In mathematics, an algebraic manifold is an algebraic variety which is also a manifold. As such, algebraic manifolds are a generalisation of the concept of smooth curves and surfaces defined by polynomials. An example is the sphere, which can be defined as the zero set of the polynomial x2 + y2 + z2 – 1, and hence is an algebraic variety.
Every sufficiently small local patch of an algebraic manifold is isomorphic to km where k is the ground field. Equivalently the variety is smooth (free from singular points). The Riemann sphere is one example of a complex algebraic manifold, since it is the complex projective line.
- Nash, John Forbes (1952). "Real algebraic manifolds". Annals of Mathematics 56 (3): 405–21. doi:10.2307/1969649. (See also Proc. Internat. Congr. Math., 1950, (AMS, 1952), pp. 516–517.)
- K-Algebraic manifold at PlanetMath
- Algebraic manifold at Mathworld
- Lecture notes on algebraic manifolds
Original source: https://en.wikipedia.org/wiki/Algebraic manifold. Read more