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.Opressor said:... I still do not know what it is!
A differentiable variety is a mathematical object that combines the concepts of a variety and a differentiable manifold. It is a space that is locally described by polynomial equations and has a well-defined notion of differentiability.
A differentiable variety is a special type of variety that has the additional property of being differentiable. This means that in addition to being described by polynomial equations, it also has a smooth structure that allows for the calculation of derivatives and other differential operators.
Some examples of differentiable varieties include algebraic curves and surfaces, as well as higher-dimensional algebraic varieties. These objects can be described by polynomial equations and also have a smooth structure that allows for the calculation of derivatives.
Differentiable varieties have many applications in mathematics and physics. They are used in the study of algebraic geometry, differential geometry, and topology. They also have applications in fields such as robotics, computer graphics, and machine learning.
Differentiable varieties are studied using a combination of algebraic and differential geometric techniques. This includes methods from commutative algebra, algebraic geometry, and differential topology. Computer algebra systems are also commonly used to study differentiable varieties.