Mathematics G4403. Modern Geometry
Spring 2016
Assignment 16
Due on Wednesday, Feburary 17, 2016
Notation: [dC] = do Carmo, Riemannian Geometry
a b
SL(2, R) = {
| a, b, c, d ∈ R, ad − bc = 1}.
c d
α β
SU (1, 1) = {
| α, β ∈ C, |α|2 − |β|2 = 1}.
β̄ ᾱ
S n−1
= {(x1 , . . . , xn ) ∈ Rn | x21 + · · · + x2n = 1}.
(1) Let g be a Riemannian metric on a manifold M , and let g̃ = e2f g, where f
˜ be the Levi-Civita connections on
is a smooth function on M . Let ∇ and ∇
(M, g) and (M, g̃), respectively. Prove that for any C ∞ vector fields X, Y
on M ,
˜ X Y = ∇X Y + X(f )Y + Y (f )X − g(X, Y )gradf
∇
where gradf is defined by g.
(2) Let (M, g) be a Riemannian manifold, and let σ : M → M be an isometry.
Let M σ = {x ∈ M | σ(x) = x} be the set of fixed points of σ. Suppose that
M σ is nonempty, and is a submanifold of M . Prove that M σ is a totally
geodesic submanifold of M .
(Indeed, the following stronger result holds: let K be any set of isometries
of a Riemanniannian manifold (M, g), and let M K be the set of points of M
which are left fixed by all elements of K. Then each connected component
of M K is a closed totally geodesic submanifold of M .)
(3) Given any positive constant K > 0, define a Riemannian metric gK on Rn
4(dx21 + · · · + dx2n )
Pn
by g =
Prove that:
(1 + K i=1 x2i )2
(a) (Rn , g) has constant sectional curvature K.
(b) (Rn , g) is not complete.
(4) Let (x, y) be coordinates of R2 , and let z = x + iy.
(a) Let H 2 = {(x, y) ∈ R2 | y > 0} be the upper half plane, equipped with
the Riemannian metric
4dzdz̄
dx2 + dy 2
=−
.
g=
y2
|z − z̄|2
az + b
a b
Prove that the map z 7→
, where
∈ SL(2, R), is an
c d
cz + d
2
isometry of (H , g).
(b) Let D2 = {(x, y) ∈ R2 | x2 + y 2 < 1}, equipped with the Riemannian
metric
4(dx2 + dy 2 )
4dzdz̄
h=
=
.
(1 − x2 − y 2 )2
(1 − |z|2 )2
αz + β
α β
Prove that the map z 7→
, where
∈ SU (1, 1), is an
β̄ ᾱ
β̄z + ᾱ
isometry of (D2 , h).