BA primer on Lie theory
The tangential distance in Section 2.4.6 as well as the construction of invariant features in Section 2.6.3 used concepts from Lie theory, but gave no formal introduction of these concepts. This section will give a concise introduction to the postponed mathematical details.
Definition B.1 (Topological Manifold). Let Π be a Hausdorff space, i. e., a set of points with a system of open sets such that for each pair of two distinct points the points can be placed in two disjoint open sets.
1.A chart (or coordinate chart or coordinate map) is a pair (U, φ) of an open subset U ⊆Π and a corresponding injective map φ: U → V to an open subset V ⊆ d of Euclidean space such that φ is a homeomorphism. To be a homeomorphism means ...