Parsing the Fibers of Critical Values

[WWGD]

Hi, All:

I am trying to better understand what happens at fibers of critical values:

Specifically, I had the fibers of the map:

f(z1,z2): ℂ²→ℂ , given by:

f(Z₁, Z₂)= Z₁²+z₂² =0

I rewrote it as a map from ℝ⁴→ℝ²=

f(x₁,y₁,x₂,x₂)=

(x₁²-y₁²+x₂²-

y₂² , x₁y₁+2x₂y₂)Which has the Jacobian:
(Sorry, I don't know how to Tex a matrix ):

[ 2x₁, 2y₁, 2x₂, 2y₂ ]

[ 2y₁, 2x₁, 2y₂, 2x₂


Now, by one of the family of theorems in { inverse function, implicit function, regular rank theorem} , the fibers of the non-zero values are submanifolds of ℂ².

After row-reducing, we get that the only critical value is (0,0) .

Now, some work on f^{-1}(0,0):

i)Not very helpful: Z₁²+Z₂²=0

implies Z₁=iZ₂.

ii) A little better:

Restricting to (x₁,x₂, 0, 0):

Then the general fiber is a circle, and the collection of all fibers is a cone .

Is there anything else we can do to get a better understanding of what the general

fibers are like ? I think all the regular fibers are isomorphic to each other--tho I cannot

find a good proof -- and then we have the critical fiber.

Any ideas?

Thanks.

 

[jvicens]

Hello,

Thank you for your post. The fibers of critical values are an important concept in understanding the behavior of a map. In the case of your map, f(z1,z2), the critical values are the points where the Jacobian matrix is not invertible, which in this case is when both z1 and z2 are equal to 0.

To get a better understanding of the fibers, we can look at the level sets of the map. These are the sets of points where the function takes on a particular value. In the case of f(z1,z2)=0, the level set is a circle, as you mentioned. However, if we consider a different level set, say f(z1,z2)=1, we get a different shape, specifically an ellipse. This is because the level sets of a map are determined by the inverse image of the values, and in this case, the inverse image of 1 is an ellipse.

In general, the regular fibers of a map are isomorphic to each other, meaning they have the same structure. This is because the map is smooth and the fibers are submanifolds of the domain. However, the critical fiber may have a different structure, as it is the only fiber that intersects the critical value of (0,0).

To better understand the general fibers, we can also look at the tangent spaces at each point on the fiber. This can give us information about the local behavior of the map and how it varies along the fiber. Additionally, we can consider the behavior of the map near the critical fiber, which may help us understand the overall behavior of the map.

I hope this helps in your understanding of the fibers of critical values. Let me know if you have any other questions or if I can provide further clarification. Thank you.