Topology/Manifolds Question:
Find for which $a$ the subset of $R^3$ given by the equations:
$$x^2+y^2+z^2=1$$
$$x^4+y^4+z^4=a$$
is a smooth curve (that is, a 1-dimensional manifold) in $R^3$ (possibly an empty one).

For various ranges of $a$, when the curve is smooth, how many ovals will the curve consist of?
You may start with a similar planar problem, in the $(x, y)$ plane: depending on the value of $a$, how many intersection points does the unit circle $x^2+y^2=1$ have with the curve $x^4+y^4=a$?

Question

Topology/Manifolds Question:
Find for which $a$ the subset of $R^3$ given by the equations:
$$x^2+y^2+z^2=1$$
$$x^4+y^4+z^4=a$$
is a smooth curve (that is, a 1-dimensional manifold) in $R^3$ (possibly an empty one).

For various ranges of $a$, when the curve is smooth, how many ovals will the curve consist of?
You may start with a similar planar problem, in the $(x, y)$ plane: depending on the value of $a$, how many intersection points does the unit circle $x^2+y^2=1$ have with the curve $x^4+y^4=a$?

experimentx · Answer

*

anonymous · Answer

What?

turingtest · Answer

that * is used like a bookmark so we can see if anyone answers the question
I am posting for the same reason

linknissan · Answer

brilliant question..m thinking on it

anonymous · Answer

Ah ok thanks @TuringTest , I look forward to seeing if anyone can find a solution.