# Proof that set is a submanifold

$$m$$ 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?

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.