Is There a Unique Torsion-Free Affine Connection on a Lie Group?

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Discussion Overview

The discussion centers around the existence and uniqueness of a torsion-free affine connection on a Lie group, specifically addressing the condition that the connection vanishes for all left invariant vector fields. Participants explore the implications of the Lie algebra being Abelian in relation to the torsion-free condition.

Discussion Character

  • Technical explanation, Homework-related, Conceptual clarification

Main Points Raised

  • One participant suggests showing the existence of a unique affine connection such that \nabla X=0 for all left invariant vector fields.
  • Another participant questions whether the discussion pertains to homework, indicating a possible misunderstanding of the context.
  • A participant mentions their background in PDEs and their interest in Riemannian Geometry, implying a focus on the theoretical aspects of the problem.
  • It is proposed that the torsion tensor can be defined in terms of the connection and the commutator, suggesting that the condition for the Lie algebra to be Abelian relates to the torsion being identically zero.
  • One participant recommends defining the connection to be zero at the identity and using left translation to extend this definition, presenting a method for constructing the connection.

Areas of Agreement / Disagreement

The discussion includes multiple viewpoints and approaches to the problem, with no clear consensus on the methods or implications presented.

Contextual Notes

Participants express varying levels of familiarity with the topic, and there are indications of differing interpretations regarding the nature of the discussion (homework vs. theoretical exploration). The relationship between the torsion-free condition and the Abelian nature of the Lie algebra remains a point of exploration.

Who May Find This Useful

Readers interested in differential geometry, Lie groups, and the properties of affine connections may find the discussion relevant.

zhangzujin
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Let G be a Lie group. Show that there exists a unique affine connection such that \nabla X=0 for all left invariant vector fields. Show that this connection is torsion free iff the Lie algebra is Abelian.
 
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Homework?
 
hamster143 said:
Homework?
Aha. Of course not. I'm just reading Riemannian Geometry by Petersen, interested in the exercises of that.

In fact, my major is PDEs.
 
The second statement shouldn't be bad, if you define the torsion tensor in terms of the connection and the commutator (i.e. show that [X,Y] is identically zero if and only if the Lie algebra is Abelian - shouldn't be too hard :) ).

For the first part, why not define the connection to be zero at the identity, and then drag all your vectors back there by left translation?
 

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