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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 28

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)

1. b) Prove that the curvature is the rate of change of angle of contingency with respect to

arc length.                                                                                                                [5]

 

1. 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)

1. 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)

1. 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]

 

1. c) Define evolute and involute. Also find their equations.

(or)

1. 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)

1. b) Prove that the first fundamental form is positive definite. [5]

 

1. c) Prove the necessary and sufficient condition for a surface to be developable.

(or)

1. d) Derive any two developables associated with a space curve. [15]

 

IV a) State the duality between  space curve and developable.

(or)

1. b) Derive the geometrical interpretation of second fundamental form. [5]

 

1. c) Find the first and second fundamental form of the curve

.

(or)

1. d) Find the principal curvature and direction of the surface

.                                                                                  [15]

 

 

V a) Derive Weingarton equation.

(or)

1. b) Show that sphere is the only surface in which all points are umbilics. [5]

 

1. 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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 29

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

    MT 1807 – DIFFERENTIAL GEOMETRY

 

 

 

Date : 11-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

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)

1. b) Show that the necessary and sufficient condition for a curve to be a straight line is that

for all points.                                                                                                      [5]

 

1. 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)

1. d) Derive the Serret-Frenet Express them in terms of  Darboux vector.      [15]

 

II a) Show that the circle , has three point contact at the

origin with a paraboloid with

(or)

1. b) Derive the necessary and sufficient condition for a space curve to be a helix.        [5]

 

1. c) If two single valued continuous functions and  of the real variable  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)

1. d) Find the intrinsic equation of the curve        [15]

 

III a) Derive the equation satisfying the principal curvature at a point on the space curve.

(or)

1. b) Prove that the metric is always positive. [5]

 

1. c) Prove that is a necessary and sufficient condition for a surface to be

developable.

(or)

1. d) Define developable. Derive polar and rectifying developables associated with a

space curve.                                                                                                             [15]

 

 

IV a) State and prove Meusnier  Theorem.

(or)

1. b) Prove that the necessary and sufficient condition for the lines of curvature to be

parametric curves is that                                                           [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)

1. d) (1) Define geodesic. State the  necessary and sufficient condition that the curve

be a geodesic .

(2) Show that the curves are geodesics on a surface with metric

.                                                                [5+10]

 

V a) Prove that the Gaussian curvature of a space curve is bending invariant.

(or)

1. b) Show that sphere is the only surface in which all points are umbilics. [5]

 

1. c) Derive the partial differential equation of surface theory. Also state Hilbert

Theorem.

(or)

1. d) State the fundamental theorem of Surface Theory and demonstrate it with an

example.                                                                                                               [15]

 

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