How does the pushforward of the inverse metric relate to the inverse of the pullback metric for an embedding?

Question

I am learning some geometry and stumbled upon these two ways to obtain a different metric.

For a smooth manifold embedding $\phi:N\to M$ suppose a non-degenerate, covariant metric $g_{ij}$ on the tangent space $TM$ which induces a pullback metric $g'_{i'j'}$ on the tangent space $TN$ by

$\quad g'_{i'j'}=\dfrac{\partial \phi^{i}}{\partial x^{i'}}\,g_{ij}\,\dfrac{\partial \phi^{j}}{\partial x^{j'}}=[\mathrm J_\phi]^{i}_{i'}\,g_{ij}\,[\mathrm J_\phi]^{j}_{j'}\quad$ (components)

$\quad [g']=[\mathrm J_\phi]^T\,[g]\,[\mathrm J_\phi]\quad$ (matrix)

$\quad g'=\phi^*\,g\quad$ (geometric)

where $\mathrm J_\phi$ is the Jacobian of $\phi$.

On $T^*M$ we have that $g_{ij}$ also induces a contravariant inverse metric $h^{kl}$ via

$\quad g_{ij}\,h^{jl}=\delta_i^l\quad$ (components)

$\quad [g][h]=[h][g]=\mathrm I\quad$ (matrix)

$\quad [h] = [g]^{-1}\quad$ (matrix inverse)

Further, on the codomain $\phi(N)$ of $\phi$, this inverse metric $h$ can be pushed forward along $\phi^{-1}$ by

$\quad h'^{k'l'}=\dfrac{\partial (\phi^{-1})^{k'}}{\partial x^{k}}\,h^{kl}\,\dfrac{\partial (\phi^{-1})^{l'}}{\partial x^{l}}=[\mathrm J_{\phi^{-1}}]^{k'}_{k}\,h^{kl}\,[\mathrm J_{\phi^{-1}}]^{l'}_{l}\quad$ (components)

$\quad [h']=[\mathrm J_{\phi^{-1}}]\,[h]\,[\mathrm J_{\phi^{-1}}]^T\quad$ (matrix)

$\quad h'={\phi^{-1}}_*\,h\quad$ (geometric)

where the Jacobian $\mathrm J_{\phi^{-1}}$ can be obtained as the Moore-Penrose pseudoinverse ${\mathrm J_{\phi^{-1}}=(\mathrm J_{\phi}^T\,\mathrm J_{\phi}^{\vphantom{T}})^{-1}\,\mathrm J_{\phi}^T}$.

(Update: here lies the mistake! As pointed out in the comments, this definition of the Moore-Penrose pseudoinverse does an orthogonal projection w.r.t. the standard metric and not w.r.t. the metric $g$.)

Now, it seems that for embeddings, this pushforward inverse metric $h'$ and the matrix inverse of the pullback metric $g'$ generally do not agree (given I have made no mistakes in my trials where they do agree for diffeomorphisms):

$\quad g'_{i'j'}\,h'^{j'l'}\neq\delta_{i'}^{l'}\quad$ (components)

$\quad [h'] \neq [g']^{-1}\quad$(matrix)

$\quad {\phi^{-1}}_*(g^{-1})\neq(\phi^*\,g)^{-1}\quad$(geometric)

Q: Which metric, the pushforward of the inverse metric or the inverse of the pullback metric is "used" on $T^*N$? (...and what is it used for in physics?)

On one hand, the pushforward of the inverse metric ${\phi^{-1}}_*\,h$ preserves inner products on covectors from the reachable subspace ${\{\omega\in T_{\phi(p)}^* M\,|\,\phi^{-1\,*}(\phi^*\,(\omega))=\omega\}}$.

When $\mathrm P_\phi$ is a projection onto that subspace

$\quad [\mathrm P_\phi]^i_k=[\mathrm J_{\phi^{\vphantom{-1}}}]^i_{i'}\; [\mathrm J_{\phi^{{-1}}}]^{i'}_k\quad$(components)

$\quad [\mathrm P_\phi]=[\mathrm J_{\phi^{\vphantom{-1}}}]\; [\mathrm J_{\phi^{{-1}}}]\quad$(matrix)

$\quad \mathrm P_\phi(\omega)=\phi^{-1\,*}(\phi^*\,(\omega))\quad$(geometric)

then we obtain

$\quad \hphantom{={}}{\langle\, \mathrm P_\phi(\omega)\,,\,\mathrm P_\phi(\mu)\,\rangle}_h$

$\quad =(\omega_i\;[\mathrm P_\phi]^i_k)\;h^{kl}\;([\mathrm P_{\phi}]^{j}_l\;\mu_j)$

$\quad =(\omega_i\;[\mathrm J_{\phi^{\vphantom{-1}}}]^i_{i'}\; [\mathrm J_{\phi^{{-1}}}]^{i'}_k)\;h^{kl}\;([\mathrm J_{\phi^{{-1}}}]^{j'}_l\;[\mathrm J_{\phi^{\vphantom{-1}}}]^j_{j'}\;\mu_j)$

$\quad =(\omega_i\;[\mathrm J_{\phi^{\vphantom{-1}}}]^i_{i'})\; ([\mathrm J_{\phi^{{-1}}}]^{i'}_k\;h^{kl}\;[\mathrm J_{\phi^{{-1}}}]^{j'}_l)\;([\mathrm J_{\phi^{\vphantom{-1}}}]^j_{j'}\;\mu_j)$

$\quad =(\phi^*\,\omega)_{i'}\;({\phi^{-1}}_*\,h)^{i'j'}\;(\phi^*\,\mu)_{j'}$

$\quad ={\langle\,\phi^*(\omega)\,,\,\phi^*(\mu)\,\rangle}_{{\phi^{-1}}_*\,h}\quad$(desirable property 1)

On the other hand, the inverse of the pullback metric $(\phi^*g)^{-1}$ or ${g'}^{i',j'}$ can be used for raising and lowering indices in invariant expressions

$\quad\langle u,v\rangle_{g'} =u^{i'}\,g'_{i'j'}\,v^{j'}=u_{k'}\,g'^{k',i'}\,g'_{i'j'}\,v^{j'}=u_{k'}\,v^{j'}\quad$(desirable property 2)

I think that both properties, preservation of covector inner products and preservation of invariants when raising and lowering indices are desirable. So currently my questions are

• Is there any advice how to deal with these two different inverse metrics on $T^*N$? E.g., do we just have both of them and use one for inner products and the other one for raising and lowering?
• Which one is used to define the hodge operator on $T^*N$ then?
• Where else is preservation of the covector inner product necessary?

Edit: I have previously displayed the pushforward along $\phi^{-1}$ wrong, it is updated to be ${\phi^{-1}}_*\,h$

Answer 1

As pointed out in the comments, the subspace projection in $TM$ needs to account for the metric $g$. I have put together a proof on matrix level to see how the pieces come together:

If we obtain the Moore-Penrose inverse w.r.t. $g$ in the following way:

$\quad[\mathrm J_{\phi\,g}^{-1}] = ([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-1} ([\mathrm J_\phi]^T\;[g])$

$\quad[\mathrm J_{\phi\,g}^{-1}]^T=([g]^T\;[\mathrm J_\phi])\;([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-T}$

then this still makes $[\mathrm J_{\phi\,g}^{-1}]$ a left inverse for $[\mathrm J_\phi]$, just as $[\mathrm J_{\phi}^{-1}]$ before:

$\quad\phantom{=} [\mathrm J_{\phi\,g}^{-1}]\;[\mathrm J_\phi]$

$\quad= ([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-1} ([\mathrm J_\phi]^T\;[g])\;[\mathrm J_\phi]$

$\quad= ([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-1} ([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])$

$\quad= \mathrm I$

And when used to transform the inverse metric

$\quad ([g]^{-1})' = [\mathrm J_{\phi\,g}^{-1}]\;[g]^{-1} [\mathrm J_{\phi\,g}^{-1}]^T$ (transformed inverse metric)

then we obtain desirable property 2: inversion commutes with transformation $([g]^{-1})' = ([g]')^{-1}$

$\quad\phantom{=} ([g]^{-1})'$

$\quad= [\mathrm J_{\phi\,g}^{-1}]\;[g]^{-1}\;[\mathrm J_{\phi\,g}^{-1}]^T$

$\quad= ([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-1}\;([\mathrm J_\phi]^T\;[g])\;[g]^{-1}\;([g]^T\;[\mathrm J_\phi])\;([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-T}$

$\quad= ([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-1}\;[\mathrm J_\phi]^T\;[g]\;[g]^{-1}\;[g]^T\;[\mathrm J_\phi]\;([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-T}$

$\quad= ([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-1}\;[\mathrm J_\phi]^T\;[g]^T\;[\mathrm J_\phi]\;([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-T}$

$\quad= ([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-1}\;([\mathrm J_\phi]^T\;[g]^T\;[\mathrm J_\phi])\;([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-T}$

$\quad= ([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-1}\;([\mathrm J_\phi]^T\;[g]^T\;[\mathrm J_\phi])\;([\mathrm J_\phi]^T\;[g]^T\;[\mathrm J_\phi])^{-1}$

$\quad= ([\mathrm J_\phi]^T\;[g]\;[\mathrm J_\phi])^{-1}$

$\quad= ([g]')^{-1}$

For the "reachable subspace" projection $\mathrm P_{\phi\,g}$ w.r.t. the metric $g$

$\quad [\mathrm P_{\phi\,g}] = [\mathrm J_\phi]\;[\mathrm J_{\phi\,g}^{-1}]$ (projection)

$\quad [\mathrm P_{\phi\,g}]^T= [\mathrm J_{\phi\,g}^{-T}]\;[\mathrm J_\phi]^T$

we also obtain desirable property 1: $[g]^{-1}$ inner product on the projection equals $([g]^{-1})'$ inner product after transformation

$\quad\phantom{=} \langle \mathrm P_{\phi\,g}(ω), \mathrm P_{\phi\,g}(μ) \rangle_{g^{-1}}$

$\quad= ([\mathrm P_{\phi\,g}]^T\;ω)^T\;[g]^{-1}\;([\mathrm P_{\phi\,g}]^T\;μ)$

$\quad= ([\mathrm J_{\phi\,g}^{-T}]\;[\mathrm J_\phi]^T\;ω)^T\;[g]^{-1}\;([\mathrm J_{\phi\,g}^{-T}]\;[\mathrm J_\phi]^T\;μ)$

$\quad= (ω^T\;[\mathrm J_\phi]\; [\mathrm J_{\phi\,g}^{-1}])\;[g]^{-1}\;([\mathrm J_{\phi\,g}^{-T}]\; [\mathrm J_\phi]^T\;μ)$

$\quad= (ω^T\;[\mathrm J_\phi])\;([\mathrm J_{\phi\,g}^{-1}] \;[g]^{-1}\;[\mathrm J_{\phi\,g}^{-T}])\;([\mathrm J_\phi]^T\;μ)$

$\quad= (ω^T)'\;([g]^{-1})'\;(μ)'$

$\quad= \langle ω' , μ' \rangle_{(g^{-1})'}$

With these definitions, there is a relation to the matrix $[\mathrm J_{\phi}^{-1}]$ which was the metric-less Moore-Penrose inverse

$\quad[\mathrm J_{\phi}^{-1}] = ([\mathrm J_\phi]^T\;[\mathrm J_\phi])^{-1}\;[\mathrm J_\phi]^T$

We have

$\quad\phantom{=}[\mathrm J_{\phi}^{-1}]\;[\mathrm P_{\phi\,g}]$

$\quad= ([\mathrm J_\phi]^T\;[\mathrm J_\phi])^{-1}\;[\mathrm J_\phi]^T\;[\mathrm J_\phi]\;[\mathrm J_{\phi\,g}^{-1}]$

$\quad= ([\mathrm J_\phi]^T\;[\mathrm J_\phi])^{-1}\;([\mathrm J_\phi]^T\;[\mathrm J_\phi])\;[\mathrm J_{\phi\,g}^{-1}]$

$\quad= [\mathrm J_{\phi\,g}^{-1}]$

which hints that our assembled $[\mathrm J_{\phi\,g}^{-1}]$ is the matrix representation occuring in a pushforward after projection $\phi_*\circ\mathrm P_{\phi\,g}$ geometrically.

This further suggests to identify "transformation" as the "pushforward after projection".

Furthermore $[\mathrm J_{\phi\,g}^{-1}]$ agrees with $[\mathrm J_{\phi}^{-1}]$ on the reachable subspace.

Any comments?