2004 Indian Statistical Institute M.Sc Mathematics Differential Geometry University Question paper for exam preparation. Question paper for 2004 Indian Statistical Institute M.Sc Mathematics Differential Geometry University Question paper, Exam Question papers 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2012 university in india question papers. SiteMap
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2004 Indian Statistical Institute M.Sc Mathematics Differential Geometry University Question paper

University Question Papers
2004 Indian Statistical Institute M.Sc Mathematics Differential Geometry University Question paper
Time: 3 hrs Date:03-05-04 Max. Marks : 50

Answer all six questions:

1. Consider the two-dimensional Riemannian manifold (M; g) where
M = f(x; y) 2 IR2 : y > 0g and g(x;y) = 1
y2 dxNdx + 1
y2 dyNdy

a) For any a 2 IR, prove that the map fa : M ! M given by
fa(x; y) = (x + a; y) is an isometry.

b) For any a 2 IR; prove that the curve ¾a : IR ! M given by
¾(t) = (a; et) is a geodesic.

c) Calculate the curvature of (M; g) at any point (x0; y0). [11]

2. Let M and N be n-manifolds with M compact and N converted. Let
f : M ! N be an immersion. Prove that f is onto (surjective). [6]

3. Let (¼ ~M ; ~ S) # (M; g) be a Riemannian Covering i.e., a smooth cov-
ering such that ¼ is orientation preserving. If the covering is a ¯nite
K-sheeted covering prove that Vol ( ~M ; ~g) = K Vol (M; g). [8]

Hint: Recall the proof when ¼ is actually an isometry.

4. Let M and N be compact oriented n- manifolds. Let ­ be an ori-
entation form for N and let f; g : M ! N be two smooth maps.
If f and g are smoothly, homotopic, (i.e. if there is a smooth map
F : M £ [0; 1] ! N with F(x; 0) = f(x) and F(x; 1) = g(x) for all x).
Then prove that RM
f¤(­) = RM
g¤(­)

5. Let (M; g) be a Riemannian manifold and \P" a point in M;Z, a
submanifold of M. Let C : [0;L] ! M be a geodesic such that
l(c) = inf
x2A
d(P; x)

a) Let Ct be a variation of C and Y be the corresponding variation
vector ¯eld. We know that inf ¡² < t²l(Ct) = l(C) for all variations
with Ct(L) 2 A and Ct(0) = P. For variations of this type, what are
the restrictions on Y (t) and Y (0)?

b) The ¯rst variation formula for the length functional is
d
dt
l(Ct)¯¯¯¯¯t=0
= hy(s); C0(s)i
2 Z0
¡
2 Z0
hy(s);rC;C0i ds
prove that C0(L) 2 (Tc(L)A)?, clearly state any result you use [6]


6. Let f : (M; g) ! IR be a smooth function on a Riemannian manifold.
The Hessian of f, at P is de¯ned as follows: Let X; y 2 TPM and
let ~X ; ~ Y be vector ¯elds extending X; y: Then
D2fjp(X; y) := ( ~XP (~y(t)) ¡ (r~X ~y; (f)

a) Prove that D2f is a tensor. i.e. D2f(X; y) doesn't depend on the
extensions ~X and ~y.

b) Prove that D2f is symmetric.

c) If P is a local minimum of f, prove that
D2fjp(X;X) ¸ 0 8X 2 TPM.
Hint: Consider a geodesic ¾ with ¾(0) = P; ¾0(0) = X. [11]



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