A description/proof of that 2 points on connected Lie group can be connected by finite left-invariant vectors field integral curve segments
Topics
About: Lie group
About: vectors field
The table of contents of this article
Starting Context
- The reader knows a definition of Lie group.
- The reader knows a definition of connected topological space.
- The reader knows a definition of path-connected topological space.
- The reader knows a definition of Lie algebra.
- The reader knows a definition of diffeomorphism.
- The reader knows a definition of left-invariant vectors field.
- The reader knows a definition of integral curve of vectors field.
- The reader knows a definition of exponential map on Lie group.
- The reader admits the proposition that any connected topological manifold is path-connected.
- The reader admits the proposition that on any Lie group there is a diffeomorphic exponential map between a neighborhood of 0 on the Lie algebra and a neighborhood of e on the Lie group.
- The reader admits the proposition that the integral curve of any left-invariant vectors field starting at any point on any Lie group is the integral curve of the vectors field starting at e, left-multiplied by the point.
Target Context
- The reader will have a description and a proof of the proposition that any 2 points on any connected Lie group can be connected by a finite number of segments each of which is of an integral curve of a left-invariant vectors field.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any connected Lie Group, G, any 2 points, \(p_1, p_2 \in G\), can be connected by a finite number of segments each of which is of an integral curve, \(\gamma _i: \mathbb{R} \rightarrow G\), of a left-invariant vectors field, \(V_i\).
2: Proof
As G is a \(C^\infty\) manifold and is connected, it is path-connected, so take a path, \(\gamma_1: [r_1, r_2] \rightarrow G\), that connects \(e\) to \(p_1\) as the terminals where e is the unit point of G. There are a neighborhood, \(U_0\), on the Lie algebra, \(\mathfrak{g}\), and a neighborhood, \(U_e\), on G such that the exponential map, \(exp: U_0 \rightarrow U_e\), is a diffeomorphism. At any point, \(p_i \in G\), \(U_{p_i} := \{\forall p| p_i^{-1} p \in U_e\}\) is an open set because it is the preimage of \(U_e\) by the \(C^\infty\) map, \(f: G \rightarrow G, p \mapsto p_i^{-1} p\). For any \(p \in U_{p_i}\), there is a vector, \(l_{p_i*, e} V_e\), at \(p_i\) such that the integral curve of the left-invariant vectors field, V, starting at \(p_i\) passes p, because by choosing \(V_e\) as \(p_i^{-1}p = exp V_e\), \(p = p_i exp V_e = \phi_1 (p_i)\) by the proposition that the integral curve of any left-invariant vectors field starting at any point on any Lie group is the integral curve of the vectors field starting at e, left-multiplied by the point. Now, \(\gamma_1\) can be covered by such open sets, and as the path is compact, there is a finite sub cover, \(U_{p_{1-0}} = U_e, U_{p_{1-1}}, . . ., U_{p_{1-k}}, U_{p_{1-k + 1}} = U_{p_1}\). \(U_{p_{1-0}}\) shares a point with one of the rest open sets, because otherwise, in the \(\gamma _1\) subspace topology, \(U_{a-0} := U_{p_{1-0}} \cap \gamma _1\) and \(U_{r-0} := (\cup _{1-i \neq 1-0} U_{p_{1-i}}) \cap \gamma _1\) are open and they would be disjoint and \(U_{a-0} \cup U_{r-0} = \gamma _1\), which means that \(\gamma _1\) would be disconnected, a contradiction (\(\gamma\) is connected as a continuous image of the connected \([r_1, r_2]\)). Denoting one of the possibly multiple sharing open sets by \(U_{p_{1-c1}}\) and the shared point by \(p_{1-s1}\), \(p_{1-s1}\) is in \(U_{p_{1-0}}\) and so can be connected to e by the integral curve of a left-invariant vectors field and is also in \(U_{p_{1-c1}}\) and so can be connected to \(p_{1-c1}\) by the integral curves of a left-invariant vectors field; if \(p_{1-c1} = p_1\), e is already connected to \(p_1\), if not, \(U_{p_{1-0}} \cup U_{p_{1-c1}}\) shares a point with one of the rest open sets and denoting one of the possibly multiple sharing open sets by \(U_{p_{1-c2}}\) and the shared point by \(p_{1-s2}\), \(p_{1-s2}\) is in \(U_{p_{1-0}}\) or \(U_{p_{1-c1}}\) and so can be connected to e or \(p_{1-c1}\) and is also in \(U_{p_{1-c2}}\) and so can be connected to \(p_{1-c2}\); if \(p_{1-c2} = p_1\), e is already connected to \(p_1\) directly or via \(p_{1-c1}\), if not, \(U_{p_{1-0}} \cup U_{p_{1-c1}} \cup U_{p_{1-c2}}\) shares a point with one of the rest open sets . . ., and so on; after all, e is connected to \(p_1\) by some finite number of segments because \(U_{p_{1-ci}}\) becomes \(U_{p_1}\) sometime in the finite iteration. Likewise, e is connected to \(p_2\) by some finite number of segments each of which is of an integral curve of a left-invariant vectors field. \(p_1\) can be connected to \(p_2\) via e.
