Lecture 7. 7.2 The tangent bundle Let M be a smooth manifold. We will associate to M a bundle T M. We will do this concretely but there are many ways of doing this. You should read about them all!!! We know what a tangent vector in Rn . Deﬁnition 7.3. A tangent vector to M at x is the equivalence class of all pairs v, (U, φ) where (U, φ) is a chart about x and v is a tangent vector to Rn at φ(x). We say that v � , (U � , φ � ) is equivalent to v, (U, φ) if v � = dφ(x) (φ � ◦ φ −1 )(v). The tangent bundleT M to M is the set of all tangent vectors. In other words the tangent bundle to M is bundle determined by choosing an atlas {(Uα , φα )|α ∈ X } and taking as transition functions gαβ (x) = dφβ (x) (φα ◦ φβ−1 )(v). Given a chart (U, φ) we get coordinates x 1 , x 2 , . . . , x n on U . A typical tan gent X vector is written as X = a1 ∂ ∂ ∂ + a2 2 + . . . an n . 1 ∂x ∂x ∂x reminding us that we can differentiate function using tangent vectors. Given f : M → R and a tangent vector at x in M we deﬁne −1 −1 ∂ f ◦ φ −1 2∂f ◦ φ n∂f ◦ φ (φ(x)). (φ(x)) + a (φ(x)) + . . . + a ∂xn ∂x1 ∂x2 (2) −1 in other word the usual directional derivative of f ◦ φ . Given a smooth map f : M → N we can deﬁne the differential of f as a map X f (x) = a 1 D f : T M → T N. Given x in M and X = (v, (U, φ)) a tangent vector and a chart (V, ψ) about f (x) set Dx f (X ) to be the equivalence class of the vector Dφ(x) (ψ ◦ f ◦ φ −1 )(v) and the chart, (V, ψ) or in terms of coordinates if we write ψ ◦ f ◦ φ −1 (x 1 , x 2 , . . . , x n ) = ( f 1 (x 1 , . . . , x n ), . . . , f m (x 1 , . . . , x n )) then the matrix of D f is �∂fi � . ∂x j

18.966 Geometry of Manifolds: Lecture 7

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