Ivancevic_Applied-Diff-Geom

28 Applied Differential Geometry: A Modern IntroductionThere are several natural geometric notions of submanifold of a symplecticmanifold. There are symplectic submanifolds (potentially of anyeven dimension), where the symplectic form is required to induce a symplecticform on the submanifold. The most important case of these is thatof Lagrangian submanifold, which are isotropic submanifolds of maximal dimension,namely half the dimension of the ambient manifold. Lagrangiansubmanifolds arise naturally in many physical and geometric situations. 381.1.7 Lie GroupsA Lie group is smooth manifold which also carries a group structure whoseproduct and inversion operations are smooth as maps of manifolds. Theseobjects arise naturally in describing symmetries.A Lie group is a group whose elements can be continuously parametrizedby real numbers, such as the rotation group SO(3), which can beparametrized by the Euler angles. More formally, a Lie group is an analyticreal or complex manifold that is also a group, such that the groupoperations multiplication and inversion are analytic maps. Lie groups areimportant in mathematical analysis, physics and geometry because theyserve to describe the symmetry of analytical structures. They were introducedby Sophus Lie in 1870 in order to study symmetries of differentialequations.While the Euclidean space R n is a real Lie group (with ordinary vectoraddition as the group operation), more typical examples are given by matrixLie groups, i.e., groups of invertible matrices (under matrix multiplication).For instance, the group SO(3) of all rotations in R 3 is a matrix Lie group.One classifies Lie groups regarding their algebraic properties 39 (simple,semisimple, solvable, nilpotent, Abelian), their connectedness (connected38 One major example is that the graph of a symplectomorphism in the product symplecticmanifold (M × M, ω × −ω) is Lagrangian. Their intersections display rigidityproperties not possessed by smooth manifolds; the Arnold conjecture gives the sum ofthe submanifold’s Betti numbers as a lower bound for the number of self intersections ofa smooth Lagrangian submanifold, rather than the Euler characteristic in the smoothcase.39 If G and H are Lie groups (both real or both complex), then a Lie–group–homomorphism f : G → H is a group homomorphism which is also an analytic map (onecan show that it is equivalent to require only that f be continuous). The composition oftwo such homomorphisms is again a homomorphism, and the class of all (real or complex)Lie groups, together with these morphisms, forms a category. The two Lie groupsare called isomorphic iff there exists a bijective homomorphism between them whoseinverse is also a homomorphism. Isomorphic Lie groups do not need to be distinguishedfor all practical purposes; they only differ in the notation of their elements.