Geometry Second Semester 2011/12 Homework 4

1. For what values of c is the level set f^{−1}(c) an n-surface for each f given below.

(a) f(x_{1}, x_{2}, …, x_{n+1}) = x_{1}^{2} + x_{2}^{2} + … + x_{n+1}^{2}

(b) f(x_{1}, x_{2}, …, x_{n+1}) = x_{1}^{2} + x_{2}^{2} + … + x_{n}^{2} − x_{n+1}^{2}

(c) f(x_{1}, x_{2}, …, x_{n+1}) = x_{1}x_{2} … x_{n+1} + 1

2. Show that the cylinder {(x_{1}, x_{2}, x_{3}) | x_{1}^{2} + x_{2}^{2} = 1} ⊂ R^{3} can be represented as the level set of each of the following functions.

(a) f(x_{1}, x_{2}, x_{3}) = x_{1}^{2} + x_{2}^{2}

(b) f(x_{1}, x_{2}, x_{3}) = −x_{1}^{2} − x_{2}^{2}

(c) f(x_{1}, x_{2}, x_{3}) = 2x_{1}^{2} + 2x_{2}^{2} + sin(x_{1}^{2} + x_{2}^{2})

3. Show that if an n-surface S is represented as both f^{−1}(c) and g^{−1}(d), for smooth functions f and g, and numbers c, d ∈ R, where ∇f(p) ≠ 0 and ∇g(p) ≠ 0 for all p ∈ S, then, for each p ∈ S, ∇f(p) = λ∇g(p) for some real number λ ≠ 0.

4. Sketch the cylinders f^{−1}(0) for the following functions f.

(a) f(x_{1}, x_{2}) = x_{1}

(b) f(x_{1}, x_{2}, x_{3}) = x_{1} − x_{2}^{2}

(c) f(x_{1}, x_{2}, x_{3}) = x_{1}^{2}/4 + x_{2}^{2}/9 − 1

5. Verify that a surface of revolution (see Example 4.6 in your notes) is a 2-surface.

6. Sketch the surface of revolution obtained by rotating C about the x_{1}-axis, where C is each of the curves defined below.

(a) {(x_{1}, x_{2}) | x_{2} = 1} (cylinder)

(b) {(x_{1}, x_{2}) | − x_{1}^{2} + x_{2}^{2} = 1, x_{2} > 0} (1-sheeted hyperboloid)

(c) {(x_{1}, x_{2}) | x_{1}^{2} + (x_{2} − 2)^{2} = 1} (torus)

7. Show that the set S of all unit vectors at all points in R^{2} forms a 3-surface in R^{4}. Hint: S = {(x_{1}, x_{2}, x_{3}, x_{4}) | x_{3}^{2} + x_{4}^{2} = 1}.

8. Let S = f^{−1}(c) be a 2-surface in R^{3} which lies in the half-space {(x_{1}, x_{2}, x_{3}) | x_{3} > 0}. Find a function g: U → R (where U an open set in R^{4}) such that g^{−1}(c) is the 3-surface obtained by rotating the 2-surface S about the (x_{1}, x_{2})-plane.

9. Let a, b, c ∈ R be such that ac − b^{2} > 0. Show that the maximum and minimum values of the function g(x_{1}, x_{2}) = x_{1}^{2} + x_{2}^{2} on the ellipse {(x_{1}, x_{2}) | ax_{1}^{2} + 2bx_{1}x_{2} + cx_{2}^{2} = 1} are of the form

1/λ_{1} and 1/λ_{2}, where λ_{1} and λ_{2} are the eigenvalues of the matrix ( \begin{matrix}a & b \\ b & c \end{matrix} ).

10. Let S be an n-surface in R^{n+1} and let p_{0} ∈ R^{n+1} \ S. Show that the shortest line segment from p_{0} to S (if one exists) is perpendicular to S. That is, show that if p ∈ S is such that ∥p_{0} − p∥^{2} ≤ ∥p_{0} − q∥^{2} for all q ∈ S, then (p, p_{0} − p) ⊥ S_{p}. [Hint: Use the Lagrange Multiplier Theorem.] Show that the same conclusion holds for the longest line segment from p_{0} to S (again, if one exists).

11. The set R^{4} may be viewed as the set of all 2 × 2 matrices with real entries by identifying the quadruple (x_{1}, x_{2}, x_{3}, x_{4}) with the matrix

( \begin{matrix}x_{1} & x_{2} \\ x_{3} & x_{4} \end{matrix} ) .

The subset consisting of those matrices with determinant equal to one forms a group under matrix multiplication, this group is called the special linear group SL(2). Show that SL(2) is a 3-surface in R^{4}.

12. The trace of a matrix

A = ( \begin{matrix}x_{1} & x_{2} \\ x_{3} & x_{4} \end{matrix} )

is defined to be tr(A) := x_{1} + x_{4}. Show that the tangent space SL(2)_{p} to SL(2) (see Question 11) at p = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) can be identified with the set of all 2 × 2 matrices of trace zero by showing that

SL(2)_{p} = \{(p, ( \begin{matrix}x_{1} & x_{2} \\ x_{3} & x_{4} \end{matrix} )) | x_{1} + x_{4} = 0\} .

[Hint: First show that if

α(t) = ( \begin{matrix}x_{1}(t) & x_{2}(t) \\ x_{3}(t) & x_{4}(t) \end{matrix} )

is a parametrised curve in SL(2) with α(0) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) then x′_{1}(0) + x′_{4}(0) = 0. Use a dimensional argument to prove the opposite inclusion.]