By Prof. P. M. Gadea, Prof. J. Muñoz Masqué (auth.)
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Extra info for Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers
Is S a regular topological space? Is S pseudometrizable? Does it satisfy all separations axioms Ti ? Is S paracompact? Can it have continuous partitions of unity? (4) Does S satisfy the properties mentioned in (3) if we constrain it to be T2 and to satisfy the second axiom of countability? H INT : Consider: (i) Urysohn’s Theorem: If S verifies the second axiom of countability, then it is equivalent for S to be pseudometrizable and to be regular. (ii) Stone’s Theorem: If S is pseudometrizable, then it is paracompact.
From (2) we have π∗p0 ◦ σ∗q0 = idTq0 N . Since the identity map is surjective, π∗ : Tp0 M → Tq0 N is surjective. 8, there exist local coordinates (x1 , . . , xm ), (y1 , . . , yn ), centered at p0 , q0 in M, N, respectively, such that yi ◦ π = xi , 1 i n. Notice that m n, as π is a submersion. Hence we can define a map σ on the domain of (y1 , . . , yn ) by setting xi ◦ σ = yi 0 if 1 i if n + 1 n i m. Then, for every i = 1, . . , n, we have yi ◦ (π ◦ σ ) = (yi ◦ π ) ◦ σ = xi ◦ σ = yi , thus proving that σ is a local section of π .
Furthermore, it suffices to consider the chart ( f −1 (0), ϕ ), where ϕ : f −1 (0) → Rn−1 , ϕ (x1 , . . , xn ) = (x1 , . . , xn−1 ). In the particular case of the paraboloids, taking into account the previous considerations, it is clear that: Case (a): It is only necessary to consider the chart (U, ϕ ) with U = f −1 (0), ϕ : f −1 (0) → R2 , ϕ (x, y, z) = (x, y). Case (b): Proceed as in (a). 8. Let F : Rn → R be any homogeneous polynomial function (with degree no less than one) with at least one positive value.
Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers by Prof. P. M. Gadea, Prof. J. Muñoz Masqué (auth.)