Ivancevic_Applied-Diff-Geom

152 Applied Differential Geometry: A Modern IntroductionT m (M)T MT (N)T ϕ(m) (N)⊙T (ϕ)✲⊙π Mπ N★M✧❄⊙m✥✦ϕ★✧✲❄⊙ Nϕ(m)✥✦i.e., the following diagram commutes:T m MT m ϕ✲ T ϕ(m) Nπ M❄M ∋ mϕπ N❄✲ ϕ(m) ∈ Nwith the natural projection π M : T M → M, given by π M (T m M) = m,that takes a tangent vector v to the point m ∈ M at which the vector v isattached i.e., v ∈ T m M.For an nD smooth manifold M, its nD tangent bundle T M is the disjointunion of all its tangent spaces T m M at all points m ∈ M, T M =⊔T m M.m∈MTo define the smooth structure on T M, we need to specify how toconstruct local coordinates on T M. To do this, let (x 1 (m), ..., x n (m)) belocal coordinates of a point m on M and let (v 1 (m), ..., v n (m)) be componentsof a tangent vector in this coordinate system. Then the 2n numbers(x 1 (m), ..., x n (m), v 1 (m), ..., v n (m)) give a local coordinate system on T M.T M =⊔T m M defines a family of vector spaces parameterized by M.m∈MThe inverse image π −1M(m) of a point m ∈ M under the natural projectionπ M is the tangent space T m M. This space is called the fibre of the tangentbundle over the point m ∈ M [Steenrod (1951)].A C k −map ϕ : M → N between two manifolds M and N induces alinear tangent map T ϕ : T M → T N between their tangent bundles, i.e.,