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

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

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

ISBN-10: 904813563X

ISBN-13: 9789048135639

ISBN-10: 9048135648

ISBN-13: 9789048135646

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

Example text

By the same reason one has y = z = 0; but (0, 0, 0) ∈ H. 6. Prove that the subset M of the Euclidean space R3 which consists of all the points (x, y, z) of R3 satisfying x2 − y2 + 2x z − 2y z = 1, 2x − y + z = 0, admits a structure of C∞ 1-manifold. Solution. The functions f1 (x, y, z) = x2 − y2 + 2x z − 2y z − 1, f2 (x, y, z) = 2x − y + z, 44 1 Differentiable manifolds are C∞ functions. The rank of the Jacobian matrix of f1 , f2 with respect to x, y, z, is less than 2 if and only if x − 2y − z = 0, but the points satisfying this equation do not belong to M = f −1 (0).

The Jacobian matrix J = (3x2 + 12x y 6y2 + 6x2 3z2 ) has rank 0 if and only if (x, y, z) = (0, 0, 0), but this point does not belong to H = f −1 (0), hence rank J = 1 and H admits a structure of C∞ manifold. 5. Prove that the subset H of the Euclidean space R3 of all the points (x, y, z) of R3 satisfying x3 + y3 + z3 − 2x y z = 1 admits a C∞ 2-manifold structure. Solution. The map f : R3 → R, f (x, y, z) = x3 + y3 + z3 − 2x y z − 1, is C∞ and its Jacobian matrix is J = (3x2 − 2y z 3y2 − 2x z 3z2 − 2x y), which vanishes only if (x, y, z) = (0, 0, 0).

5. Consider the C∞ manifold Rn and a submanifold L given by a vector subspace of Rn with dim L n − 1. Prove that L has zero measure. Solution. Let dim L = k n − 1. Consider the map f : Rk → Rn , f (x1 , x2 , . . , xk ) = xi ei , where {ei } is a basis of L. 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.

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


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