Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann,
is a model of the extended complex plane : the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the Riemann model, the point is near to very large numbers, just as the point is near to very small numbers.
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 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, the projective space of all complex lines in. 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.
Extended complex numbers
The extended complex numbers consist of the complex numbers together with. The set of extended complex numbers may be written as, and is often denoted by adding some decoration to the letter, such asThe notation has also seen use, but as this notation is also used for the punctured plane, it can lead to ambiguity.
Geometrically, the set of extended complex numbers is referred to as the Riemann sphere.
Arithmetic operations
of complex numbers may be extended by defining, for,and multiplication may be defined by
for all nonzero complex numbers, with. Note that,, and are left undefined. Unlike the complex numbers, the extended complex numbers do not form a field, since does not have an additive nor multiplicative inverse. Nonetheless, it is customary to define division on by
for all nonzero complex numbers with and. The quotients and are left undefined.
Rational functions
Any rational function can be extended to a continuous function on the Riemann sphere. Specifically, if is a complex number such that the denominator is zero but the numerator is nonzero, then can be defined as. Moreover, can be defined as the limit of as, which may be finite or infinite.The set of complex rational functions—whose mathematical symbol is —form all possible holomorphic functions from the Riemann sphere to itself, when it is viewed as a Riemann surface, except for the constant function taking the value everywhere. The functions of form an algebraic field, known as the field of rational functions on the sphere.
For example, given the function
we may define, since the denominator is zero at, and since as. Using these definitions, becomes a continuous function from the Riemann sphere to itself.
As a complex manifold
As a one-dimensional complex manifold, the Riemann sphere can be described by two charts, both with domain equal to the complex number plane. Let be a complex number in one copy of, and let be a complex number in another copy of. Identify each nonzero complex number of the first with the nonzero complex number of the second. Then the mapis called the transition map between the two copies of —the so-called charts—glueing them together. Since the transition maps are holomorphic, they define a complex manifold, called the Riemann sphere. As a complex manifold of 1 complex dimension, this is also called a Riemann surface.
Intuitively, the transition maps indicate how to glue two planes together to form the Riemann sphere. The planes are glued in an "inside-out" manner, so that they overlap almost everywhere, with each plane contributing just one point missing from the other plane. In other words, every point in the Riemann sphere has both a value and a value, and the two values are related by. The point where should then have -value ""; in this sense, the origin of the -chart plays the role of in the -chart. Symmetrically, the origin of the -chart plays the role of in the -chart.
Topologically, the resulting space is the one-point compactification of a plane into the sphere. However, the Riemann sphere is not merely a topological sphere. It is a sphere with a well-defined complex structure, so that around every point on the sphere there is a neighborhood that can be biholomorphically identified with.
On the other hand, the uniformization theorem, a central result in the classification of Riemann surfaces, states that every simply-connected Riemann surface is biholomorphic to the complex plane, the hyperbolic plane, or the Riemann sphere. Of these, the Riemann sphere is the only one that is a closed surface. Hence the two-dimensional sphere admits a unique complex structure turning it into a one-dimensional complex manifold.
As the complex projective line
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 : two non-null vectors and are equivalent iff for some non-zero coefficient.In this case, the equivalence class is written using projective coordinates. Given any point in the complex projective line, one of and must be non-zero, say. Then by the notion of equivalence,, which is in a chart for the Riemann sphere manifold.
This treatment of the Riemann sphere connects most readily to projective geometry. For example, any line 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.
As a sphere
The Riemann sphere can be visualized as the unit sphere in the three-dimensional real space. To this end, consider the stereographic projection from the unit sphere minus the point onto the plane which we identify with the complex plane by. In Cartesian coordinates and spherical coordinates on the sphere, the projection isSimilarly, stereographic projection from onto the plane identified with another copy of the complex plane by is written
The inverses of these two stereographic projections are maps from the complex plane to the sphere. The first inverse covers the sphere except the point, and the second covers the sphere except the point. The two complex planes, that are the domains of these maps, are identified differently with the plane, because an orientation-reversal is necessary to maintain consistent orientation on the sphere.
The transition maps between -coordinates and -coordinates are obtained by composing one projection with the inverse of the other. They turn out to be and, as described above. Thus the unit sphere is diffeomorphic to the Riemann sphere.
Under this diffeomorphism, the unit circle in the -chart, the unit circle in the -chart, and the equator of the unit sphere are all identified. The unit disk is identified with the southern hemisphere, while the unit disk is identified with the northern hemisphere.
Metric
A Riemann surface does not come equipped with any particular Riemannian metric. The Riemann surface's conformal structure does, however, determine a class of metrics: all those whose subordinate conformal structure is the given one. In more detail: The complex structure of the Riemann surface does uniquely determine a metric up to conformal equivalence. Conversely, any metric on an oriented surface uniquely determines a complex structure, which depends on the metric only up to conformal equivalence. Complex structures on an oriented surface are therefore in one-to-one correspondence with conformal classes of metrics on that surface.Within a given conformal class, one can use conformal symmetry to find a representative metric with convenient properties. In particular, there is always a complete metric with constant curvature in any given conformal class.
In the case of the Riemann sphere, the Gauss–Bonnet theorem implies that a constant-curvature metric
must have positive curvature. It follows that the metric must be isometric to the sphere of radius in via stereographic projection. In the -chart on the Riemann sphere, the metric with is given by
In real coordinates, the formula is
Up to a constant factor, this metric agrees with the standard Fubini–Study metric on complex projective space.
The two non-vanishing Christoffel symbols of its Levi-Civita connection are
and its conjugate.
This metric is therefore equal to its own Ricci curvature,
Up to scaling, this is the only metric on the sphere whose group of orientation-preserving isometries is 3-dimensional ; that group is called SO|. In this sense, this is by far the most symmetric metric on the sphere.
Conversely, let denote the sphere. By the uniformization theorem there exists a unique complex structure on up to conformal equivalence. It follows that any metric on is conformally equivalent to the round metric. All such metrics determine the same conformal geometry. The round metric is therefore not intrinsic to the Riemann sphere, since "roundness" is not an invariant of conformal geometry. The Riemann sphere is only a conformal manifold, not a Riemannian manifold. However, if one needs to do Riemannian geometry on the Riemann sphere, the round metric is a natural choice. That is because only a round metric on the Riemann sphere has its isometry group be a 3-dimensional group. |, a continuous