Lie theory in the context of "Lie group"

Jules Henri Poincaré (UK: /ˈpwæ̃kɑːreɪ/, US: /ˌpwæ̃kɑːˈreɪ/; French: [ɑ̃ʁi pwɛ̃kaʁe] ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. He has further been called "the Gauss of modern mathematics". Due to his success in science, along with his influence and philosophy, he has further been called "the philosopher par excellence of modern science".

As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. Poincaré is regarded as the creator of the field of algebraic topology, and is further credited with introducing automorphic forms. He also made important contributions to algebraic geometry, number theory, complex analysis and Lie theory. He famously introduced the concept of the Poincaré recurrence theorem, which states that a state will eventually return arbitrarily close to its initial state after a sufficiently long time, which has far-reaching consequences. Early in the 20th century he formulated the Poincaré conjecture, which became, over time, one of the famous unsolved problems in mathematics. It was eventually solved in 2002–2003 by Grigori Perelman. Poincaré popularized the use of non-Euclidean geometry in mathematics as well.

In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.

In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (M, g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space ${\displaystyle T_{p}M}$ as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about the endpoints). Both descriptions can also naturally be extended to the setting of pseudo-Riemannian manifolds.