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

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**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.

Solution. 18). We recall that σ (t0 ) = σ∗t0 d dt , t0 that is, σ (t0 ) is the image of the canonical vector at t0 ∈ R. (b) σ is a differentiable map, and since σ (t) = (3t 2 − 4, 2t) = (0, 0) for all t, the map σ is an immersion. But for t = ±2, it has a self-intersection, so it is not an injective immersion. 38 1 Differentiable manifolds Fig. 18 (a) σ is not an immersion. (b) σ is a non-injective immersion. Fig. 19 (c) σ is an embedding. (d) σ is a non-injective immersion. (c) σ is an immersion as σ (t) = (−2π sin 2π t, 2π cos 2π t, 1) = (0, 0, 0), ∀t ∈ R.

This is the case for C ⊂ R2 , as it is a finite union of 1-submanifolds of R2 . 2. (1) Let N = (x, y) ∈ R2 : y = 0 and M = R2 . We define f : M → R by f (x, y) = y2 . Prove that the set of critical points of f |N is the intersection with N of the set of critical points of f . (2) Let N = (x, y) ∈ R2 : x2 + y2 = 1 and M = R2 . We define f : M → R by f (x, y) = x2 + y2 . Is the set of critical points of f |N the same as the one of f ? Solution. (1) The set of critical points of f is N and f |N is the zero map.