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

[wacki]

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)?

 

[Orodruin]

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.

 

[wacki]

Ahh yes, thanks Orodruin.
I clearly lack experience and intuition with non-Levi-Civita connections.

 

[martinbn]

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.