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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 \( n \).

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Most popular questions from this chapter

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 \(x>0\).

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