George Washington University Differential Geometry Smooth Mapping Practice

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Please solve the Question and show all the steps. you may need the note(pdf) but you cannot copy from it

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3.
In the questions below you need to justify your answers rigorously.
(i) Let 0 :RRM be a smooth map. Define the term singular point of a map
0. Give an example of a map 0 : R + R that has a unique singular point.
(ii) Can a diffeomorphism D : R2 + R2 have singular points?
(ii) Give an example of a smooth map 0 : R2 + R2 whose only singular point is
(0,0).
(iv) Give an example of a non-constant smooth map 0 : R² + R2 such that every
point (2,4) € R2 is singular.
(v) Give an example of a smooth map Ø : R2 + R2 such that the set of its
singular points is the parabola {(x,y) € R2 : y = 2²}.
(i) Let 0 : R4 + R2 be a smooth map of the form,
(21, 12, 13, 14) = (x122 +2314, f(x1, 22, 23, 24)),
where f is an unspecified smooth function. Find a function f : R4 + R
such that the map and the pre-image 0-1(1, 2) satisfy the conditions of the
Regular Value Theorem. Verify that your answer satisfies these conditions.
(ii) Choose any point pe 0-1(1, 2) in your answer for (b)(i). Write down its
coordinates and construct a basis for the tangent space TΣ, where Σ =
0-1(1,2). Justify your answer.

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Tags:
singular points

Differential geometry

singular point

smooth mapping

minimun dimensions

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