How Is the Exponential Map Defined for Lie Groups Without a Metric?

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Discussion Overview

The discussion centers on the definition of the exponential map for Lie groups, particularly in the absence of a metric. Participants explore the relationship between the exponential map, geodesics, and connections, questioning how the exponential function can be understood in this context.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that the exponential map is typically associated with the geodesic ODE, which requires Christoffel symbols and a metric.
  • Another participant argues that a connection is sufficient for defining the exponential map, and that metric compatibility is irrelevant without a metric.
  • A third participant acknowledges their lack of experience with non-Levi-Civita connections, suggesting a personal gap in understanding.
  • It is pointed out that the exponential map for a Lie group is defined via one-parameter subgroups associated with a given tangent vector, rather than relying solely on geodesics.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of a metric for defining the exponential map, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

The discussion highlights the distinction between the use of geodesics with a connection versus the specific requirements of the exponential map in the context of Lie groups, which may depend on the definitions and assumptions made by participants.

wacki
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I’ve read about the exponential map that for Lie groups the exponential map is actually the exponential function. But the exponential map is based on the geodesic ODE, so you need Christoffel symbols and thus the metric. But usually nobody gives you a metric with a Lie group. So how can I get the exponential map (and finally see that it’s just the exponential function)?
 
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wacki said:
But the exponential map is based on the geodesic ODE, so you need Christoffel symbols and thus the metric.
No you do not. You need a connection, not necessarily the Levi-Civita connection. In fact, asking for metric compatibility is senseless without a metric.
 
Ahh yes, thanks Orodruin.
I clearly lack experience and intuition with non-Levi-Civita connections.
 
You need to be careful when you say the exponential map. The exponential map for a Lie group is defined using the one parameter subgroup with a given tangent vector. For a manifold with a connection the geodesic is used.
 

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