Download MATH 782 Differential Geometry : homework assignment five 1. A

MATH 782 Differential Geometry : homework assignment five
1. A ray is a geodesic γ : [0, ∞) → M which minimizes the distance from γ(0) to γ(t) for
all t ∈ [0, ∞). If M is complete and non-compact, prove that there is a ray starting at
γ(0) = p for all p ∈ M .
2. Let M and M be Riemannian manifolds and let f : M → M be a diffeomorphism.
Assume that M is complete and that there exists a constant c > 0 such that
|v| ≥ c|dfp (v)|
for all p ∈ M and all v ∈ Tp M . Prove that M is complete.
3. Let M be the upper half-plane R2+ with the metric
ds2 =
dx2 + dy 2
.
yk
For which values of k is M complete?
4. a) Let
A=
a b
c −a
∈ sl(2, R)
be an arbitrary matrix in the Lie algebra of SL(2, R), and let d = −detA = a2 + bc. Prove
that
√

√
ω
A
if
d
>
0,
where
ω
=
d=
a2 + bc,
 cosh ωI + sinh
ω
√
√
A
e =
cos ωI + sinω ω A
if d < 0, where ω = −d = −a2 − bc,

I +A
if d = 0.
b) Using part a), prove that the image of the exponential map applied to sl(2, R) is
precisely the subset
{B ∈ SL(2, R) | traceB > −2 or B = −I}
of SL(2, R).
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