2.2 KiB
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.