Transformations

Let \(\mathrm{M}\) be a differential manifold with \(\dim \mathrm{M} = n \in \mathbb{N}\) used throughout the section. Let \(\mathrm{TM}\) and \(\mathrm{T^*M}\) denote the tangent and cotangent bundle.

Push forward and pull back

Definition 1: let \(\mathrm{M}, \mathrm{N}\) be two differential manifolds with \(\dim \mathrm{N} \geq \dim \mathrm{M}\) and let \(\psi: \mathrm{M} \to \mathrm{N}\) be the diffeomorphism between the manifolds. Then we define the pull back \(\psi^*\) and push forward \(\psi_*\) operators, such that for \(\mathbf{v} \in \mathrm{T}_x \mathrm{M}\) and \(\bm{\omega} \in \mathrm{T}_{\psi(x)}^* \mathrm{M}\) we have

\[ \mathbf{k}_x(\psi^* \bm{\omega}, \mathbf{v}) = \mathbf{k}_{\psi(x)}(\bm{\omega}, \psi_* \mathbf{v}), \]

for all \(x \in \mathrm{M}\).

Which indicates the proper separation between the elements of both spaces.

Basis transformation

Let \(\psi: \mathscr{D}(\mathrm{M}) \to \mathrm{M}: x \mapsto \psi(x) \overset{\text{def}}{=} \overline{x}\) be an active coordinate transformation from a point \(x\) to a point \(\overline{x}\) on \(\mathrm{M}\). Then we have a basis \(\{\partial_i\}_{i=1}^n \subset \mathrm{T}_x\mathrm{M}\) for the tangent space \(\mathrm{T}_x\mathrm{M}\) at \(x\) and a basis \(\{\overline{\partial_i}\}_{i=1}^n \subset \mathrm{T}_{\overline{x}}\mathrm{M}\) for the tangent space \(\mathrm{T}_{\overline{x}}\mathrm{M}\) at \(\overline{x}\). Which are related by

\[ \partial_i = J^j_i \overline{\partial_j} = \partial_i \psi^j(x) \overline{\partial_j}, \]

with \(J^j_i = \partial_i \psi^j(x)\) the Jacobian at \(x \in \mathrm{M}\). For it to make sense, it helps to change notation to

\[ \frac{\partial}{\partial x_i} = \frac{\partial \overline{x}^j}{\partial x_i} \frac{\partial}{\partial \overline{x}_j} = \frac{\partial \psi^j}{\partial x_i} \frac{\partial}{\partial \overline{x}_j}. \]

Similarly, we have a basis \(\{dx^i\}_{i=1}^n \subset \mathrm{T}_x^*\mathrm{M}\) for the cotangent space \(\mathrm{T}_x\mathrm{M}\) at \(x\) and a basis \(\{d\overline{x}^i\}_{i=1}^n \subset \mathrm{T}_{\overline{x}}^*\mathrm{M}\) for the cotangent space \(\mathrm{T}_{\overline{x}}^*\mathrm{M}\) at \(\overline{x}\). Which are related by

\[ d\overline{x}^i = J^i_j dx^j = \partial_j \psi^i(x) dx^j. \]