gtsam 4.1.1
gtsam

A manifold defines a space in which there is a notion of a linear tangent space that can be centered around a given point on the manifold.
These nonlinear spaces may have such properties as wrapping around (as is the case with rotations), which might make linear operations on parameters not return a viable element of the manifold.
We perform optimization by computing a linear delta in the tangent space of the current estimate, and then apply this change using a retraction operation, which maps the change in tangent space back to the manifold itself.
There may be multiple possible retractions for a given manifold, which can be chosen between depending on the computational complexity. The important criteria for the creation for the retract and localCoordinates functions is that they be inverse operations.