surfacemathematics

...The basic incentive in this regard was to find topological invariants associated with different structures. The simplest example is the Euler characteristic, which is a number associated with a surface. In 1750 the Swiss mathematician Leonhard Euler proved the polyhedral formula V – E + F = 2, or Euler characteristic, which relates...

...where f is a polynomial in x whose coefficients are polynomials in y. When x and y are complex variables, the locus can be thought of as a real surface spread out over the x plane of complex numbers (today called a Riemann surface). To each value of x there correspond a finite number of values of y. Such surfaces are not...

branch of mathematics that studies the geometry of curves, surfaces, and manifolds (the higher-dimensional analogs of surfaces). The discipline owes its name to its use of ideas and techniques from differential calculus, though the modern subject often uses algebraic and purely geometric techniques instead. Although basic definitions, notations, and analytic descriptions vary widely, the...

...up the task and looked for ways in which such manifolds could be distinguished, thus opening up the whole subject of topology, then known as analysis situs. Riemann had shown that in two dimensions surfaces can be distinguished by their genus (the number of holes in the surface), and Enrico Betti in Italy and Walther von Dyck in Germany had extended this work to three dimensions, but much...

...of a surface is intrinsic, and he argued that one should therefore ignore Euclidean space and treat each surface by itself. A geometric property, he argued, was one that was intrinsic to the surface. To do geometry, it was enough to be given a set of points and a way of measuring lengths along curves in the surface. For this, traditional ways of applying the calculus to the study of...