The term "manifold" comes from German Mannigfaltigkeit, by Bernhard Riemann(who made profound contributions to analysis, number theory, and differential geometry.1826-1866). He expressed in his lecture "On the Hypotheses which lie at the bases of Geometry". It is a very broad and abstract generalization of the differential geometry of surfaces $R^3$. The concept of manifold is a concept that allows us to deal with different geometries outside of Euclidean space, that is, planar (linear) spaces. The main motivation for the concept of manifold is Gaussian curvature. Gauss(was a German mathematician, astronomer, geodesist, and physicist who lived between 1777-1855 and contributed to many fields in mathematics and science) observed that not every surface is flat, but when we look at it pointwise, it can behave like a plane. Gaussian curvature is the measure of deviation of a surface from its tangent plane at that point. For example, of course, our Earth is not flat, but when we look at a flat plain on the Earth, you may have a hard time noticing that the Earth is not flat. Because we are a point on the Earth and the Earth is flat as a point, but not in its general structure. Curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being its tangent plane. In 1905, Albert Einstein(was a German-born theoretical physicist who is widely held as one of the most influential scientists. Best known for developing the theory of relativity and lived between 1879-1955) published special relativity, which he had been thinking about for a long time. However, this required strong mathematical foundations. Of course, Euclidean geometry was not sufficient for this. more was needed. Here manifolds were needed.