An analytic manifold G that has a group structure such that that the map (x,y) - xy from G x G to G is analytic (i. e. infinitely differentiable). The general linear groups GL(), GL(), and their subgroups, are the most common examples. There is an extensive and deep theory of Lie groups, with many theoretical and applied uses.

A differentiable manifold that has a group structure on its elements, with the property that the group operations (multiplication and inversion) are continuous. A standard and useful example is the set of nonsingular real matrices. Other examples are the group of rigid motions in Euclidean space and the group of equivalence classes of transformations that agree on some neighborhood of the identity. The last was the original object of study by Sophus Lie around 1890.

a differentiable manifold obeying the group properties and that

a differentiable manifold with an additional structure that allows us to multiply the points on the manifold, and define the inverse of a point

a group object in the category of smooth manifolds with differentiable maps

a group whereof the product of group elements is differentiable

a group which is also a C manifold or a complex manifold

a manifold which is also a group

a smooth manifold (real and finite dimensional) that is also a group

In mathematics, a Lie group (IPA pronunciation: , sounds like "Lee") is a smooth group, in the sense that the set of group elements has topology and smooth structure of a smooth manifold, and the group operations are smooth functions of the elements.