfying the principal curvature at a point on the space curve.
(or)
- b) Prove that the first fundamental form is positive definite. [5]
- c) Prove the necessary and sufficient condition for a surface to be developable.
(or)
- d) Derive any two developables associated with a space curve. [15]
IV a) State the duality between space curve and developable.
(or)
- b) Derive the geometrical interpretation of second fundamental form. [5]
- c) Find the first and second fundamental form of the curve
.
(or)
- d) Find the principal curvature and direction of the surface
. [15]
V a) Derive Weingarton equation.
(or)
- b) Show that sphere is the only surface in which all points are umbilics. [5]
- c) Derive Gauss equation in terms of Christoffel’s symbol.
(or)
(d) State the fundamental theorem of Surface Theory and demonstrate it in the case
of unit sphere . [15]
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