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In mathematics, the closed subgroup theorem is a theorem in the theory of Lie groups. It states that if is a closed subgroup of a Lie group , then is an embedded Lie group with the relative topology being the same as the group topology.〔 Theorem 20.10. Lee states and proves this theorem in all generality.〕〔 Theorem 1, Section 2.7 Rossmann states the theorem for linear groups. The statement is that there is an open subset such that is an analytic bijection onto an open neighborhood of in .〕 One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan,〔 See § 26.〕 who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations.〔; .〕 == Informal description == Let be a Lie group, whose associated Lie algebra under the Lie correspondence is . If is any subgroup of ''G'', not necessarily a closed subgroup, then } is a Lie subalgebra of ; here is the exponential map from the Lie algebra to the Lie group. As a consequence of the Lie correspondence, the connected component of is the closed subgroup of ''G'' generated by .〔 Section 2.6.〕 An open set that contains the origin is said to be "small enough" when the exponential map has an inverse function on satisfying the following two conditions: #For every in , , and #For every in , . Then the sets of the form for group elements in and for small enough open sets form a basis for a topology on , called the ''group topology''.〔 At each fixed , the sets form a neighborhood basis. Willard's theorem 5.4 then guarantees that as both and varies form a basis for some topology.〕 Also, the function that maps each element of to the element of provides coordinates in small enough neighborhoods of the identity of , and more generally the function that maps in to in provides smooth coordinates around any element of .〔 Section 4.1.〕 These are called ''exponential coordinates''. Their values belong to the Lie algebra rather than being real numbers, but this is immaterial because is a vector space so (given a basis for this space) its elements can be expanded into tuples of real-number coordinates. In exponential coordinates the group with the group topology is a Lie group, i.e. an analytic manifold with analytic group operations with respect to the topology. Embedded submanifolds are characterized by so called ''slice charts'' on the embedding manifold. If is a slice chart in on , then , where is the dimension of . If each is in the domain of a slice chart on in , then is an embedded submanifold.〔 Embedded manifolds are defined in Chapter 8.〕 Embedded submanifolds always have the subspace topology. It is not necessarily true that the subspace topology is the same as the group topology. The group topology is often finer than the relative topology. If this is the case, then is not an embedded submanifold of , but may instead be an immersed submanifold. The closed subgroup theorem provides a sufficient condition for a subgroup to be an embedded Lie subgroup. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「closed subgroup theorem」の詳細全文を読む スポンサード リンク
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