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I am learning differential geometry and a basis of the theory of smooth manifolds but i'm feeling a lack of practice in solving problems on submanifolds in $\mathbb{R}^n$ (problems like 'prove that $SL_n(\mathbb{R})$ is a closed submanifold' or 'prove that preimage of a regular value is a submanifold' etc). Books i am reading are also poor in such problems.
Can you offer such problems or give a link to any problem sheets? Thank-you!

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  • $\begingroup$ What sources are you using? I like John Lee's "Introduction to Smooth Manifolds". Chapter 8 of this book is fully devoted to submanifolds and it includes a section with problems. $\endgroup$ May 11, 2013 at 22:24

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As a set, the special linear group $SL(n,\Bbb R)$ is the subset of $GL(n,\Bbb R)$ consisting of matrices of determinant 1. Since $det(AB) = (det A)(det B)$ and $det(A^{−1}) = 1/det A$

$SL(n, \Bbb R)$ is a subgroup of $GL(n,\Bbb R)$.

To show that it is a regular submanifold, we let $f : GL(n, \Bbb R) → \Bbb R$ be the determinant map $f (A) = det A$, and apply the regular level set theorem to $f^{−1}(1) = SL(n,\Bbb R)$.

We need to check that 1 is a regular value of $f$. Let $a_{ij}$,

$1 ≤ i ≤ n, 1 ≤ j ≤ n$, be the standard coordinates on $\Bbb R^{n×n}$, and

let $S_{ij}$ denote the submatrix o $A = [a_{i j} ] ∈ \Bbb R^{n×n}$ obtained by deleting its ith row and jth column. Then $m_{ij} := det S_{ij}$ is the $(i, j)$-minor of $A$. From linear algebra we have a formula for computing the determinant by expanding along any row or any column: if we expand along the ith row, we obtain

$f(A) = det A = (−1)^{i+1}a_{i1}m_{i1} +(−1)^{i+2}a_{i2}m_{i2} +···+(−1)^{i+n}a_{in}m_{in}$. Therefore

$∂f/∂aij =(−1)^{i+j}m_{ij}$.

Hence, a matrix $A ∈ GL(n,\Bbb R)$ is a critical point of $f$ if and only if all the $(n− 1) × (n − 1)$ minors $m_{i j}$ of $A$ are $0$. By the above queation, such a matrix A has determinant 0. Since every matrix in SL(n,R) has determinant 1, all the matrices in $SL(n,\Bbb R)$ are regular points of the determinant function. By the regular level set theorem, $SL(n,\Bbb R)$ is a regular submanifold of $GL(n,\Bbb R)$ of codimension 1; i.e., $dimSL(n,\Bbb R) = dimGL(n,\Bbb R)−1 = n^2 −1$.

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  • $\begingroup$ Thanks a lot! But i meant these problems were examples of problems i need. $\endgroup$
    – user74574
    May 8, 2013 at 18:08

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