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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T ) as above and moreover ϕi j (t 1 , . . ,t n ) = t1 t i−1 t i+1 t j−1 1 t j tn , . . , , , . . , , , , . . , ti ti ti ti ti ti ti = (x1 , . . , xn ). The equations x1 = t1 t i−1 i t i+1 i−1 , . . , ti ti t t j−1 1 tj tn x j−2 = i , x j−1 = i , x j = i , . . , xn = i , t t t t correspond to differentiable functions on Ui j . ) (3) Rn+1 − {0} is an open submanifold of Rn+1 . 7 Submersions. Quotient Manifolds ϕi ◦ π ◦ id−1 : 49 π −1 (Ui ) → Rn (x1 , . . , xn+1 ) → x1 xi−1 xi+1 xn+1 , .

3 3 R EMARK . f∗ cannot be injective at any point since dim R2 > dim R. 2. Let f : R2 → R2 , (x, y) → (x2 − 2y, 4x3 y2 ), g : R2 → R3 , (x, y) → (x2 y + y2 , x − 2y3 , yex ). (1) Compute f∗(1,2) and g∗(x,y) . (2) Find g∗ 4 ∂ ∂ − ∂x ∂y . 3 Differentiable Functions and Mappings λ ∂ ∂ ∂ +μ +ν ∂x ∂y ∂z to be the image of some vector by g∗ . ⎛ Solution. (1) f∗(1,2) ≡ ⎝ 2 −2 48 23 ⎞ ⎠, 16 g(0,0) ⎛ ⎞ 2x y x2 + 2y ⎜ ⎟ g∗(x,y) ≡ ⎜ −6y2 ⎟ ⎝ 1 ⎠. x x ye e (2) g∗ 4 ∂ ∂ − ∂x ∂y (0,1) ⎛ ⎞ 0 2 ⎛ ⎞ ⎜ ⎟ 4 ⎟⎝ ⎠ ≡⎜ ⎝1 −6⎠ −1 1 1 ⎛ ⎞ −2 ⎜ ⎟ ∂ ∂ ∂ ⎟ =⎜ ≡ −2 + 10 + 3 ⎝ 10⎠ ∂x ∂y ∂z 3 .

By virtue of Sard’s theorem, f (Rk ) = L has zero measure. 6. Let M1 and M2 be two C∞ manifolds. Give an example of differentiable mapping f : M1 → M2 such that all the points of M1 are critical points and the set of critical values has zero measure. Solution. Let f : M1 → M2 defined by f (p) = q, for every p ∈ M1 and q a fixed point of M2 . Then the rank of f is zero, hence all the points of M1 are critical. On the other hand, the set of critical values is reduced to the point q, and the set {q} has obviously zero measure.

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