Algebraic manifold

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.

For an algebraic manifold, the ground field will be the real numbers or complex numbers; in the case of the real numbers, the manifold of real points is sometimes called a Nash manifold.

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.

Examples
gollark: How can you be in weather that hot without imploding? It's been something like 25 good degrees here lately and even that is very unpleasant.
gollark: Well, they say exercise is good for your health.
gollark: Dunbar's number is someone's wild guess at maximum social relationships per person or something, not average friend count.
gollark: How is fatigue not a symptom?
gollark: Really? Because apparently post-viral fatigue happens with the flu sometimes.

See also

References
  • Nash, John Forbes (1952). "Real algebraic manifolds". Annals of Mathematics. 56 (3): 405–21. doi:10.2307/1969649. MR 0050928. (See also Proc. Internat. Congr. Math., 1950, (AMS, 1952), pp. 516–517.)


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