PushPullTensor(Phi, Psi, M) will apply the transformation Phi to the vector, form or tensor T on  M and return an object of the same type on N.

Applied to tensors on M, PushPullTensor acts as a ring homomorphism, that is, if T1 and T2 are tensors on M, then PushPullTensor(Phi, Psi, T1 &tensor T2) = PushPullTensor(Phi, Psi, T1) &tensor PushPullTensor(Phi, Psi, T2).  Therefore to completely describe the action of PushPullTensor on a tensor T, it suffices to note that [i] if T is a vector field on M, then PushPullTensor(Phi , Psi, T) = Pushforward(Phi, Psi, T) and [ii], if T is a 1-form on M, then PushPullTensor(Phi, Psi, T) = Pullback(Psi, T).

Let A be a connection on M and let B = PushPullTensor(Phi, Psi, A).  Let X, Y be vector fields on M and let Z = DirectionalCovariantDerivative(X, Y, A).  Let U, V, W be the vector fields on N which are the pushforwards of X, Y and Z using Phi and Psi.  The defining property of the connection B is the identity W = DirectionalCovariantDerivative(U, V, B).

Let A be a connection on M and let B = PushPullTensor(Phi, Psi, A).  Let R be the curvature tensor for A and let S be the curvature tensor for B.  We remark that PushPullTensor(Phi, Psi, R) = S.

Let S be a differential form or a covariant tensor field defined on the manifold N.  Then the command PushPullTensor(Phi, S) will Pullback S to a differential form or tensor on M.  With this calling sequence, the transformation Phi from M to N need not be invertible.  For example, suppose M is a submanifold of N and Phi : M -> N is the inclusion mapping.  Then, if g is a Riemannian metric on N, PushPullTensor(Phi, g) computes the induced metric on M.  

The PushPullTensor command works with arbitrary frames and can therefore be used to re-express any tensor in the coordinate frame in terms of the user defined frame.

The PushPullTensor command may also be applied to tensor fields associated  with a vector bundle E -> M.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form PushPullTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-PushPullTensor.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$\mathrm{DGsetup}\left(\left[u\,v\,w\right]\,N\right)\:$

$\mathrm{\Φ1}≔\mathrm{Transformation}\left(M\,N\,\left[u=xy\,v=y\,w=xz\right]\right)$

$\mathrm{\Φ1}:=\left[u\=xy\,v\=y\,w\=xz\right]$

$\mathrm{\Ψ1}≔\mathrm{InverseTransformation}\left(\mathrm{\Φ1}\right)$

$\mathrm{\Ψ1}:=\left[x\=\frac{u}{v}\,y\=v\,z\=\frac{wv}{u}\right]$

$\mathrm{Pushforward}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{D\_x}\right),\mathrm{Pushforward}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{D\_y}\right),\mathrm{Pushforward}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{D\_z}\right)$

$\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{D\_x}\right),\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{D\_y}\right),\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{D\_z}\right)$

$\mathrm{Pullback}\left(\mathrm{\Ψ1}\,\mathrm{dx}\right),\mathrm{Pullback}\left(\mathrm{\Ψ1}\,\mathrm{dy}\right),\mathrm{Pullback}\left(\mathrm{\Ψ1}\,\mathrm{dz}\right)$

$\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{dx}\right),\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{dy}\right),\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{dz}\right)$

$\mathrm{W1}≔\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{D\_x}\&t\mathrm{dx}\right)$

$\mathrm{W1}:=\mathrm{D\_u}\mathrm{du}-\frac{u\mathrm{D\_u}\mathrm{dv}}{v}\+\frac{w\mathrm{D\_w}\mathrm{du}}{u}-\frac{w\mathrm{D\_w}\mathrm{dv}}{v}$

$\mathrm{W2}≔\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{D\_x}\right)\&tensor\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,\mathrm{dx}\right)$

$\mathrm{W2}:=\mathrm{D\_u}\mathrm{du}-\frac{u\mathrm{D\_u}\mathrm{dv}}{v}\+\frac{w\mathrm{D\_w}\mathrm{du}}{u}-\frac{w\mathrm{D\_w}\mathrm{dv}}{v}$

$\mathrm{W1}\&minus\mathrm{W2}$

$0\mathrm{D\_u}\mathrm{du}$

$A≔\mathrm{Connection}\left(y\left(\mathrm{D\_y}\&t\mathrm{dy}\right)\&t\mathrm{dx}\right)$

$A:=y\mathrm{D\_y}\mathrm{dy}\mathrm{dx}$

$B≔\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,A\right)$

$B:=-\frac{\mathrm{D\_u}\mathrm{du}\mathrm{dv}}{v}\+\frac{2w\mathrm{D\_u}\mathrm{dw}\mathrm{du}}{u^2}-\frac{2w\mathrm{D\_u}\mathrm{dw}\mathrm{dv}}{vu}-\frac{\mathrm{D\_u}\mathrm{dw}\mathrm{dw}}{u}\+\frac{\left(u-1\right)\mathrm{D\_v}\mathrm{du}\mathrm{du}}{v}-\frac{u\left(u-2\right)\mathrm{D\_v}\mathrm{du}\mathrm{dv}}{v^2}\+\mathrm{D\_v}\mathrm{dv}\mathrm{du}-\frac{u\mathrm{D\_v}\mathrm{dv}\mathrm{dv}}{v}-\frac{2w\mathrm{D\_v}\mathrm{dw}\mathrm{du}}{vu}\+\frac{2w\mathrm{D\_v}\mathrm{dw}\mathrm{dv}}{v^2}\+\frac{\mathrm{D\_v}\mathrm{dw}\mathrm{dw}}{v}-\frac{\mathrm{D\_w}\mathrm{dw}\mathrm{du}}{u}\+\frac{\mathrm{D\_w}\mathrm{dw}\mathrm{dv}}{v}$

$X≔\mathrm{D\_x}\:$$Y≔\mathrm{D\_y}\:$$Z≔\mathrm{DirectionalCovariantDerivative}\left(X\,Y\,A\right)$

$Z:=y\mathrm{D\_y}$

$U≔\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,X\right)$

$U:=v\mathrm{D\_u}\+\frac{wv\mathrm{D\_w}}{u}$

$V≔\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,Y\right)$

$V:=\frac{u\mathrm{D\_u}}{v}\+\mathrm{D\_v}$

$W≔\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,Z\right)$

$W:=\mathrm{D\_u}u\+\mathrm{D\_v}v$

$\mathrm{DirectionalCovariantDerivative}\left(U\,V\,B\right)$

$\mathrm{D\_u}\+\frac{\left(u^2\+v^2-u\right)\mathrm{D\_v}}{v}$

$R≔\mathrm{CurvatureTensor}\left(A\right)$

$R:=-\mathrm{D\_y}\mathrm{dx}\mathrm{dy}\mathrm{dy}\+\mathrm{D\_y}\mathrm{dx}\mathrm{dy}\mathrm{dy}$

$S≔\mathrm{CurvatureTensor}\left(B\right)$

$S:=\frac{4w\mathrm{D\_u}\mathrm{dw}\mathrm{du}\mathrm{dv}}{v{u}^2}-\frac{2\mathrm{D\_u}\mathrm{dw}\mathrm{du}\mathrm{dw}}{u^2}-\frac{4w\mathrm{D\_u}\mathrm{dw}\mathrm{dv}\mathrm{du}}{v{u}^2}\+\frac{4\mathrm{D\_u}\mathrm{dw}\mathrm{dv}\mathrm{dw}}{vu}\+\frac{2\mathrm{D\_u}\mathrm{dw}\mathrm{dw}\mathrm{du}}{u^2}-\frac{4\mathrm{D\_u}\mathrm{dw}\mathrm{dw}\mathrm{dv}}{vu}-\frac{\left(u-2\right)\mathrm{D\_v}\mathrm{du}\mathrm{du}\mathrm{dv}}{v^2}\+\frac{\left(u-2\right)\mathrm{D\_v}\mathrm{du}\mathrm{dv}\mathrm{du}}{v^2}-\frac{\mathrm{D\_v}\mathrm{dv}\mathrm{du}\mathrm{dv}}{v}\+\frac{\mathrm{D\_v}\mathrm{dv}\mathrm{dv}\mathrm{du}}{v}-\frac{4w\mathrm{D\_v}\mathrm{dw}\mathrm{du}\mathrm{dv}}{v^{2}u}\+\frac{4\mathrm{D\_v}\mathrm{dw}\mathrm{du}\mathrm{dw}}{vu}\+\frac{4w\mathrm{D\_v}\mathrm{dw}\mathrm{dv}\mathrm{du}}{v^{2}u}-\frac{6\mathrm{D\_v}\mathrm{dw}\mathrm{dv}\mathrm{dw}}{v^2}-\frac{4\mathrm{D\_v}\mathrm{dw}\mathrm{dw}\mathrm{du}}{vu}\+\frac{6\mathrm{D\_v}\mathrm{dw}\mathrm{dw}\mathrm{dv}}{v^2}$

$\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,\mathrm{\Ψ1}\,R\right)\&minusS$

$-\frac{u\mathrm{D\_u}\mathrm{dv}\mathrm{du}\mathrm{dv}}{v^2}\+\frac{u\mathrm{D\_u}\mathrm{dv}\mathrm{dv}\mathrm{du}}{v^2}-\frac{4w\mathrm{D\_u}\mathrm{dw}\mathrm{du}\mathrm{dv}}{v{u}^2}\+\frac{2\mathrm{D\_u}\mathrm{dw}\mathrm{du}\mathrm{dw}}{u^2}\+\frac{4w\mathrm{D\_u}\mathrm{dw}\mathrm{dv}\mathrm{du}}{v{u}^2}-\frac{4\mathrm{D\_u}\mathrm{dw}\mathrm{dv}\mathrm{dw}}{vu}-\frac{2\mathrm{D\_u}\mathrm{dw}\mathrm{dw}\mathrm{du}}{u^2}\+\frac{4\mathrm{D\_u}\mathrm{dw}\mathrm{dw}\mathrm{dv}}{vu}\+\frac{\left(u-2\right)\mathrm{D\_v}\mathrm{du}\mathrm{du}\mathrm{dv}}{v^2}-\frac{\left(u-2\right)\mathrm{D\_v}\mathrm{du}\mathrm{dv}\mathrm{du}}{v^2}\+\frac{4w\mathrm{D\_v}\mathrm{dw}\mathrm{du}\mathrm{dv}}{v^{2}u}-\frac{4\mathrm{D\_v}\mathrm{dw}\mathrm{du}\mathrm{dw}}{vu}-\frac{4w\mathrm{D\_v}\mathrm{dw}\mathrm{dv}\mathrm{du}}{v^{2}u}\+\frac{6\mathrm{D\_v}\mathrm{dw}\mathrm{dv}\mathrm{dw}}{v^2}\+\frac{4\mathrm{D\_v}\mathrm{dw}\mathrm{dw}\mathrm{du}}{vu}-\frac{6\mathrm{D\_v}\mathrm{dw}\mathrm{dw}\mathrm{dv}}{v^2}$

$S≔\mathrm{evalDG}\left(\mathrm{du}\&t\mathrm{dw}\right)$

$S:=\mathrm{du}\mathrm{dw}$

$\mathrm{PushPullTensor}\left(\mathrm{\Φ1}\,S\right)$

$yz\mathrm{dx}\mathrm{dx}\+xy\mathrm{dx}\mathrm{dz}\+xz\mathrm{dy}\mathrm{dx}\+x^2\mathrm{dy}\mathrm{dz}$

$\mathrm{DGsetup}\left(\left[u\,v\,w\right]\,\mathrm{R3}\right)\:$

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\,\mathrm{R4}\right)\:$

$\mathrm{\Φ2}≔\mathrm{Transformation}\left(\mathrm{R3}\,\mathrm{R4}\,\left[\mathrm{x2}=\frac{2v}{v^2+w^2+1+u^2}\,\mathrm{x1}=\frac{2u}{v^2+w^2+1+u^2}\,\mathrm{x4}=\frac{u^2+w^2+v^2-1}{v^2+w^2+1+u^2}\,\mathrm{x3}=\frac{2w}{v^2+w^2+1+u^2}\right]\right)$

$\mathrm{\Φ2}:=\left[\mathrm{x1}\=\frac{2u}{u^2\+v^2\+w^2\+1}\,\mathrm{x2}\=\frac{2v}{u^2\+v^2\+w^2\+1}\,\mathrm{x3}\=\frac{2w}{u^2\+v^2\+w^2\+1}\,\mathrm{x4}\=\frac{u^2\+v^2\+w^2-1}{u^2\+v^2\+w^2\+1}\right]$

$f≔\mathrm{Pullback}\left(\mathrm{\Φ2}\,{\mathrm{x1}}^2+{\mathrm{x2}}^2+{\mathrm{x3}}^2+{\mathrm{x4}}^2\right)$

$f:=\frac{4u^2}{{\left(u^2\+v^2\+w^2\+1\right)}^2}\+\frac{4v^2}{{\left(u^2\+v^2\+w^2\+1\right)}^2}\+\frac{4w^2}{{\left(u^2\+v^2\+w^2\+1\right)}^2}\+\frac{{\left(u^2\+v^2\+w^2-1\right)}^2}{{\left(u^2\+v^2\+w^2\+1\right)}^2}$

$\mathrm{simplify}\left(f\right)$

$1$

$g≔\mathrm{evalDG}\left(\mathrm{dx1}\&t\mathrm{dx1}+\mathrm{dx2}\&t\mathrm{dx2}+\mathrm{dx3}\&t\mathrm{dx3}+\mathrm{dx4}\&t\mathrm{dx4}\right)$

$g:=\mathrm{dx1}\mathrm{dx1}\+\mathrm{dx2}\mathrm{dx2}\+\mathrm{dx3}\mathrm{dx3}\+\mathrm{dx4}\mathrm{dx4}$

$\mathrm{PushPullTensor}\left(\mathrm{\Φ2}\,g\right)$

$\frac{4\mathrm{du}\mathrm{du}}{{\left(u^2\+v^2\+w^2\+1\right)}^2}\+\frac{4\mathrm{dv}\mathrm{dv}}{{\left(u^2\+v^2\+w^2\+1\right)}^2}\+\frac{4\mathrm{dw}\mathrm{dw}}{{\left(u^2\+v^2\+w^2\+1\right)}^2}$

$\mathrm{ChangeFrame}\left(M\right)$

$\mathrm{R3}$

$\mathrm{FR}≔\mathrm{FrameData}\left(\left[x\mathrm{D\_x}+y\mathrm{D\_y}\,x\mathrm{D\_y}\,y\mathrm{D\_z}\right]\,\mathrm{M1}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{FR}\right)\:$

$\mathrm{\Φ3}≔\mathrm{Transformation}\left(M\,\mathrm{M1}\,\left[x=x\,y=y\,z=z\right]\right)$

$\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[M\,0\right]\,\left[\mathrm{M1}\,0\right]\right]\,\left[\right]\,\left[\mathrm{Matrix}\left(\mathrm{\%id}\=4553395650\right)\right]\right]\,\left[\left[x\,x\right]\,\left[y\,y\right]\,\left[z\,z\right]\right]\right]\right)$

$\mathrm{\Ψ3}≔\mathrm{Transformation}\left(\mathrm{M1}\,M\,\left[x=x\,y=y\,z=z\right]\right)$

$\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{M1}\,0\right]\,\left[M\,0\right]\right]\,\left[\right]\,\left[\mathrm{Matrix}\left(\mathrm{\%id}\=4535464194\right)\right]\right]\,\left[\left[x\,x\right]\,\left[y\,y\right]\,\left[z\,z\right]\right]\right]\right)$

$\mathrm{PushPullTensor}\left(\mathrm{\Φ3}\,\mathrm{\Ψ3}\,\mathrm{D\_x}\&t\mathrm{dy}\right)$

$\frac{y\mathrm{E1}\mathrm{\Θ1}}{x}\+\mathrm{E1}\mathrm{\Θ2}-\frac{y^2\mathrm{E2}\mathrm{\Θ1}}{x^2}-\frac{y\mathrm{E2}\mathrm{\Θ2}}{x}$

$\mathrm{PushPullTensor}\left(\mathrm{\Ψ3}\,\mathrm{\Φ3}\,\left(\mathrm{E3}\&t\mathrm{E2}\right)\&t\mathrm{E1}\right)$

$\mathrm{D\_x}\mathrm{D\_y}\mathrm{D\_z}{x}^{2}y\+\mathrm{D\_y}\mathrm{D\_y}\mathrm{D\_z}x{y}^2$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\left[u\,v\right]\,E\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\left[p\,q\right]\,F\right)\:$

$\mathrm{\Φ4}≔\mathrm{Transformation}\left(E\,F\,\left[x=x\,y=y\,z=z\,p=xu+yv\,q=yu-xv\right]\right)$

$\mathrm{\Φ4}:=\left[x\=x\,y\=y\,z\=z\,p\=ux\+vy\,q\=uy-vx\right]$

$\mathrm{\Ψ4}≔\mathrm{InverseTransformation}\left(\mathrm{\Φ4}\right)$

$\mathrm{\Ψ4}:=\left[x\=x\,y\=y\,z\=z\,u\=\frac{px\+qy}{x^2\+y^2}\,v\=-\frac{-py\+qx}{x^2\+y^2}\right]$

$\mathrm{PushPullTensor}\left(\mathrm{\Φ4}\,\mathrm{\Ψ4}\,y\mathrm{D\_u}\right)$

$\mathrm{D\_p}xy\+\mathrm{D\_q}{y}^2$

$\mathrm{PushPullTensor}\left(\mathrm{\Φ4}\,\mathrm{\Ψ4}\,\mathrm{convert}\left(\mathrm{dv}\,\mathrm{DGtensor}\right)\right)$

$\frac{y\mathrm{dp}}{x^2\+y^2}-\frac{x\mathrm{dq}}{x^2\+y^2}$

$\mathrm{ChangeFrame}\left(E\right)\:$

$\mathrm{ConE}≔\mathrm{Connection}\left(\left(\mathrm{D\_u}\&t\mathrm{du}\right)\&t\mathrm{dx}\right)$

$\mathrm{ConE}:=\mathrm{D\_u}\mathrm{du}\mathrm{dx}$

$\mathrm{ConF}≔\mathrm{PushPullTensor}\left(\mathrm{\Φ4}\,\mathrm{\Ψ4}\,\mathrm{ConE}\right)$

$\mathrm{ConF}:=\frac{x\left(x-1\right)\mathrm{D\_p}\mathrm{dp}\mathrm{dx}}{x^2\+y^2}-\frac{y\mathrm{D\_p}\mathrm{dp}\mathrm{dy}}{x^2\+y^2}\+\frac{y\left(x\+1\right)\mathrm{D\_p}\mathrm{dq}\mathrm{dx}}{x^2\+y^2}-\frac{x\mathrm{D\_p}\mathrm{dq}\mathrm{dy}}{x^2\+y^2}\+\frac{y\left(x-1\right)\mathrm{D\_q}\mathrm{dp}\mathrm{dx}}{x^2\+y^2}\+\frac{x\mathrm{D\_q}\mathrm{dp}\mathrm{dy}}{x^2\+y^2}-\frac{\left(-y^2\+x\right)\mathrm{D\_q}\mathrm{dq}\mathrm{dx}}{x^2\+y^2}-\frac{y\mathrm{D\_q}\mathrm{dq}\mathrm{dy}}{x^2\+y^2}$

$\mathrm{CurvatureTensor}\left(\mathrm{ConF}\right)$

$\frac{4y{x}^2\mathrm{D\_p}\mathrm{dp}\mathrm{dx}\mathrm{dy}}{{\left(x^2\+y^2\right)}^2}-\frac{4y{x}^2\mathrm{D\_p}\mathrm{dp}\mathrm{dy}\mathrm{dx}}{{\left(x^2\+y^2\right)}^2}-\frac{2\left(x^2-y^2\right)x\mathrm{D\_p}\mathrm{dq}\mathrm{dx}\mathrm{dy}}{{\left(x^2\+y^2\right)}^2}\+\frac{2\left(x^2-y^2\right)x\mathrm{D\_p}\mathrm{dq}\mathrm{dy}\mathrm{dx}}{{\left(x^2\+y^2\right)}^2}-\frac{2\left(x^2-y^2\right)x\mathrm{D\_q}\mathrm{dp}\mathrm{dx}\mathrm{dy}}{{\left(x^2\+y^2\right)}^2}\+\frac{2\left(x^2-y^2\right)x\mathrm{D\_q}\mathrm{dp}\mathrm{dy}\mathrm{dx}}{{\left(x^2\+y^2\right)}^2}-\frac{4y{x}^2\mathrm{D\_q}\mathrm{dq}\mathrm{dx}\mathrm{dy}}{{\left(x^2\+y^2\right)}^2}\+\frac{4y{x}^2\mathrm{D\_q}\mathrm{dq}\mathrm{dy}\mathrm{dx}}{{\left(x^2\+y^2\right)}^2}$