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 \(\mathbf{u}, \mathbf{v} \in \Gamma(\mathrm{TM})\).

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 \(\mathbf{T}: \mathrm{T}_x^* \mathrm{M} \times \mathrm{T}_x \mathrm{M} \times \mathrm{T}_x \mathrm{M} \to \mathbb{K}\) with \(x \in \mathrm{M}\) 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 \(\mathbf{u}, \mathbf{v} \in \mathrm{T}_x \mathrm{M}\).