Mathematics G4402. Modern Geometry
Spring 2016
Assignment 14
Due Monday, February 1, 2016
(1) Let (M , g) be a Riemannian manifold of dimension n + 1. Let f :
M → R be a smooth function, and let a ∈ R be a regular value of
f such that M = f −1 (a) is nonempty. Then M is an n-dimensional
submanfold of M . Let i : M → M be the inclusion map, and let
g = i∗ g, so that i : (M, g) → (M , g) is an isometric embedding.
Let Hessf = ∇df be the Hessian of f , where ∇ is the Levi-Civita
connection on (M , g). Prove the following statements.
(a) For any p ∈ M and x, y ∈ Tp M , we have
gradf (p) ∈ (Tp M )⊥ ,
Hgradf (p) (x, y) = −Hessf (p)(x, y).
(b) Let H be the mean curvature of M in M with respect to the
gradf
unit normal |gradf
| . Then
gradf
1
H = − div
n
|gradf |
where the gradient and the divergence are defined by g.
(2) Let (M, g) be a Riemannian manifold of dimension n, and let f ∈
C ∞ (M ). Let M × R be equipped with the product metric ḡ. In local
coordinates (x1 , . . . , xn ) on M ,
g = gij dxi dxj , ḡ = gij dxi dxj + dt2 ,
df = fi dxi , Hess(f ) = fij dxi dxj
where the Hessian is defined by the metric g. The map φf : M →
M ×R defined by x 7→ (x, f (x)) is a smooth embedding, so the image
Γf = φf (M ) is an n-dimensional submanifold of M × R.
(a) Let g̃ = φ∗f ḡ, and write g̃ = g̃ij dxi dxj . Show that
g̃ij = gij + fi fj ,
g̃ ij = g ij −
f if j
1 + |df |2
where f i = g ij fj and |df |2 = g ij fi fj .
(b) Let H be the mean curvature of Γf ⊂ M × R with respect to
the upward unit normal. Show that
g̃ ij fij
nφ∗f H = p
1 + |df |2
(3) Recall that the cartesian coordinates (x, y, z) and the spherical coordinates (r, φ, θ) on R3 are related by
x = r sin φ cos θ,
y = r sin φ sin θ,
z = r cos φ.
2
Consider a Riemannian metric on R3 of the form
g = u(r)2 dr2 + r2 (dφ2 + sin2 φdθ2 )
where u = u(r) is a positive, smooth function on R3 which depends
only on r. Note that u = 1 corresponds to the Euclidean metric.
Given any ρ > 0, let Sρ be the sphere defined by r = ρ.
(a) Find the scalar curvature of (R3 , g).
(b) Find the second fundamental form and the mean curvature of
Sρ with respect to the inner unit normal.
(c) Find the scalar curvature of Sρ .