H4/PG2/13/A16 Reg. No

St. Joseph’s College of Arts & Science (Autonomous) St. Joseph’s College Road, Cuddalore – 607001 PMT1018S - DIFFERENTIAL GEOMETRY

Time : 3 hrs

Max Marks :75 SECTION – A (5X2=10) Answer ALL Questions

1. Write Serret—Frenet formulae. 2. Define Surface. 3. State Minding’s Theorem. 4. Define naval point. 5. State Fundamental existence theorem for surfaces. SECTION – B (3X5=15) Answer any THREE Questions .

6. Prove that

7. Find the coefficients of the direction which makes an angle

with

the direction whose coefficients are (l, m). 8. Find the geodesic curvature of the parametric curve

.

9. Derive the principal curvature. 10. Prove that the only compact surfaces with constant Gaussian curvature are spheres. SECTION- C (5X10=50) Answer ALL Questions 11. a) Show that that the curve be plane.

is a necessary and sufficient condition

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H4/PG2/13/A16 (or) b) Calculate the curvature and torsion of the cubic curve given by 12. a) A helicoids is generated by the screw motion of a straight line which meets the axis at an angle . Find the metric of the surface referred to the generators and their orthogonal trajectories as parametric curves. (or) b) On the paraboloid , find the orthogonal trajectories of the sections by the z = constant. 13. a) Prove that every helix on a cylinder is a geodesic. (or) b) Prove that, on the general surface, a necessary and sufficient condition that the curve be a geodesic is when , for all values of u. 14. a) Prove that a necessary and sufficient condition for a surface to be developable is that its Gaussian curvature shall be zero. (or) b) Derive Rodriques’ formula. 15. a) Show that in terms of E, F, G, L, M, N, the Weingarten equations are ,

H N2  ( FN  GM )r1  ( FM  EN )r2 . (or) b) Prove that the only compact surfaces of class every point is an umbilic are spheres. 2

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for which