Does $\Gamma$ intersect $SL(2, \mathbb{R})$ transversely at $I$?

Question

Identify the space of all $2 \times 2$ real matrices with $\mathbb{R}^4$ so that the matrix $\left( \begin{array}{cc} a & b\\ c & d\end{array} \right)$

corresponds to $(a, b, c, d)$. Let $\Gamma$ denote the hyperplane in $\mathbb{R}^ 4$ with equation $x_1 +x_2 +x_3 - x_4 = 0$. Does $\Gamma$ intersect $SL(2, \mathbb{R})$ transversely at $I$?�

What does it mean for two spaces to intersect transversely at a point? — Chris Gerig, Jun 27 '13 at 20:21
I know that there tangent planes at the point of intersection must span the whole of $\mathbbR}^4$, but I am having a difficult time figuring out what those planes are. — kiwi, Jun 27 '13 at 20:23

Hagen von Eitzen · Accepted Answer · 2013-06-27T20:27:21.257

1

At $\begin{pmatrix}a&b\\c&d\end{pmatrix}=I$, we have $$\begin{align}\frac{\partial}{\partial a}\det\begin{pmatrix}a&b\\c&d\end{pmatrix}&=d=1,\\\frac{\partial}{\partial b}\det\begin{pmatrix}a&b\\c&d\end{pmatrix}&=-c=0, \\\frac{\partial}{\partial c}\det\begin{pmatrix}a&b\\c&d\end{pmatrix}&=-b=0,\\\frac{\partial}{\partial d}\det\begin{pmatrix}a&b\\c&d\end{pmatrix}&=a=1.\end{align}$$ hence the tangent space to $SL$ at $I$ is the orthogonal space to $(1,0,0,1)$. On the other hand $\Gamma$ is th eorthogonal space to $(1,1,1,-1)$, hence these two tangent sapces span all of $\mathbb R^4$.

edited Jun 27 '13 at 20:27

answered Jun 27 '13 at 19:27

Hagen von Eitzen

374,180

Sorry. I wrote the wrong thing down. – kiwi Jun 27 '13 at 20:12

Does $\Gamma$ intersect $SL(2, \mathbb{R})$ transversely at $I$?

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