1.7 KiB
Torsion
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, V and V^* the fiber and dual fiber bundle and \mathscr{B} the tensor fiber bundle.
Torsion operator
Definition 1: the torsion operator
\Theta: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to \Gamma(\mathrm{TM})is defined as
\Theta(\mathbf{u}, \mathbf{v}) = \nabla_\mathbf{u} \mathbf{v} - \nabla_\mathbf{v} \mathbf{u} - \mathscr{L}_\mathbf{u} \mathbf{v},for all
\mathbf{u}, \mathbf{v} \in \Gamma(\mathrm{TM})and\mathscr{L}the Lie derivative.
Using this definition we obtain the following results.
Proposition 1: the decomposition of the torsion operator results into
\mathbf{k}(\bm{\omega}, \Theta(\mathbf{u}, \mathbf{v})) = \omega_i u^j v^k (\Gamma^i_{kj} - \Gamma^i_{jk}),for all
\bm{\omega} \in \Gamma(\mathrm{T}^*\mathrm{M})and\mathbf{u}, \mathbf{v} \in \Gamma(\mathrm{TM}).
??? note "Proof:"
Will be added later.
Torsion tensor
As a result of proposition 1 we may view torsion as a locally defined mixed tensor of type \mathbf{T} \in \mathrm{T}_x \mathrm{M} \otimes \mathrm{T}_x^* \mathrm{M} \otimes \mathrm{T}_x^* \mathrm{M}.
Definition 2: the torsion tensor
\mathbf{T}: \mathrm{T}_x^* \mathrm{M} \times \mathrm{T}_x \mathrm{M} \times \mathrm{T}_x \mathrm{M} \to \mathbb{K}withx \in \mathrm{M}is defined as
\mathbf{T}(\bm{\omega}, \mathbf{u}, \mathbf{v}) = \mathbf{k} \big(\bm{\omega}, \Theta(\mathbf{u}, \mathbf{v}) \big),for all
\bm{\omega} \in \mathrm{T}^*_x\mathrm{M}and\mathbf{u}, \mathbf{v} \in \mathrm{T}_x \mathrm{M}.