1.6 KiB
Curvature
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.
Curvature operator
Definition 1: the curvature operator
\Omega: \Gamma(\mathrm{TM})^3 \to \Gamma(\mathrm{TM})is defined as
\Omega(\mathbf{v}, \mathbf{w}) \mathbf{u} = [\nabla_\mathbf{v}, \nabla_\mathbf{w}] \mathbf{u} - \nabla_{[\mathbf{v}, \mathbf{w}]}\mathbf{u},for all
\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM})with[\cdot, \cdot]denoting the Lie bracket.
It then follows from the definition that the curvature operator \Omega can be decomposed.
Proposition 1: the decomposition of the curvature operator
\Omegarelative to a basis\{\partial_i\}_{i=1}^nof\Gamma(\mathrm{TM})results into
\Omega(\mathbf{v}, \mathbf{w}) \mathbf{u} = v^i w^j [D_i, D_j] u^l \partial_l,for all
\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM}).
??? note "Proof:"
Will be added later.
Curvature tensor
Definition 2: the Riemann curvature tensor
\mathbf{R}: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}is defined as
\mathbf{R}(\bm{\omega}, \mathbf{u}, \mathbf{v}, \mathbf{w}) = \mathbf{k}(\bm{\omega}, \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u}),for all
\bm{\omega} \in \Gamma(\mathrm{T}^*\mathrm{M})and\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM}).