Nikolai Invanovich Lobachevsky
Euclid's "parallel postulate" --- the existence of a unique parallel
to a fixed line through a fixed point --- defied many attempts to
deduce it from the other axioms. Many results were proved equivalent
to it, for example the sum of angles in a triangle equals 180 degrees.
Investigations into the study of parallel lines led Janos Bolyai
(1802-1860), Carl Friedrich Gauss (1777-1855) and Nikolai Invanovich
Lobachevsky (1792-1856) to models of geometry in which either no parallel
lines exist at all or infinitely many lines through a fixed point can miss
a given line.
Carl Friedrich Gauss
Nikolai Invanovich Lobachevsky
Janos Bolyai
Groups of symmetry were exploited in the early 19th century by Galois, and Jordan to prove that fifth-degree equations could not be solved by radicals, as in the quadratic formula. Although more complicated "cubic" and "quartic" formulas akin the quadratic formula can be used to solve equations of the third and fourth degree, no such formulas exist.
Inspired by the success of Galois in algebraic equations, Marius Sophus Lie formulated a theory of "continuous groups" to play a role for differential equations analogous to the role that finite groups play for algebraic equations. In particular the group of symmetries of a "geometry" forms a continuous group. Influenced by the ideas of his friend Lie, Felix Klein proposed in a lecture delivered at Erlangen that geometry can be unified and approached in terms of Lie's continuous groups of transformations. In Klein's approach, a geometry is the study of quantities which are unchanged by the transformations preserving the geometry, such as rigid motions in Euclidean geometry, or angle-preserving transformations in conformal geometry. This lead to a great unification of Euclidean geometry, projective geometry, non-Euclidean geometry and many other geometries.
Sophus Lie
In his 1854 Inagaural Lecture, "On the hypotheses underlying the foundations of geometry," George Bernhard Riemann () boldly generalized Gauss's theory of surfaces. He imagined a space of arbitrary dimension, a "many-fold" space with a notion of length infinitesimally described by a quadratic function such as dx^2 + dy^2 + dz^2. Such Riemannian manifolds included the elliptic and hyperbolic spaces of Bolyai, Gauss and Lobachevsky, but included many more. Riemann's revolutionary ideas were given a further twist and application by Albert Einstein, who proposed a model for space-time which had 4 total dimensions --- 3 of space and 1 of time --- but the infinitesimal measure of "distance" between "events" was given by a quadratic function like dx^2 + dy^2 + dz^2 - c^2 dt^2 (where c is the speed of light). A space with such a structure is called a "Lorentzian manifold" and the theory of Lie groups provides many analogues of non-Euclidean geometry among such Lorentzian manifolds.
Albert Einstein
Elie Joseph Cartan and Charles Ehresmann took this generalization further. Using the theory of Lie groups as the basis for the infinitesimal geometry (like the quadratic functions defined infinitesimally on Riemannian and Lorentzian manifolds), geometric structures could be defined using extremely general local symmetries. Many of these structures have been used as alternate geometric models for the universe.
Ehresmann's "locally homogeneous geometric structures" could be considered like an atlas of local maps of a loosely organized collection of points. The qualitative organization of a continuous aggregate of points is what is called a "topological space" and one might want to impart a geometric structure to a topological space modelled on a classical geometry, such as projective geometry. Thus, each point has a "neighborhood" or "coordinate patch" with a "coordinate chart" mapping into, say, projective space. On overlapping patches, the coordinate charts are related by some transformation in the geometry. The study of such geometric structures was enriched when in the late 1970's, William Thurston suggested that all 3-dimensional manifolds could be decomposed into pieces each with a local geometry. In particular the most interesting and "typical" 3-dimensional manifold would support hyperbolic non-Euclidean geometry. Although much evidence exists for this conjectural picture, Thurston's conjectures are still unproved.
Charles Ehresmann
William Thurston
