What is the difference between manifold and variety?

What is the difference between manifold and variety?

In English, “manifold” refers to spaces with a differentiable or topological structure, while “variety” refers to spaces with an algebraic structure, as in algebraic varieties.

Are all manifolds varieties?

There can be varieties that are not manifolds, for instance, y2−x2(x+1)=0 is a “nodal cubic” and so it has a singularity at (0,0). It can’t be a manifold because it looks like “X” a cross at the origin so is not homeomorphic locally to R.

What is a manifold in space?

A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in. ). To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round.

Are all algebraic varieties manifolds?

Every non-singular algebraic variety over C is a smooth manifold. See for instance: http://en.wikipedia.org/wiki/Manifold under “Generalizations of Manifolds”.

Are algebraic varieties manifolds?

Many algebraic varieties are manifolds, but an algebraic variety may have singular points while a manifold cannot. Algebraic varieties can be characterized by their dimension. Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces.

What do you understand by manifold classification explain?

If a population is divided into a number of mutually exclusive classes according to some given characteristic and then each class is divided by reference to some second, third, etc. characteristic, the final grouping is called a manifold classification.

How are manifolds used in physics?

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. A Riemannian metric on a manifold allows distances and angles to be measured.

What is a projective manifold?

A projectively flat manifold (orbifold) is a manifold (orbifold) with an atlas of charts to the projective space with transition maps in the projective automorphism group. These objects are closely related to the representations of groups into the projective groups PGL(n + 1, R).

Are varieties irreducible?

Definition An affine variety is reducible if it is the union of proper subvarieties . Otherwise, is irreducible. That is, an affine variety is irreducible if whenever is written in the form , where and are affine varieties, then either or .