Analysis and Algebra on Differentiable Manifolds: A Workbook by Prof. P. M. Gadea, Prof. J. Muñoz Masqué (auth.)

By Prof. P. M. Gadea, Prof. J. Muñoz Masqué (auth.)

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Additional resources for Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers

Example text

Xn , one has σ∗ d dt = s ∂ d(xi ◦ σ ) (s) i dt ∂x . σ (s) 54 1 Differentiable manifolds Fig. 24 The tangent vector field σ to a curve σ as a curve in T M. Thus, σ (s) = (x1 ◦ σ )(s), . . , (x1 ◦ σ )(s), d(xn ◦ σ ) d(x1 ◦ σ ) (s), . . , (s) . dt dt The coordinate functions {xi } and σ are C∞ . Hence the composition xi ◦ σ is C∞ for each i = 1, . . , n; and the functions d(xi ◦ σ )/dt, which are coordinate functions on the open subset π −1 (U) of T M, are also C∞ . 2. Let M and N be C∞ manifolds and let p ∈ M, q ∈ N.

0. ⎜. . ⎠⎝ ⎠ ⎝ μ1k · · · μkk x1n · · · xkn As rank(xij ) = k, the previous equality implies (μba ) = 0. Finally, let us show that M/∼ can be identified to the Grassmannian. We have a natural surjective map Ψ : M → Gk (Rn ), Ψ (v1 , . . , vk ) = v1 , . . , vk . We have dim v1 , . . , vk = k as v1 , . . , vk are linearly independent. Moreover, Ψ (v1 , . . , vk ) = Ψ (v1 , . . 8 The Tangent Bundle 53 if and only if {v1 , . . , vk } and {v1 , . . , vk } are two bases of V . Hence there exists A ∈ GL(k, R) such that (v1 , .

Notice that ϕi (Vi ) = ϕi (Vi ), but ϕi− = ϕi+ ; in fact, we have ϕi− ([x]) = −ϕi+ ([x]). Since ϕi− = −ϕi+ (they differ by the diffeomorphism t → −t of Rn ) we shall forget the charts (Vi , ϕi− ), and we / shall consider only the charts (Vi , ϕi+ ), i = 1, . . , n + 1. If i = j, then Vi ∩ V j = 0, and moreover −1 + + ϕi+ ◦ (ϕ + j ) : ϕ j (Vi ∩V j ) → ϕi (Vi ∩V j ) 50 1 Differentiable manifolds is given by −1 −1 ϕi+ ◦ (ϕ + = h+ ◦ (π |V + ) ◦ (h+j )−1 j ) i ◦ (π |V + ) i j −1 = h+ ◦ π )|V + ∩V + i ◦ (π i j ◦ h+j −1 + −1 = h+ i ◦ (h j ) , which is differentiable since it is a change of coordinates in Sn , known to be differentiable.

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