91Ó°ÊÓ

Let \(R^{2}=\left\\{(x, y, z) \in R^{3} ; z=-1\right\\}\) be identified with the complex plane \(\mathbb{C}\) by setting \((x, y,-1)=x+i y=\zeta \in \mathbb{C}\). Let \(P: \mathbb{C} \rightarrow \mathbb{C}\) be the complex polynomial $$ P(\zeta)=a_{0} \zeta^{n}+a_{1} \zeta^{n-1}+\cdots+a_{n}, \quad a_{0} \neq 0, a_{i} \in \mathbb{C}, i=0, \ldots, n $$ Denote by \(\pi_{N}\) the stereographic projection of \(S^{2}=\left\\{(x, y, z) \in R^{3}\right.\); \(\left.x^{2}+y^{2}+z^{2}=1\right\\}\) from the north pole \(N=(0,0,1)\) onto \(R^{2}\). Prove that the map \(F: S^{2} \rightarrow S^{2}\) given by $$ \begin{aligned} &F(p)=\pi_{N}^{-1} \circ P \circ \pi_{N}(p), \quad \text { if } p \in S^{2}-\\{N\\} \\ &F(N)=N \end{aligned} $$ is differentiable.

Step by step solution

01

Understand Stereographic Projection

The stereographic projection \(\pi_{N}\) maps a point \((x, y, z)\) on the sphere \(\displaystyle S^{2}\) (excluding the north pole \(\displaystyle N=(0,0,1)\)) onto the plane \(\displaystyle \mathbb{C}\) at \(\displaystyle z = -1\). Points are identified as: \((x, y, -1) = x + i y = \zeta \in \displaystyle \mathbb{C}\).

02

Polynomial Mapping

Given the polynomial \(\displaystyle P(\zeta) = a_{0} \zeta^{n} + a_{1} \zeta^{n-1} + \cdots + a_{n}\), map the complex plane \(\math\bb{C}\) under this polynomial. For a given \(\displaystyle \zeta \in R^{2}\), it transforms as \(\displaystyle P(\zeta)\).

03

Map Back to the Sphere

Employ the inverse stereographic projection \(\displaystyle \pi_{N}^{-1}\) to take the resulting point \(\displaystyle P(\zeta)\) and map it back onto the sphere \(\displaystyle S^{2}\).

04

Define the Map F

The map \(\displaystyle F: S^{2} \rightarrow S^{2}\) is defined by \(\displaystyle F(p) = \pi_{N}^{−1} \circ P \circ \pi_{N}(p) \) for \(\displaystyle p \in S^{2} - \{ N \} \). At the north pole \(\displaystyle N\), set \(\displaystyle F(N) = N\).

05

Prove Differentiability

Since \(\displaystyle P\) is a polynomial, it is differentiable on the complex plane. The stereographic projection \(\displaystyle \pi_{N}\) and its inverse \(\displaystyle \pi_{N}^{−1}\) are also differentiable. The composition of differentiable functions is differentiable. Thus, \(\displaystyle F\) is differentiable everywhere on \(\displaystyle S^{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stereographic Projection

Stereographic projection is a way to project points from a sphere \(S^2\) onto a plane \(\mathbb{C}\). Imagine a sphere resting on a plane. We project a point \(p\) on the sphere (not including the North Pole \(N\)) to a point \(\zeta\) on the plane where \(z = -1\). The coordinates \(x + iy \) form \(\zeta \). This mapping is essential for understanding how to transition between spherical and planar coordinates in differential geometry.

Complex Polynomial

A complex polynomial \(P(\zeta)\) is an expression involving powers of a complex variable \(\zeta\). Consider \[ P(\zeta) = a_0 \zeta^n + a_1 \zeta^{n-1} + \cdots + a_n \] where \(a_i\) are complex constants and \(a_0 eq 0\). This polynomial maps each point \(\zeta\) in the complex plane to another point in the plane. Complex polynomials play a crucial role in many areas of mathematics, including this exercise where they help transform points after projection.

Differentiability Proof

To prove F is differentiable, we need to confirm that each component of our mapping function \( F(p) = \pi_N^{-1} \circ P \circ \pi_N(p) \) is differentiable. Since \(P(\zeta)\) is a polynomial, it is inherently differentiable in the complex plane \(\mathbb{C}\). Both \(\pi_N\) and its inverse \(\pi_N^{-1}\) are differentiable mappings. The composition of differentiable functions remains differentiable. Hence, the map \(F\) is differentiable on the entire sphere \(S^2\).

Inverse Mapping

The inverse stereographic projection \(\pi_N^{-1}\) helps to map points from the complex plane back onto the sphere. After applying the polynomial \(P(\zeta)\), the transformed point in \(\mathbb{C}\) is again projected back onto \(S^2\). This ensures a smooth and consistent transition between the sphere and the plane. The process of applying \(\pi_N^{-1}\) after \(P\) confirms that the points transformed by \(P\) return to their rightful places on the sphere, maintaining the differentiability of the overall map.