mathematics-physics-wiki/docs/en/mathematics/differential-geometry/linear-connections.md

Linear connections

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.

Definition 1: a linear connection on the fiber bundle \mathscr{B} is a map

\nabla: \Gamma(\mathrm{TM}) \times \Gamma(\mathscr{B}) \to \Gamma(\mathscr{B}): (\mathbf{v}, \mathbf{T}) \mapsto \nabla_\mathbf{v} \mathbf{T},

satisfying the following properties, if f,g \in C^\infty(\mathrm{M}), \mathbf{v} \in \mathrm{TM} and \mathbf{T}, \mathbf{S} \in \mathscr{B} then

1. \nabla_{f\mathbf{v}} \mathbf{T} = f \nabla_\mathbf{v} \mathbf{T}
2. \nabla_\mathbf{v} (f \mathbf{T} + g \mathbf{S}) = (\nabla_\mathbf{v} f) \mathbf{T} + f \nabla_\mathbf{v} \mathbf{T} + (\nabla_\mathbf{v} g) \mathbf{S} + g \nabla_{\mathbf{v}} \mathbf{S},
3. \nabla_\mathbf{v} f = \mathbf{v} f = \mathbf{k}(df, \mathbf{v}).

From property 3 it becomes clear that \nabla_\mathbf{v} is an analogue of a directional derivative.