Indian Statistical Institute (ISI) 2004 M.Sc Mathematics Differential Geometry - Question Paper

Time: three hrs Date:03-05-04 Max. Marks : 50

ans all 6 questions:

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

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

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

c) compute 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 truly 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 2 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) two A and Ct(0) = P. For variations of this type, elaborate
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) two (Tc(L)A)?, clearly state any outcome 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 two 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 two TPM.
Hint: Consider a geodesic ¾ with ¾(0) = P; ¾0(0) = X. [11]

Earning:   Approval pending.