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https://en.wikipedia.org/wiki/History_of_manifolds_and_varieties
History of manifolds and varieties

Nomenclature
The term "manifold" comes from German Mannigfaltigkeit, by Bernhard Riemann.

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.

In Romance languages, manifold is translated as "variety" ? such spaces with a differentiable structure are literally translated as "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties". Thus for example, the French word "variete topologique" means topological manifold. In the same vein, the Japanese word "多様体" (tay?tai) also encompasses both manifold and variety. ("多様" (tay?) means various.)

Background
Lagrangian mechanics and Hamiltonian mechanics, when considered geometrically, are naturally manifold theories. All these use the notion of several characteristic axes or dimensions (known as generalized coordinates in the latter two cases), but these dimensions do not lie along the physical dimensions of width, height, and breadth.

In the early 19th century the theory of elliptic functions succeeded in giving a basis for the theory of elliptic integrals, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen?

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