Tutorial 10—Global Analysis
1. Suppose (M, g) ⊂ (R3
, g) = (R3
, geuc
) = (R3
, < −,−>) is a surface in Euclidean
space. Let u : U → u(U) be a local chart for M with corresponding local
parametrization
v = u−1
: u(U) → U.
With respect to the frame { ∂
∂x1 , ∂
∂x2 } of TR2
, we can write v∗
g and v∗
II as matrices
E F
F G
and
˜E ˜F
˜F ˜G
,
where
E = g(
∂
∂u1
,
∂
∂u1
) ◦ v F = g(
∂
∂u1
,
∂
∂u2
) ◦ v G = g(
∂
∂u2
,
∂
∂u2
) ◦ v,
and
˜E = II(
∂
∂u1
,
∂
∂u1
) ◦ v ˜F = II(
∂
∂u1
,
∂
∂u2
) ◦ v ˜G = II(
∂
∂u2
,
∂
∂u2
) ◦ v.
Compute in terms of E, F, G, ˜E, ˜F and ˜G, the Weingarten map L ◦ v, the Gauß
curvature K ◦ v, the mean curvature H ◦ v, and the principal curvatures κ1 ◦ v and
κ2 ◦ v.
2. Let us write (x1
, x2
, x3
) for the coordinates in R3
. Take a circle of radius r > 0 in
the (x1
, x3
)-plane and rotate it around a circle of radius R > r in the (x1
, x2
)-plane.
The result is a 2-dimensional torus M in R3
. If I ⊂ R is an open interval of length
< 2π the map v : I × I → R3
given by
v(φ, θ) = ((R + r cos θ) cos φ, (R + r cos θ) sin φ, r sin θ)
defines a local parametrization of M. With respect to v, compute, using the previous
exercise, the metric g on M induced by the Euclidean metric on R3
, the 2nd
fundamental form, the Gauß and the mean curvature, the principal curvatures and
the principal curvature directions of the surface (M, g) in R3
.
Hint: Note that ν(φ, θ) = (cos φ cos θ, sin φ cos θ, sin θ) defines a local unit normal
vector field for M.
1
2
3. Suppose (M, g) ⊂ (Rm+1
, g) = (Rm+1
, geuc
) is a connected oriented hypersurface
in Euclidean space. Show that all points in M are umbilic if and only if M is part
of an affine hyperplane or a sphere.
Hint: For , =⇒ show the following:
• Fix a global unit normal vector field ν : M → Rm+1
. Then, by assumption,
for any x ∈ M there exists λ(x) ∈ R such that
Lx = λ(x)IdTxM .
Since λ = g(L(ξ),ξ)
g(ξ,ξ)
for any local vector field ξ on M, λ : M → R is smooth.
Show that λ is constant, by, for instance, picking a chart and computing the
left-hand-side of [ ∂
∂ui , ∂
∂uj ] · ν = 0.
• If λ = 0, show that any curve in M is contained in an affine hyperplane with
(constant) normal vector ν.
• If λ = 0, show that f : M → Rm+1
, given by f(x) = x − 1
λ
ν(x), is constant.