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$m$ and $k$ are natural numbers satisfying $1\leq k\leq m$,$M_{k,m}$ is the set of the entire $k\times m$ matrix, and its components are real numbers,and the subset of $M_{k,m}$ is defined as follows $$V_{k,m}= \{A\in M_{k,m} | AA^T = I_k\}$$ $I_k$ is the k-dimensional identity matrix, $A^T$ is the transposed matrix, take $M_{k,m}$ as $km$-dimesional Euclidean space $\mathbb{R}^{k\times m}$, $V_{k, m}$ is a subset of $\mathbb{R}^{k\times m}$

Problem: Prove that $V_{2, m}$ is a $(2m-3)$-dimensional $C^\infty$ submanifold of $\mathbb{R}^{2m}$

My knowledge of manifolds is not very good. I think need find out the Jacobian and prove that it is a regular value, but I don’t have any ideas and I don’t know how to start,can anyone show me how to do this?

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1 Answer 1

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In a lot of problems using matrices like this, it's helpful to use velocity vectors of curves. For example, if $F: M_{k,m} \to M_{k, k}$ is defined by $F(A) = A A^T$, then you can realize an element $v \in T_AM_{k,m}$ of some tangent space of $M_{k,m}$ as an element of $M_{k,m}$ which is realized by a curve $\gamma(t) = A + vt$ for $t \in (-\epsilon, \epsilon)$. Then, $\gamma'(0) = v$ and by the chain rule \begin{align*} d_AF(v) = (F \circ \gamma)'(0)& = \frac{d}{dt}|_{0}(A + vt)(A^T + v^T t) \\&= \frac{d}{dt}|_{0}AA^T + (vA^T + Av^T)t + vv^Tt^2\\&= vA^T + Av^T.\end{align*}

This formula should give you a place to start with regards to proving that $F^{-1}(I_k)$ is a comprised of regular values.

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