Proving a set is not a embedded submaifold.
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So I am asked to find out which level sets of the function $f:mathbb{R}^2 to mathbb{R}$ given by $f(x,y)=x^3+xy+y^3+1$ are embedded submanifolds of $mathbb{R}^2$.
You can see that the points in the set $X=mathbb{R}-{1,frac{28}{27}}$ are regular values ($(0,0)$ and $(-frac{1}{3},-frac{1}{3})$ are not regular points).
So $f^{-1}(q)$ is a embedded submanifold for every $q$ in $X$.
My question is, how can I find out if $f^{-1}(1)$ is/is not an embedded submanifold? Likewise with $frac{28}{27}$.
Thanks in advanced.
differential-topology
asked Nov 16 at 14:50
Bajo Fondo
421213
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0
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favorite
So I am asked to find out which level sets of the function $f:mathbb{R}^2 to mathbb{R}$ given by $f(x,y)=x^3+xy+y^3+1$ are embedded submanifolds of $mathbb{R}^2$.
You can see that the points in the set $X=mathbb{R}-{1,frac{28}{27}}$ are regular values ($(0,0)$ and $(-frac{1}{3},-frac{1}{3})$ are not regular points).
So $f^{-1}(q)$ is a embedded submanifold for every $q$ in $X$.
My question is, how can I find out if $f^{-1}(1)$ is/is not an embedded submanifold? Likewise with $frac{28}{27}$.
Thanks in advanced.
differential-topology
asked Nov 16 at 14:50
Bajo Fondo
421213
$f$ is defined on $mathbb{R}^2$.
– Paul Frost
Nov 16 at 16:14
My bad... its a typo though. My mind was someplace else when I was writting, evidently. Thanks
– Bajo Fondo
Nov 16 at 17:49
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
So I am asked to find out which level sets of the function $f:mathbb{R}^2 to mathbb{R}$ given by $f(x,y)=x^3+xy+y^3+1$ are embedded submanifolds of $mathbb{R}^2$.
You can see that the points in the set $X=mathbb{R}-{1,frac{28}{27}}$ are regular values ($(0,0)$ and $(-frac{1}{3},-frac{1}{3})$ are not regular points).
So $f^{-1}(q)$ is a embedded submanifold for every $q$ in $X$.
My question is, how can I find out if $f^{-1}(1)$ is/is not an embedded submanifold? Likewise with $frac{28}{27}$.
Thanks in advanced.
differential-topology
asked Nov 16 at 14:50
Bajo Fondo
421213
So I am asked to find out which level sets of the function $f:mathbb{R}^2 to mathbb{R}$ given by $f(x,y)=x^3+xy+y^3+1$ are embedded submanifolds of $mathbb{R}^2$.
You can see that the points in the set $X=mathbb{R}-{1,frac{28}{27}}$ are regular values ($(0,0)$ and $(-frac{1}{3},-frac{1}{3})$ are not regular points).
So $f^{-1}(q)$ is a embedded submanifold for every $q$ in $X$.
My question is, how can I find out if $f^{-1}(1)$ is/is not an embedded submanifold? Likewise with $frac{28}{27}$.
Thanks in advanced.
differential-topology
differential-topology
asked Nov 16 at 14:50
Bajo Fondo
421213
asked Nov 16 at 14:50
Bajo Fondo
421213
edited Nov 16 at 17:49
asked Nov 16 at 14:50
Bajo Fondo
421213
asked Nov 16 at 14:50
Bajo Fondo
421213
asked Nov 16 at 14:50
Bajo Fondo
421213
421213
$f$ is defined on $mathbb{R}^2$.
– Paul Frost
Nov 16 at 16:14
My bad... its a typo though. My mind was someplace else when I was writting, evidently. Thanks
– Bajo Fondo
Nov 16 at 17:49
add a comment |
$f$ is defined on $mathbb{R}^2$.
– Paul Frost
Nov 16 at 16:14
My bad... its a typo though. My mind was someplace else when I was writting, evidently. Thanks
– Bajo Fondo
Nov 16 at 17:49
$f$ is defined on $mathbb{R}^2$.
– Paul Frost
Nov 16 at 16:14
$f$ is defined on $mathbb{R}^2$.
– Paul Frost
Nov 16 at 16:14
My bad... its a typo though. My mind was someplace else when I was writting, evidently. Thanks
– Bajo Fondo
Nov 16 at 17:49
My bad... its a typo though. My mind was someplace else when I was writting, evidently. Thanks
– Bajo Fondo
Nov 16 at 17:49
add a comment |
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$f$ is defined on $mathbb{R}^2$.
– Paul Frost
Nov 16 at 16:14
My bad... its a typo though. My mind was someplace else when I was writting, evidently. Thanks
– Bajo Fondo
Nov 16 at 17:49