DIFFERENTIAL MANIFOLDS: HW9 (DUE TO 12/20)
YI LI
1. Let S be the south pole of Sm . For any point x ∈ Sm \ {S}, its
stereographic projection is the point y at the subspace
Rm = {z m+1 : z m+1 = 0},
which belongs to the straight line through x and S.
(a) Show that the stereographic projection is a bijection x ↔ y between
Sm \ {S} and Rm given by
y=
x0
,
1 + xm+1
where x = (x1 , · · · , xm , xm+1 ) and x0 = (x1 , · · · , xm ).
(b) In the Cartesian coordinates y 1 , · · · , y n , the canonical sphere metric
has the form
4
gRm ,
gSm =
(1 + |y|2 )2
P
where |y|2 = 1≤i≤m (y i )2 and gRm = (dy 1 )2 + · · · + (dy m )2 is the
canonical Euclidean metric.
2. Consider in Rm+1 a hyperboloid H given by the equation
(xn+1 )2 − (x0 )2 = 1,
where x0 = (x1 , · · · , xn ) ∈ Rm and xn+1 > 0. Observe that H is a submanifold of Rm+1 of dimension m. Consider in Rm+1 the Minkowski metric
gMink = (dx1 )2 + · · · + (dxm )2 − (dxm+1 )2 ,
which is a bilinear symmetric form in any tangent space Tx Rm+1 but not
positive definite (so, gMink is not a Riemannian metric, but a pseudoRiemannian metric). Let gH be the restriction of the gMink to H. We
will prove in this exercise that gH is positive definite so that (H, gH ) is a
Riemannian manifold. By definition, this manifold is called the hyperbolic
space and is denoted by Hm , and the metric gH is called the canonical
hyperbolic metric and is denoted also by gHm .
(a) Show that the equation
x0
y=
xm+1 + 1
determines a bijection of the hyperboloid Hm onto the unit ball
Bm = {y ∈ Rm : |y| < 1}.
1
2
YI LI
(b) In the Cartesian coordinates y 1 , · · · , y m in Bm , the canonical hyperbolic metric has the form
4
gHm =
gRm ,
(1 − |y|2 )2
P
where |y|2 = 1≤i≤m (y i )2 and gRm = (dy 1 )2 + · · · + (dy m ) is the
canonical Euclidean metric.
4
The ball Bm with the metric (1−|y|
2 )2 gRm is called the Poincaré model of
the hyperbolic space.
3. For any two-dimensional Riemannian manifold (M, g), the Gauss curvature KM,g (x) is defined in a certain way as a function on M. It is known
that if the metric g has in coordinates x1 , x2 the form
g=
(dx1 )2 + (dx2 )2
,
f 2 (x)
where f is a positive smooth function on M, then the Gauss curvature can
be computed in this chart as follows
KM,g = f 2 ∆ ln f,
2
2
where ∆ = (∂x∂ 1 )2 + (∂x∂ 2 )2 is the Laplace operator of the metric (dx1 )2 +
(dx2 )2 .
(a) Evaluate the Gauss curvature of R2 , S2 , H2 .
(b) Consider the half-plane R2+ := {(x1 , x2 ) ∈ R2 : x2 > 0} the metric
gR2+ =
(dx1 )2 + (dx2 )2
.
(x2 )2
Evaluate the Gauss curvature of this metric.
(c) The witten’s black hole or Hamilton’s cigar soliton is the complete Riemannian surface (R2 , gΣ ), where
gΣ :=
(dx1 )2 + (dx2 )2
.
1 + (x1 )2 + (x2 )2
Evaluate the Gauss curvature of this metric, all Christoffel symbols
Γkij , and all components Rijk` of Riemann curvature (4, 0)-tensor
field.
(d) Consider a new metric g̃ = h12 g, where h is a positive smooth function
on M. Prove that
KM,g̃ = (KM,g + ∆g ln h) h2 ,
where ∆g is the Laplace operator of the metric g, that is,
p
1
∂
ij ∂
√
∆g =
, g = (gij )1≤i,j≤2 .
det gg
∂xj
det g ∂xi
In particular, if g̃ = ev g, then
KM,g̃ = e−v (KM,g − ∆g v) .
DIFFERENTIAL MANIFOLDS: HW9 (DUE TO 12/20)
3
(e) In this case, we will show in the class that
R
Rijk` = (gi` gjk − gik gj` ) , R := KM,g .
2
The Ricci curvature Rij is defined by Rij := g k` Rik`j . Prove that
Rij = R2 gij .
(Bonus) The well-known Hamilton’s Ricci flow is given by
∂
g(t) = −2Ricg(t) , g(0) = g,
∂t
where g(t) is a family of Riemannian metrics. In surface case, the
Ricci flow can be reduced to
∂
g(t) = −Rg(t) g(t), g(0) = g.
∂t
Let gS2 denote the canonical spherical metric on S2 , and consider a
family of metrics g(t) := r2 (t)gS2 , where r(t) is to be determined.
Show that g(t) is a solution of the Ricci flow if and only if
p
√ √
1
r0 (t) = − , r(t) = r2 (0) − 2t = 2 T − t,
r
2
where T := r (0)/2. Consequently, this Ricci flow exists for the time
interval (−∞, T ). Such a solution is called an ancient solution.
(f) Suppose g has in polar coordinates (r, θ) the form
g = dr2 + ψ 2 (r) dθ2 ,
where ψ(r) is positive
R dr smooth function. Show that, under the change
of variable ρ := ψ(r) , we have
ψ 00 (r)
.
ψ(r)
Furthermore, find all metrics g of the above form with constant
Gauss curvature.
g = ψ 2 (r) dρ2 + dθ2 ,
KM,g = −
Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan
Road, Shanghai, 200240 China
E-mail address: [email protected]