6
$\begingroup$

As the title,my question is why $SU(2)$ is diffeomorphic to $S^3$ ? How can I structure a diffeomorphic map between $SU(2)$ and $S^3$?

Please help me,and give me more details!

Thank you very much!

$\endgroup$
2
  • 3
    $\begingroup$ Write out the elements of $SU(2)$. They satisfy the equation of the sphere $\endgroup$
    – leibnewtz
    Nov 2, 2018 at 4:30
  • $\begingroup$ I can not understand what you mean,please give sufficient evidence. $\endgroup$ Nov 2, 2018 at 4:45

2 Answers 2

7
$\begingroup$

By definition, we have that $\displaystyle SU(2)=\left\{\begin{pmatrix}a &-\overline{b}\\ b &\overline{a}\end{pmatrix}: a, b\in\mathbb{C}, |a|^2+|b|^2=1\right\}.$

Since $\mathbb{R}^4\cong\mathbb{C}^2 $, we may think of $S^3$ as $\displaystyle S^3=\{(a,b)\in\mathbb{C}^2: |a|^2+|b|^2=1\}$.

Define a map from $S^3$to $SU(2)$ as: \begin{aligned}F: S^3&\to SU(2)\\ F(a,b)&=\begin{pmatrix}a &-\overline{b}\\ b &\overline{a}\end{pmatrix}. \end{aligned}

Can you show that $F$ is well defined, $F$ is injective, and $F$ is surjective (i.e. an isomorphism)? After that, you should show that $F$ and $F^{-1}$ are smooth (or $C^{\infty}$). (Hint: Show $\displaystyle SU(2)$ is a submanifold of of the set of 2 x 2 matrices with complex entries, or better yet $\mathbb{R}^8$. What can you say about $F$ and its inverse restricted to these submanifolds?)

$\endgroup$
2
  • $\begingroup$ Hi. This answer is very clear but could you maybe help me understanad why $SU(2)$ is a submanifold of $\mathbb{R}^8 \cong \mathbb{C}^4 \cong M_2(\mathbb{C})$? $\endgroup$
    – user661541
    Dec 18, 2019 at 14:23
  • $\begingroup$ Does the condition $|a|^2+|b|^2=1$ not reduce the degrees of freedom by 1, to 3. Afterall, the lie algebra of SU(2) has 3 degrees of freedom:$x\sigma_x+y\sigma_y+z\sigma_z$. Wouldn't SU(2) then map better to just a normal three-dimensional sphere? $\endgroup$
    – Anon21
    Jan 30, 2020 at 23:16
-3
$\begingroup$

Every SU2 matrix is a linear combination of Pauli matrices p1, p2, p3 + unity matrix I which is a base of the associated Lie algebra of the SU2 group. So every SU2 matrix X can be written a X=x1.p1 +x2.p2 +x3.p3 +x4.I and the condition that detX = 1 gives x1power2 + x2power2 +x3power2 +x4power2 = 1 which is the S3 sphere. and every point on that sphere represents an SU2 matrix

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.