Theorem 5-2 Spivak's Calulus on Manifolds

Question

My question is :

1) How to show $H(f(y)=y$?$H(f(y))=(h^1(f(y)),\dots,h^k(f(y)))=?$

2)How we get $det g'(a,b)=det(D_jf^i(a))$

Thanks in advance!!

Answer 1

\begin{align*} H(f(y)) &= H(h^{-1}(\underbrace{y}_{\in \Bbb{R}^k},\underbrace{0}_{\in \Bbb{R}^{n-k}})) \\ &= (h^1(h^{-1}(y,0)), \dots, h^k(h^{-1}(y,0))) \\ &= (y^1, \dots, y^k) \\ &= y \text{.} \end{align*}

For the other, note that $\frac{\partial g}{\partial a} = \begin{pmatrix} (D_j f^i(y) ) \\ 0 \end{pmatrix}$ and $\frac{\partial g}{\partial b} = \begin{pmatrix} 0 \\ \Bbb{I}_{n-k} \end{pmatrix}$, so $g' = \begin{pmatrix} (D_j f^i(y) )_{i=1}^k & (D_j f^i(y) )_{i=k+1}^n \\ 0 & \Bbb{I}_{n-k} \end{pmatrix}$ \begin{align*} \det g' &= \det (D_j f^i(y) )_{i=1}^k \det \Bbb{I}_{n-k} - \det (D_j f^i(y) )_{i=k+1}^n \cdot 0 \\ &= \det (D_j f^i(y) )_{i=1}^k \\ &= \det (D_j f^i(y) ) \text{,} \end{align*}

where the last equality is perhaps most easily seen by the evaluation of the determinant by recursive minors starting with all the $1$s in the $\Bbb{I}_{n-k}$.

It might be helpful to mentally tag the various objects as they are made:

• $U$ is an open neighborhood in $\Bbb{R}^n$ of $x \in M$,
• $V$ is an open neighborhood in $\Bbb{R}^n$, with the convenient property that
• $h$ maps $U$ to $V$, flattening $M$ out so that the image of $M$ lies on the subspace with last $n-k$ coordinates equal to zero, and maps points of $U \smallsetminus M$ "above" and "below" that subspace,
• $W$ is that image of $U \cap M$ in $\Bbb{R}^k$ where we drop the $n-k$ superfluous zero coordinates,
• $f$ maps the $W$ patch of $\Bbb{R}^k$ to $M$ in $\Bbb{R}^n$,
• $H$ is the canonical projection of $h$ onto its first $k$ coordinates; this is another way to get from $U$ to $\Bbb{R}^k$, but it is far from invertible (it is not injective) and is not required to land exactly on $W$. However, $W$ is mapped to itself pointwise by this projection, so $H$ must have $\mathrm{rank}\, H \geq \dim W = k$ on points of $U \cap M$ (and it can't be greater because $H$ maps to $\Bbb{R}^k$),
• $g$ maps the (filled) cylinder over $W$ by taking the $W$-part to its point of $U \cap M$ and just adding the last $n-k$ coordinates of its input to the last $n-k$ coordinates of that point of $U \cap M$. This makes $g'$ a block matrix with $f'$ as the upper-left block, the identity matrix in the lower right, and zeroes in the other two blocks,
• ...