In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Lie subgroup H of a Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 G is a Lie group that is a subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

 of G and such that the inclusion map

Inclusion map

In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....

 from H to G is an injective immersion and group homomorphism

Group homomorphism

In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

. According to Cartan's theorem

Cartan's theorem

In mathematics, three results in Lie group theory are called Cartan's theorem, named after Élie Cartan:See also Cartan's theorems A and B, results of Henri Cartan, and Cartan's lemma for various other results attributed to Élie and Henri Cartan....

, a closed subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

 of G admits a unique smooth structure

Smooth structure

In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....

 which makes it an embedded

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

 Lie subgroup of G -- i.e. a Lie subgroup such that the inclusion map is a smooth embedding.

Examples of non-closed subgroups are plentiful; for example take G to be a torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

 of dimension ≥ 2, and let H be a one-parameter subgroup of irrational slope, i.e. one that winds around in G. Then there is a Lie group homomorphism

Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

 φ : R → G with H as its image. The closure

Closure (topology)

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...

 of H will be a sub-torus in G.

In terms of the exponential map

Exponential map

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....

of G, in general, only some of the Lie subalgebras of the Lie algebra g of G correspond to closed Lie subgroups H of G. There is no criterion solely based on the structure of g which determines which those are.