Riemann sphere

The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as ${\displaystyle 1/0=\infty }$ well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.


In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry, the sphere is an example of a complex projective space and can be thought of as the complex projective line ${\displaystyle \mathbf {P} ^{1}(\mathbf {C} )}$, the projective space of all complex lines in ${\displaystyle \mathbf {C} ^{2}}$. As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve, making it a fundamental example in algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as the Bloch sphere of quantum mechanics and in other branches of physics.

As the complex projective line[edit]

The Riemann sphere can also be defined as the complex projective line. The points of the complex projective line can be defined as equivalence classes of non-null vectors in the complex vector space ${\displaystyle \mathbf {C} ^{2}}$: two non-null vectors ${\displaystyle (w,z)}$ and ${\displaystyle (u,v)}$ are equivalent iff ${\displaystyle (w,z)=(\lambda u,\lambda v)}$ for some non-zero coefficient ${\displaystyle \lambda \in \mathbf {C} }$.


In this case, the equivalence class is written ${\displaystyle [w,z]}$ using projective coordinates. Given any point ${\displaystyle [w,z]}$ in the complex projective line, one of ${\displaystyle w}$ and ${\displaystyle z}$ must be non-zero, say ${\displaystyle w\neq 0}$. Then by the notion of equivalence, ${\displaystyle [w,z]=\left[1,z/w\right]}$, which is in a chart for the Riemann sphere manifold.^[3]


This treatment of the Riemann sphere connects most readily to projective geometry. For example, any line (or smooth conic) in the complex projective plane is biholomorphic to the complex projective line. It is also convenient for studying the sphere's automorphisms, later in this article.