The Slice theorem
Lie Group, proper action and Slice.
Our goal in this work is to study the classic and celebrated Slice Theorem. Initially proven by Koszul, the Slice Theorem states that given a proper action of a Lie group on a manifold, there exists a slice passing through each point in M , i.e., a submanifold transversal to the orbit passing through the given point with some special properties. This theorem is a fundamental tool in the Theory of Transformation Groups. This result allows us to reduce the study of a Lie group action near an orbit to the study of the
geometry transversal to the orbit.