SOME REMARKS ON THE EXPONENTIAL MAP ON THE GROUPS SO(n) AND SE(n)

The problem of describing or determining the image of the exponential map exp : g -> G of a Lie group G is important and it has many applications. If the group G is compact, then it is well-known that the exponential map is surjective, hence the exponential image is G. In this case the problem is reduced to the computation of the exponential and the formulas strongly depend on the group G. In this paper we discuss the generalization of Rodrigues formulas for computing the exponential map of the special orthogonal group SO(n), which is compact, and of the special Euclidean group SE(n), which is not compact but its exponential map is surjective, in the case n >= 4.