The exponential map for Lie groups is defined using one-parameter subgroups associated with given tangent vectors, rather than relying on a metric or the Levi-Civita connection. While traditional definitions involve the geodesic ordinary differential equation (ODE) and Christoffel symbols, these are not necessary for defining the exponential map in the context of Lie groups. The discussion emphasizes the importance of understanding connections in this framework, highlighting that metric compatibility is irrelevant without a metric.
PREREQUISITESMathematicians, physicists, and students of differential geometry seeking to deepen their understanding of Lie groups and the exponential map without relying on traditional metric-based approaches.
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 said:But the exponential map is based on the geodesic ODE, so you need Christoffel symbols and thus the metric.