Manifolds 11 | Projective Space is a Manifold [dark version]

Real projective space P^n\mathbb{R} is shown to be a well-defined n-dimensional manifold by building an explicit atlas from the sphere S^n and the standard identification x \sim -x. The construction starts with the quotient map Q: S^n \to P^n\mathbb{R}, sending each point to its equivalence class. Two points on the sphere represent the same projective point exactly when they are antipodal, so the topology on P^n\mathbb{R} is the quotient topology induced by Q.

To prove manifold structure, the key move is to cover P^n\mathbb{R} with n+1 open sets that correspond to where a chosen homogeneous coordinate is nonzero. For each i = 1,\dots,n+1, define V_i \subset P^n\mathbb{R} as the set of equivalence classes [x] for which the i-th coordinate of x is not zero. Because Q^{-1}(V_i) consists of points on S^n where x_i \neq 0, it becomes a union of two open hemispheres on the sphere (the regions where x_i is positive or negative). Since those preimages are open in S^n and the topology is quotient, each V_i is open in P^n\mathbb{R}. Together, the V_i cover all of P^n\mathbb{R} because every point on the sphere has at least one nonzero coordinate.

The atlas then comes from charts H_i that turn each V_i into Euclidean space. The transcript illustrates the idea first for n=1, where P^1\mathbb{R} can be visualized as lines through the origin in the plane. On the circle S^1, the chart domain V_1 corresponds to lines excluding the one where x_1=0; the chart uses slope as the coordinate. For a point on the sphere with x_1 \neq 0, the slope is x_2/x_1, and this ratio is well-defined on equivalence classes because flipping x \mapsto -x cancels the sign. The chart map is bijective onto an open subset of \mathbb{R}, and its inverse is built by taking the vector (1, m), scaling it to lie on the unit circle, and then passing to its equivalence class. Continuity of both directions yields a homeomorphism, establishing local Euclidean behavior.

For general n, the same mechanism scales up: each chart H_i maps an equivalence class [x] (with x_i \neq 0) to a vector in \mathbb{R}^n by dividing every coordinate by x_i and omitting the i-th slot. Concretely, the image is (x_1/x_i, \dots, x_{i-1}/x_i, x_{i+1}/x_i, \dots, x_{n+1}/x_i), producing an n-component vector. The normalization ensures the representative lies on S^n, and the inverse is constructed by inserting a 1 in the i-th position and scaling back to the sphere before taking the equivalence class. Each (V_i, H_i) therefore provides a homeomorphism with \mathbb{R}^n, so P^n\mathbb{R} is locally Euclidean of dimension n.

With this atlas in place—and using standard background properties like second countability and the Hausdorff condition—the result is that real projective space is a bona fide n-dimensional manifold. The next step, hinted for a later installment, is adding differentiability structure to these manifolds.