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Lie Groups (Definition)

Groups that are also smooth manifolds.
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The statement of the theorem

Let GG be a set equipped with a smooth manifold structure such that the underlying topological space is Hausdorff and second-countable. GG is a Lie Group if it is also a group (G,m,i)(G, \text{m}, \text{i}) such that the group multiplication map m:G×G→G\text{m}: G \times G \to G and the inversion map i:G→G\text{i}: G \to G are smooth maps. Specifically, if \phi: U \to \bb{R}^n are local coordinate charts for GG, the maps m\text{m} and i\text{i} must induce smooth functions on the coordinate representations. Formally, for any point g∈Gg \in G, there exist charts ϕ1,ϕ2,ϕ3\phi_1, \phi_2, \phi_3 around g,g1,g2g, g_1, g_2 respectively, such that the coordinate representations of the operations are smooth: \n\nm∗:C∞(U1)×C∞(U2)→C∞(U3)\text{m}_{*}: \text{C}^{\infty}(U_1) \times \text{C}^{\infty}(U_2) \to \text{C}^{\infty}(U_3) \ni∗:C∞(U)→C∞(U)\text{i}_{*}: \text{C}^{\infty}(U) \to \text{C}^{\infty}(U) \nwhere C∞\text{C}^{\infty} denotes the space of smooth functions, and the maps m∗\text{m}_{*} and i∗\text{i}_{*} are defined by the local coordinates of the group operations. Equivalently, GG is a Lie Group if and only if it is a smooth manifold such that the group operations m:G×G→G\text{m}: G \times G \to G and i:G→G\text{i}: G \to G are smooth maps in the manifold topology.