Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 147then E is called the model space and M is referred to as a C k −Banachmanifold modelled on E. Similarly, if a covering by charts takes their valuesin a Hilbert space H, then H is called the model space and M is referredto as a C k −Hilbert manifold modelled on H. If not otherwise specified, wewill consider M to be an Euclidean manifold, with its covering by chartstaking their values in R n .For a Hausdorff C k −manifold the following properties are equivalent[Kolar et al. (1993)]: (i) it is paracompact; (ii) it is metrizable; (iii) itadmits a Riemannian metric; 2 (iv) each connected component is separable.3.4 Smooth Maps Between Smooth ManifoldsA map ϕ : M → N between two manifolds M and N, with M ∋ m ↦→ϕ(m) ∈ N, is called a smooth map, or C k −map, if we have the followingcharting:2 Recall the corresponding properties of a Euclidean metric d. For any three pointsx, y, z ∈ R n , the following axioms are valid:M 1 : d(x, y) > 0, for x ≠ y; and d(x, y) = 0, for x = y;M 2 : d(x, y) = d(y, x); M 3 : d(x, y) ≤ d(x, z) + d(z, y).