LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2012

MT 1813 – DIFFERENTIAL GEOMETRY

 

 

Date : 03-05-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer all the questions:

All questions carry equal marks:

 

 

I a) Obtain the equation of the tangent at any point on the circular helix.

(or)

1. b) Derive the equation of osculating plane at a point on the circular helix.                   [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) Find the plane that has three point of contact at origin with the curve

 

(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) (1) How can you find whether the given equation represents a curve or a surface?

(2) State and prove Euler’s Theorem.

(3) Define oblique, normal, principal sections of a surface.                            [3+6+6]

 

 

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.

(or)

(d) State the fundamental theorem of Surface Theory and demonstrate it in the case

of unit sphere .                                                                                                      [15]

 

 

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