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 \(\Omega\) relative to a basis \(\{\partial_i\}_{i=1}^n\) of \(\Gamma(\mathrm{TM})\) results into

\[ \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u} = v^i w^j [D_i, D_j] u^l \partial_l, \]

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

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

The Riemann curvature defines the curvature of the differential manifold at a certain point \(x \in \mathrm{M}\).

Proposition 2: let \(\mathbf{R}: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}\) be the Riemann curvature tensor, with its decomposition given by

\[ \mathbf{R} = R^i_{jkl} \partial_i \otimes dx^j \otimes dx^k \otimes dx^l, \]

then we have that its holor is given by

\[ R^i_{jkl} = \partial_k \Gamma^i_{jl} + \Gamma^m_{jl} \Gamma^i_{mk} - \partial_k \Gamma^i_{jk} - \Gamma^m_{jk} \Gamma^i_{ml}, \]

for all \((i,j,k,l) \in \{1, \dots, n\}^4\) with \(\Gamma^i_{jk}\) denoting the linear connection symbols.

Proof:

Will be added later.

It may then be observed that \(R^i_{jkl} = - R^i_{jlk}\) such that

\[ \mathbf{R} = \frac{1}{2} R^i_{jkl} \partial_i \otimes dx^j \otimes (dx^k \wedge dx^l). \]