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!

  • 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


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?)

  • $\begingroup$ 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

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


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