LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 21

FIRST SEMESTER – NOV 2006

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)

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

is  = 0.                                                                                                     [5]

 

1. c) Derive the equation of the osculating plane at a point on the curve of intersection of

two surfacesin terms of the  parameter u.                          [15]

(or)

1. 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)

1. b) Prove that a curve is of constant slope if and only if the ratio of curvature to torsion

is  constant .                                                                                                                [5]

 

1. c) State and prove the fundamental theorem for space curve. [15]

(or)

1. d) Find the intrinsic equations of the curve given by

 

III a) What is metric? Prove that the first fundamental form is invariant under the

transformation of parameters.

(or)

1. b) Derive the condition for a proper transformation from regular point. [5]

 

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

that the Gaussian curvature is zero.                                                                       [15]

(or)

1. d) Define envelope and developable surface. Derive rectifying developable associated

with a space curve.

 

IV a) State and prove Meusnier  Theorem.

(or)

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

parametric curve is that                                                             [5]

 

1. c) Prove that on the general surface, a necessary and sufficient condition that the curve

be a geodesic is  for all values of the parameter .  [15]

(or)

1. d) Find the principal curvature and principal direction at any point on a surface

 

 

V a)  Derive Weingarten equation.                                                                                   [5]

(or)

1. 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.

 

1. c) Derive Gauss equation. [15]

(or)

1. d) State the fundamental theorem of Surface Theory and illustrate with an example

 

 

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