Q10/13U/04-13 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. Define functions of class m. 2. Define singularity and classify. 3. Find the area of the geoderic triangle. 4. State and prove Meusnier’s theorem. 5. State fundamental existence theorem for surfaces. SECTION – B (3X5=15) Answer any THREE Questions 6. Show that the involutes of a circular helix are plane curves.  = r 7. For the paraboloid

( u ,ϑ , u

2

)

− ϑ 2 . Find E,F,G and H.

8. Derive the normal property of geodesics. 9. Derive Rodrigne’s formula that characterizes the lines of curvature. 10. Prove that the only compact surfaces whose Gaussian curvature is positive and mean curvature constant are spheres.

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Q10/13U/04-13 SECTION – C (5X10=50) Answer ALL Questions 11. a) Derive serret - Frenet formulae. (or) b) State and prove fundamental excistence theorem for space surves. 12. a) A helicoids is generated by the screw motion of a straight line skew to the axis. Find the curve coplanar with the axis which generates the same helicoid. (or) b) Find the area of the anchor ring corresponding to the domain 0 ≤ u ≤ 2π and 0 ≤ ϑ ≤ 2π . 13. a) Obtain the geodesic equations of a surface. (or) b) State and prove Gauss – Bonnet theorem. 14. a) State and prove Mongc’s theorem for lines of curvature on a surface. (or) b) Prove that the edge of regression  of the rectifying developede    (τ t + kb ) has equation, R= r + k k 'τ − kτ ' 15. a) State and prove Hilbert’s lemma. (or) b) Prove that the only compact surfaces of class≥2 for which every point is an umbilic are spheres.

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