Riemann sphere

In mathematics, the Riemann number sphere is the Riemann surface, which results from the addition of a point at infinity to the complex plane. It goes back to Bernhard Riemann.

Next is a topology on the Riemann number sphere is defined as follows: Open quantities on the one hand, the open sets in and the other, the respect formed complements of compact subsets of. The topological space so defined is a compactification of the complex plane dar. topology, it is equivalent to the unit sphere. With the chordal metric, the Riemann sphere is a metric space. This metric induced the same topology that is induced by the number Einpunktkompaktifizierung ball.

The complex structure of the Riemann number sphere is given by two cards. The first is to define and is the identity. The second is defined on the neighborhood of the infinitely distant point by

Clearly there is a sphere of radius 1, whose north pole is at ( 0,0,1 ) ( one can choose arbitrarily the ball as long as its north pole ( 0,0,1 ) is ). The point at infinity is the north pole of the sphere and assigned to each point of the complex plane of the different intersection of the sphere with the line through ( stereographic projection).

The automorphisms, ie biholomorphic images of the Riemann number sphere onto itself, constitute the group of Möbius transformations.