Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 151Fig. 3.3 A sketch of a tangent bundle T M of a smooth manifold M (see text forexplanation).that the following limit exists and is finite [Abraham et al. (1988)]:ddt (φ ◦ γ)(a) ≡ (φ ◦ (φ ◦ γ)(t) − (φ ◦ γ)(a)γ)′ (a) = lim. (3.10)t→a t − aGeneralizing (3.10), we get the notion of the curve on a manifold. Fora smooth manifold M and a point m ∈ M a curve at m is a C 0 −mapγ : I → M from an interval I ⊂ R into M with 0 ∈ I and γ(0) = m.Two curves γ 1 and γ 2 passing though a point m ∈ U are tangent at mwith respect to the chart (U, φ) if (φ ◦ γ 1 ) ′ (0) = (φ ◦ γ 2 ) ′ (0). Thus, twocurves are tangent if they have identical tangent vectors (same directionand speed) in a local chart on a manifold.For a smooth manifold M and a point m ∈ M, the tangent space T m Mto M at m is the set of equivalence classes of curves at m:T m M = {[γ] m : γ is a curve at a point m ∈ M}.A C k −map ϕ : M ∋ m ↦→ ϕ(m) ∈ N between two manifolds M and Ninduces a linear map T m ϕ : T m M → T ϕ(m) N for each point m ∈ M, calleda tangent map, if we have: