3 Manifolds

Definition 3.1 (Locally Euclidean) A topologically space \(M\) is locally Euclidean if for all point \(p \in M\), there is a homeomorphism \[ \phi: U \to \mathbb H^n \] defined on a neighborhood of \(p\).

The pair \((U, \phi)\) (or simply \(\phi\)) is called a chart of \(M\).

Definition 3.2 (Topological Manifold) A topological manifold is a Hausdorff, second countable, locally Euclidean space.

Definition 3.3 (Compactible Charts) Two charts of a topological manifold \((U, \phi), (V, \psi)\) are compactible, if \[ \phi \circ \psi^{-1}: \psi(U \cap V) \to \phi(U \cap V), \quad \psi \circ \phi^{-1}: \phi(U \cap V) \to \psi(U \cap V) \] are smooth (in half spaces).

Definition 3.4 (Altas) A (smooth) altas on a locally Euclidean space \(M\) is a collection of pairwise compactible charts.

Definition 3.5 (Manifold) A smooth manifold is a topological manifold \(M\) together with a maximal atlas. This atlas is also called a differnetial structure of \(M\).

Remark. Henceforth refer

1. manifolds as smooth manifolds
2. charts of a manifold as charts in the differential structure of the manifold

Proposition 3.1 Any altas is contained in a unique maximal altas. Therefore it is enough to define the differential structure of manifold with an arbitrary atlas.

Theorem 3.1 (Smooth invariance of dimension) If \(\phi: U \to \mathbb R^n\), \(\psi: V \to \mathbb R^m\) are two charts of a smooth manifold \(M\), then \(n = m\).

Proof. Apply chain rule on the diffeomorphism \(\phi \circ \psi^{-1}\) and its inverse.