MA/MSCMT-04
December Examination 2017
M.A./M.Sc. (Previous) Mathematics Examination
Differential Geometry and Tensors
Paper MA/MSCMT-04
Time 3 Hours Max. Marks 80
Note: The question paper is divided into three sections B and C. Write answers as per given instruction. Use of non-programmable scientific calculator is allowed in this paper.
Section A 8 × 2 16
(Very Short Answer Type Questions)
Note: Section contain 08 Very Short Answer Type Questions. Examinees have to attempt all questions. Each question is of
02 marks and maximum word limit is thirty words.
Write the equation of oscillating plane.
Write down the formula of curvature of evolutes.
Write down the equation of tangent plane to a ruled surface in vector notation.
Write the parametric equation of anchor ring.
Define Normal section and oblique section of a surface.
Define zero tensor.
Explain divergent of a covariant vector.
(viii) Define flat space.
402
MA/MSCMT-04 1900 3 (P.T.O.)
MA/MSCMT-04 1900 3 (Contd.)
402
Section B 4 × 8 32
(Short Answer Type Questions)
Note: Section contain Eight Short Answer Type Questions.
Examinees will have to answer any four questions. Each
question is of 08 marks. Examinees have to delimit each answer
in maximum 200 words.
Find the inflexional tangent at on the surface y z 4ax 2
The necessary and sufficient condition for the curve to be a straight
line is that K 0 at all points of the curve.
Prove that the torsion of the two Bertrand curves have the same sign
and their product is constant.
On the paraboloid x y z 2 2 find the orthogonal trajectories of
the section by the planes z constant.
Prove that osculating plane at any point of a curved asymptotic lines
is the tangent plane to the surface.
Show that on the surface of a sphere, all great circles are geodesics
while no other circle is a geodesic.
If Aij is the curl of a covariant vector then prove that
Aij,k Ajk,i Aki, j 0
A covariant tensor of first order has components xy, 2y z xz 2 in
rectangular coordinates. Determine its covariant components in
spherical coordinate.
MA/MSCMT-04 1900 3
402
Section C 2 × 16 32
(Long Answer Questions)
Note: Section contain 4 Long Answer Type Questions. Examinees
will have to answer any two questions. Each question is of 16
marks. Examinees have to delimit each answer in maximum 500
words.
10) Explain edge of regression. Find the equation of the developable
surface whose generating line passes through the curve
y 4ax, z x 4ay, z c 2 2 and show that its edge of
regression is given by cx 2ayz 0 cy 3ax(c 2 2 .
11) Find the curvature of a normal section of the right helicoids
x u cosz, y u sinz, z cz.
12) Explain geodesic and derive geodesic on surface of revolution given
by x u cos y u sin z f .
13) If a Riemannian space VN is isotropic at each point in
a region, then prove that the Riemannian curvature is constant
throughout that region.
If surface of sphere is a two dimensional Riemannian space
then Compute the Christoffel Symbols.