A What is a differentiable variety?

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In mathematics, variety is a generalization of the surface idea. There are several types of varieties, according to the properties they possess. The most usual are the topological varieties and the differentiable varieties. but I still do not know what it is!
 

fresh_42

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To answer the title question: It is called a manifold and locally a Euclidean space such that we can apply analytical and topological methods.

Variety is usually called "algebraic variety" in algebraic geometry. They are defined as the zeros of multivariate polynomials, which lead to ideals in rings, so that ring theory is applicable to investigate the geometry of these surfaces. However, they are called manifolds, when similar is done with means of analysis and topology to investigate the (differential) geometry. I haven't seen these concept mixed, it's a bit of a stretch.
... I still do not know what it is!
This isn't the place, neither for a lecture in algebraic geometry, nor in differential geometry. How could I answer such a question? They are surfaces as you have said, defined by equations. So this answers your question and probably simultaneously doesn't. Can you be more specific? Otherwise I have to recommend our Science and Math Textbooks forum, where you can find or ask for appropriate book recommendations.

Btw, if you add algorithm varieties to your list, you'll get a third kind and computer science as toolbox.
 
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