Vinayaka Missions University 2008 M.Sc Mathematics DIFFERENTIAL GEOMETRY : t : Correspondence - Question Paper

COURSE CODE –2030504
PG DEGREE exam – SEP 2008
M.SC (MATHS)
DIFFERENTIAL GEOMETRY
(For Candidate Admitted from Calendar 2007 Onwards)
Time: three Hours Max. Marks: 75
part - A
ans all the questions:- 15 X 1=15
1. Definition of curve of class m
2. Show that the involutes of a circular helix are plane curve
3. obtain the condition for 2 directions to the orthogonal
4. Definition of familiars of curves.
5. Definition of locally isometric.
6. Definition of parametric transformation.
7. Write a note an intrinsic formula
8. obtain the Gaussian curvature at every point of a sphere of radius a
9. Definition of Edge of regression.
10. Write a note an mainardi- codazz, Equations.
11. Show that the geodesic an a right circular cylinder is helix.
12. Definition of Dupin in cicatrix
13. Definition of metric on the surface
14. Definition of curvature vector
15. Definition of conformal mapping.
part - B
ans any 5 questions:- five X six = 30
16. a) It the torsion is zero at all points of a curve, show that the curve is a
place curve.
(Or)
b) Prove that ser ret- Frenet formula
17. a) obtain the involutes and evolutes of the circular helix
? = (a cos ?, a sin ?, b ?)
(Or)
b) If ? = ? (s) is the provided curve ?, then the centre C and radius R of
spherical curvature at a point pon ? are provided by
C= ? + ?1 b ?, R = ?2+s2? '2
18. a) The 2 directions provided by
Pdu2 two Q dudv + Rdv2 = 0 are orthogonal an a surface, if and
only if ER- 2QF + GP = 0
(Or)
b)Write a note on intrinsic formula with an example.
19. a) obtain the parametric directions and the angle ranging from the
parametric curves.
(Or)
b) 2 surfaces other identical constant curvature are locally
isometric.
20. Show that the surface ez Cosn = Cosy is minima 1
(Or)
b) Prove that Hilbert’s lemma.
part – C
ans any 2 questions:- two X 15 = 30
21. obtain the curvature and torsion of the curve of intersection of the
quadratic surfaces.
a?2 + by2 + CZ2 = one and a1 ?2 + b1y2 + C1z2 =1
22. obtain the surface of revolution which is isometric with the region
of the right helicoids
23. Prove that liouville’s formula
24. If the is a surface of minimum area passing through a closed
curve, then it necessarily a minimal surface in the sense that it is
of zero mean curvature.
25. If ?1 one ?2 and N are 3 linearly independent vectors at a point P
on the surface ? = ? (u, v), then
1 2
r11 = 11 r1+ 11 r2 +LN
1 2
r12 = 12 r1 + 12 r2+MN
1 2
r22= 22 r1 + 22 r2 +NN

Earning:   Approval pending.