checking $C^infty$-manifold analytically











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We look at the set $C : =lbrace (x,y) in mathbb{R}^2 mid y^2 = x^2+x^3 rbrace$.



We want to know if $C$ is a $C^infty$-manifold. When looking at the plot it is easy to see that $(0,0)$ is the problem here but we are interested in solving this analytically.



With $lbrace (x,y) in mathbb{R}^2 mid y^3 = x^2 rbrace$ we can see that a projection $phi$ onto the $x$-axis is homeomorph but $phi^{-1}$ is not continuously differentiable. So there is no chart.



How would you argument on $C$ while dont have any idea how the set looks?










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    down vote

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    We look at the set $C : =lbrace (x,y) in mathbb{R}^2 mid y^2 = x^2+x^3 rbrace$.



    We want to know if $C$ is a $C^infty$-manifold. When looking at the plot it is easy to see that $(0,0)$ is the problem here but we are interested in solving this analytically.



    With $lbrace (x,y) in mathbb{R}^2 mid y^3 = x^2 rbrace$ we can see that a projection $phi$ onto the $x$-axis is homeomorph but $phi^{-1}$ is not continuously differentiable. So there is no chart.



    How would you argument on $C$ while dont have any idea how the set looks?










    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      We look at the set $C : =lbrace (x,y) in mathbb{R}^2 mid y^2 = x^2+x^3 rbrace$.



      We want to know if $C$ is a $C^infty$-manifold. When looking at the plot it is easy to see that $(0,0)$ is the problem here but we are interested in solving this analytically.



      With $lbrace (x,y) in mathbb{R}^2 mid y^3 = x^2 rbrace$ we can see that a projection $phi$ onto the $x$-axis is homeomorph but $phi^{-1}$ is not continuously differentiable. So there is no chart.



      How would you argument on $C$ while dont have any idea how the set looks?










      share|cite|improve this question













      We look at the set $C : =lbrace (x,y) in mathbb{R}^2 mid y^2 = x^2+x^3 rbrace$.



      We want to know if $C$ is a $C^infty$-manifold. When looking at the plot it is easy to see that $(0,0)$ is the problem here but we are interested in solving this analytically.



      With $lbrace (x,y) in mathbb{R}^2 mid y^3 = x^2 rbrace$ we can see that a projection $phi$ onto the $x$-axis is homeomorph but $phi^{-1}$ is not continuously differentiable. So there is no chart.



      How would you argument on $C$ while dont have any idea how the set looks?







      calculus manifolds






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      asked Nov 22 at 14:58









      Arjihad

      378111




      378111






















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          It won't be a manifold although if you remove the point $(0,0)$ it will be a 1-dimensional manifold.



          When $x$ is very close to 0, the $x^2$ term dominates, and so the set looks like the graph of $y=pm x$, a cross, which is not locally homeomorphic to an interval of $mathbb{R}$.



          For points away from $(0,0)$ you can use the regular level set theorem (aka the preimage theorem).






          share|cite|improve this answer





















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            up vote
            0
            down vote













            It won't be a manifold although if you remove the point $(0,0)$ it will be a 1-dimensional manifold.



            When $x$ is very close to 0, the $x^2$ term dominates, and so the set looks like the graph of $y=pm x$, a cross, which is not locally homeomorphic to an interval of $mathbb{R}$.



            For points away from $(0,0)$ you can use the regular level set theorem (aka the preimage theorem).






            share|cite|improve this answer

























              up vote
              0
              down vote













              It won't be a manifold although if you remove the point $(0,0)$ it will be a 1-dimensional manifold.



              When $x$ is very close to 0, the $x^2$ term dominates, and so the set looks like the graph of $y=pm x$, a cross, which is not locally homeomorphic to an interval of $mathbb{R}$.



              For points away from $(0,0)$ you can use the regular level set theorem (aka the preimage theorem).






              share|cite|improve this answer























                up vote
                0
                down vote










                up vote
                0
                down vote









                It won't be a manifold although if you remove the point $(0,0)$ it will be a 1-dimensional manifold.



                When $x$ is very close to 0, the $x^2$ term dominates, and so the set looks like the graph of $y=pm x$, a cross, which is not locally homeomorphic to an interval of $mathbb{R}$.



                For points away from $(0,0)$ you can use the regular level set theorem (aka the preimage theorem).






                share|cite|improve this answer












                It won't be a manifold although if you remove the point $(0,0)$ it will be a 1-dimensional manifold.



                When $x$ is very close to 0, the $x^2$ term dominates, and so the set looks like the graph of $y=pm x$, a cross, which is not locally homeomorphic to an interval of $mathbb{R}$.



                For points away from $(0,0)$ you can use the regular level set theorem (aka the preimage theorem).







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 22 at 22:37









                user25959

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                1,559816






























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