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Differential Geometry, imbedded submanifold.

Let phi : R^2 --> R be a function given by

phi(x,y) = x^3 + xy + y^3 +1

For which points p =(0,0) , p=(1/3,1/3), p =(-1/3,-1/3) is the subset
phi^-1 ( phi(p)) an imbedded sub-manifold of R^2.

( This question was given as part of Differential Geometry course, we use Do Carmo's book,Riemannian Geometry)

Please justify all your steps and claims. And try to solve the problem in a manner similar to the book mentioned above.

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Solution Summary

Differential geometry for imbedded submanifolds are determined. Problems in a manner similar to the books mentioned are determined.