MA/MSCMT-04
June Examination 2016
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. Use of non-programmable scientific calculator is allowed in this paper.
Section A 8 × 2 16
(Very Short Answer Questions)
Note: Section contain Very Short Answer Type Questions. Examinees have to attempt all questions. Each question is of 02 marks and maximum word limit may be thirty words.
Write equation of osculating plane in terms of generating parameter
Define Torsion.
Define Indicatrix.
Write criterion for a surface to be developable.
402
MA/MSCMT-04 2800 3 (P.T.O.)
402
MA/MSCMT-04 2800 3 (Contd.)
Write first fundamental form.
Write the relation between three fundamental form.
Write differential equation of geodesics in a VN.
(viii) State Gauss's characteristics equation.
Section B 4 × 8 32
(Short Answer Questions)
Note: Section contain Short Answer Type Questions.
Examinees have to answer any four questions. Each
question is of 08 marks. Examinees have to delimit each
answer in maximum 200 words.
Prove that curvature and torsion of either associate Bertrand
curves are connected by a linear equation.
Find the equation to the conoid generated by lines parallel to the
plane XOY are drawn to intersect OZ and the curve x2 y2 r2
2z
a
x
b
y
2
2
2
2
c
Prove that the envelope of a family of surfaces touches each
member of the family at all points of its characteristic.
Show that for the right helicoid r cos u sin cv i


l m n u 0
n c
u v 2 2


If Aijk is a skew-symmetric tensor; show that
Aji
ijk g x
1 j A g
ijk
2
2 a k
402
MA/MSCMT-04 2800 3
Obtain differential equations of geodesics for the metric
ds2 dx2 dy2 dz2 dt2
Contract the Riemann Christoffel tensor and find Ricci Tensor.
If the metric of a two dimensional flat space is
dx1 dx
2
2
2
i i C where show that
where c and k are constants.
Section C 2 × 16 32
(Long Answer Questions)
Note: Section contain 04 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) State and prove theorem related to geometrical significance of
second fundamental form and derive Weingarten equations.
11) Find the principal sections and principal curvatures of the
surface x y b z uv
12) State and prove Gauss Bonnet theorem.
13) Show that the metric of a Euclidean plane referred to
cylindrical co-ordinates is given by
ds2 dr2 (rd θ)2 dz2
Show that Divergence of Einstein Tensor Vanishes.