Harmonic Analysis and Convex Geometry, Fall 2013, November 22.

Instructor: Dmitry Ryabogin

Assignment 11.

1. Problem 1.

a) Find the principle curvatures of the paraboloid z = a(x^{2} + y^{2}) at the point (0, 0, 0).

b) Find the lines of curvature of the Helicoid

x = u cos v, y = u sin v, z = cv.

c) Find the mean and the Gaussian curvature of the paraboloid z = axy at the point x = y = 0.

2. Problem 2. Let N(p) = \frac{α_{u}×α_{v}}{|α_{u}×α_{v}|} be a Gauss map N : S → S^{2} of a regular surface S, associated to a point p ∈ S. Prove Rodrigues’s formula: γ(t) is a line of curvature if and only if

\frac{d}{dt}N(γ(t)) = λ(t)γ′(t)

for some differentiable function λ(t). Prove further that the principle curvature in this case is −λ(t).

3. Problem 3.

Let α, β, γ be real numbers such that α^{2} + β^{2} + γ^{2} = 1, and let K be a convex body in ℝ^{3}.

a) Use the fact that h_{K}(λ(α, β, γ)) = λh_{K}((α, β, γ)) for any λ > 0 to obtain

αH_{α} + βH_{β} + γH_{γ} = H,

and

\{ \begin{matrix}αH_{αα} + βH_{αβ} + γH_{αγ} = 0, \\ αH_{βα} + βH_{ββ} + γH_{βγ} = 0, \\ αH_{γα} + βH_{γβ} + γH_{γγ} = 0, \end{matrix}

where H = h_{K} and, for example, H_{α} = \frac{∂h_{K}}{∂α}, H_{αβ} = \frac{∂^{2}h_{K}}{∂α∂β}. Here K is such that all the derivatives make sense.

b) Let (x, y, z) be a point of tangency of the plane αx + βy + γz = h_{K}((α, β, γ)) with K, i.e.,

x = H_{α}, y = H_{β}, x = H_{γ},

and let R be a principle radius of curvature of the convex surface F which is a boundary of K near (x, y, z), i.e., the coordinates (ξ, η, ζ) of the corresponding center of curvature are

ξ = H_{α} − Rα, η = H_{β} − Rβ, ζ = H_{γ} − Rγ.

Show that the vector (dξ, dη, dζ) is parallel to the vector (α, β, γ).

Hint: Move along the corresponding line of curvature of F near (x, y, z) = ∇h_{K}((α, β, γ)). Use Problem 2 assuming the normal to F is directed inward.

Conclude that

\{ \begin{matrix}H_{αα}dα + H_{αβ}dβ + H_{αγ}dγ = Rdα + λα, \\ H_{βα}dα + H_{ββ}dβ + H_{βγ}dγ = Rdβ + λβ, \\ H_{γα}dα + H_{γβ}dβ + H_{γγ}dγ = Rdγ + λγ, \end{matrix}

for some real number λ.

c) Multiply the above equalities by α, β and γ to observe that λ = 0. Then prove that R satisfies

| \begin{matrix}H_{αα} − R & H_{αβ} & H_{αγ} \\ H_{βα} & H_{ββ} − R & H_{βγ} \\ H_{γα} & H_{γβ} & H_{γγ} − R \end{matrix} | = 0.

Hint: Use a).

d) Reduce the above cubic (in R) equation to the quadratic one,

R^{2} − (R_{1} + R_{2})R + R_{1}R_{2} = 0.

Prove that

R_{1} + R_{2} = H_{αα} + H_{ββ} + H_{γγ} = Δh_{K}, R_{1}R_{2} = K_{αα} + K_{ββ} + K_{γγ},

where the matrix

(\begin{matrix}K_{αα} & K_{αβ} & K_{αγ} \\ K_{βα} & K_{ββ} & K_{βγ} \\ K_{γα} & K_{γβ} & K_{γγ} \end{matrix})

is the one of the algebraic complements of the matrix of the second derivatives of H,

(\begin{matrix}H_{αα} & H_{αβ} & H_{αγ} \\ H_{βα} & H_{ββ} & H_{βγ} \\ H_{γα} & H_{γβ} & H_{γγ} \end{matrix}) .

Observe that R_{1}((α, β, γ))R_{2}((α, β, γ)) = \frac{1}{K(x,y,z)} is the reciprocal of the Gaussian curvature K at the point (x, y, z) = ∇h_{K}.

4. Problem 4*. Prove that on the surface of every C^{2}-regular convex body in ℝ^{3} there are at least two umbilical points.