271: Point on Connected Lie Group Can Be Expressed as Finite Product of Exponential Maps

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A description/proof of that point on connected Lie group can be expressed as finite product of exponential maps

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Topics

About: Lie group
About: vectors field

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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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 Lie algebra.
• The reader knows a definition of exponential map on Lie group.
• The reader admits 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.
• 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.

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Target Context

• The reader will have a description and a proof of the proposition that any point on any connected Lie group can be expressed as the product of some finite number of exponential maps.

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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.

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Main Body

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1: Description

For any connected Lie Group, G, any point, $p\in G$, can be expressed as $p=exp(V_{0,e})exp(V_{1,e})...exp(V_{k,e})$.

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2: Proof

p can be connected with e by a finite number of segments each of which is an integral curve of a left-invariant vectors field via $p_0:=e\to p_1\to ...\to p_{k+1}:=p$. $p_{i+1}=p_{i}exp{V}_{i,e}$. $p=p_{k+1}=p_{0}exp{V}_{0,e}exp{V}_{1,e}...exp{V}_{k,e}$.

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References

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