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.
The exponential map for Lie groups is a mathematical tool that maps elements of a Lie group (a type of mathematical group) to elements of its corresponding Lie algebra (a vector space associated with the group). It is a fundamental concept in the study of Lie groups and their applications in physics, engineering, and other fields.
The exponential map is defined as a mapping from the tangent space of a Lie group at the identity element to the group itself. In other words, it takes a vector from the tangent space and "exponentiates" it to obtain an element of the group. The specific formula for the exponential map varies depending on the specific Lie group and its corresponding Lie algebra.
The exponential map plays a crucial role in Lie group theory as it allows for the translation of algebraic operations on the Lie algebra to the corresponding Lie group. This enables the study of Lie groups using tools and techniques from linear algebra and differential geometry.
The exponential map has numerous applications in various fields of mathematics, physics, and engineering. It is used in the study of differential equations, dynamical systems, and optimization problems. It is also essential in the fields of robotics, computer graphics, and geometric modeling.
While the exponential map is a powerful and useful tool, it is not without limitations. In some cases, the exponential map may not be well-defined, leading to difficulties in its application. Additionally, the computation of the exponential map can be computationally expensive, making it challenging to use in certain applications.