Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 15: Problem 3

Show that the real projective \(n\)-space \(P^{\text {* }}\) defined in Example \(10.15\) as the set of straight lines through the origin in \(\mathbb{R}^{n+1}\) is a differentiable manifold of dimension \(n\), by finding an atlas of compatible charts that cover it.

Short Answer

Expert verified
An atlas for \( P^* \) is constructed by partitioning \( P^* \) into \( n+1 \) sets \( U_i \), each corresponding to the lines intersecting an upper-half sphere in a different open half-space. Charts are then defined by mapping each line into its \( n \) nth coordinates in \( R^{n+1} \). These charts are shown to be compatible with one another, forming an atlas. Thus, \( P^* \) is a differentiable manifold of dimension \( n \).

Step by step solution

01

Partition the Projective Space

Divide \( P^* \) into \( n+1 \) sets \( U_i \) for \( i = 0, 1, ..., n \), where \( U_i \) consists of all lines in \( P^* \) which intersect the \( (n+1) \)-dimensional upper-half sphere in the \( (i+1) \)-th open half space of \( R^{n+1} \). Note that these sets \( U_i \) cover \( P^* \), since any line in \( P^* \) will intersect the upper-half sphere in exactly one point.
02

Definition of Charts

Define charts \( \phi_i: U_i \to \mathbb{R}^n \) by mapping the line through a point in \( U_i \) to its \( n \) nth coordinates in \( R^{n+1} \). Care should be taken to avoid the \( i \)-th coordinate for each \( U_i \). These charts are bijective and have continuous, constant rank derivatives, so are indeed diffeomorphisms from \( U_i \) to \( \mathbb{R}^n \).
03

Compatibility of Charts

Now it needs to be shown that these charts are compatible. For each pair \( i \ne j \), consider \( U_i \cap U_j \). The corresponding map \( \phi_j \circ \phi_i^{-1} \) between open subsets of \( \mathbb{R}^n \) is clearly differentiable, as it is a ratio of polynomials where the denominator doesn't vanish, hence \( \phi_i \) and \( \phi_j \) are compatible.
04

Conclusion

Since the \( U_i \)'s cover the projective space and the corresponding charts are compatible, \( \{ (U_i, \phi_i) \} \) forms an atlas for \( P^* \). Hence, \( P^* \) is a differentiable manifold of dimension \( n \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that the Lie derivative \(\mathcal{L}_{x}\) commules with all operations of contraction \(C_{t}^{y}\) on a tensor field \(T\), $$ \mathcal{L}_{X} C_{J} T=C_{j} \mathcal{L}_{\mathrm{Y}} T $$

Show that the curve $$ 2 x^{2}+2 y^{2}+2 x y=1 $$ can be converted by a rotation of axcs to the standand form for an ellipse $$ x^{\prime 2}+3 y^{2}=1 $$ If \(x^{\prime}=\cos \psi, v^{\prime}=\frac{1}{\sqrt{3}} \sin \psi\) is used as a parametrization of this curve, show that $$ x=\frac{1}{\sqrt{2}}\left(\cos \psi+\frac{1}{\sqrt{3}} \sin \psi\right), \quad y=\frac{1}{\sqrt{2}}\left(-\cos \psi+\frac{1}{\sqrt{3}} \sin \psi\right) $$ Compute the components of the tangent vector $$ X=\frac{\mathrm{d} x}{\mathrm{~d} \psi} \partial_{x}+\frac{\mathrm{d} y}{\mathrm{~d} \psi} \partial_{y} $$ Show that \(X(f)=(2 / \sqrt{3})\left(x^{2}-y^{2}\right)\).

For any real positive number \(n\) show that the vector field \(X=x^{n} \partial_{x}\) is differentiable on the manifold \(\mathbb{R}^{+}\)consisting of the positive real line \([x \in \mathbb{R} \mid x>0]\). Why is this not true in general on the enture real line \(\mathrm{R}\) ? As done for the case \(n=2\) in Example 15,13, find the maximal one-parameter subgroup \(\sigma_{i}\) generated by this vector field at any point \(x>0\).

Express the vector field \(\partial_{\varphi}\) in polar coordinates \((\theta, \phi)\) on the unit 2 -sphere in terms of stereographic coordinates \(X\) and \(Y\).

Show that the set of real \(m \times m\) matrices \(M(m, n ; \mathbb{R})\) is a manifold of dimension \(m n\). Show that the matrix multiplication map \(M(m, k ; \mathbb{R}) \times M(k, n ; \mathbb{R}) \rightarrow M f(m, n ; \mathbb{R})\) is differentiable.
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free