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
D) S being pseudometrizable, it is paracompact by Stone’s Theorem. (e) S being paracompact, it admits continuous partitions of unity. 10. 13) with the topology induced by the usual one of R3 . Prove that the algebraic manifold S is not even a locally Euclidean space. Fig. 13 The cone is not a locally Euclidean space because of the origin. Solution. The point (0, 0, 0) ∈ S does not have a neighborhood homeomorphic to an open subset of R2 . 3 Differentiable Functions and Mappings 21 in V . If we drop the point h(0) in B(h(0), ε ) the remaining set is connected.
Vn ) in Rn . Let k v j,h = ∑ aij,h vi + i=1 n ∑ bij,h vi , 1 j k, ( ) i=k+1 be the expression of v j,h in this basis. As lim h→∞ v j,h = v j , we obtain lim h→∞ aij,h = δi j , for i, j = 1, . . , k, and lim h→∞ bij,h = 0, for k + 1 i n, 1 j k. Set X = (vk+1 , . . ,k 1 j k . ,k , Then, ( ) can be rewritten as Xh = XAh + XBh ; hence Xh Zh = XAh Zh + XBh Zh , and passing to the limit we obtain XZ = X lim (Ah Zh ) + X lim (Bh Zh ). h→∞ h→∞ Taking components we have Z = lim h→∞ (Ah Zh ) and lim h→∞ (Bh Zh ) = 0.
We have σ (1) = (0, 0), σ (−1) = (0, 0), and σ (1) ≡ (2t, 3t 2 − 1)t=1 = (2, 2), σ (−1) ≡ (−2, 2). 15). 6. 1). Consider the vector v = (d/ds)0 tangent at the origin p = (0, 0) to E and let j : E → R2 be the canonical injection of E in R2 . (1) Compute j∗ v. (2) Compute j∗ v if E is given by the chart (sin 2s, sin s) → s, s ∈ (−π , π ). Solution. (1) The origin p corresponds to s = π , so ⎛ ∂ sin 2s ⎜ ∂s j∗p ≡ ⎝ As v = d ds ⎞ ∂ sin s ∂s ⎛ ⎞ 2 = ⎝ ⎠. −1 ⎟ ⎠ s=π we have p ⎛ ⎞ 2 j∗p v ≡ ⎝ ⎠ (1) −1 ⎛ ⎞ 2 ∂ =⎝ ⎠≡2 ∂ x −1 − p ∂ ∂y .
Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers by Prof. P. M. Gadea, Prof. J. Muñoz Masqué (auth.)