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, V and V^* the fiber and dual fiber bundle and \mathscr{B} the tensor fiber 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 \Gamma(\mathrm{TM}) and \mathbf{T}, \mathbf{S} \in \Gamma(\mathscr{B}) then

\nabla_{f\mathbf{v}} \mathbf{T} = f \nabla_\mathbf{v} \mathbf{T}

\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},

\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. The linear connection can also be defined in terms of the cotangent bundle and the dual fiber bundle.

Covariant derivative

Definition 2: let \mathbf{v} = v^i \mathbf{e}_i\in \Gamma(\mathscr{B}) then the covariant derivative on \mathbf{v} is defined as

D_k \mathbf{v} \overset{\text{def}}= \nabla_{\partial_k} \mathbf{v} = (\partial_k v^i) \mathbf{e}i + v^i \Gamma^j{ik} \mathbf{e}j = (\partial_k v^i + \Gamma^i{jk} v^j)\mathbf{e}_i,

with formally \mathbf{k}(\mathbf{\hat e}^j, \nabla_{\partial_k} \mathbf{e}_i) = \Gamma^j_{ik} the linear connection symbols, in this case \nabla_{\partial_k} \mathbf{e}_i = \Gamma^j_{ik} \mathbf{e}_j.

The covariant derivative can thus be seen as a linear connection for which only the basis is used of the tangent vector. The covariant derivative can also be applied on higher, mixed rank tensors \mathbf{T} = T^{ij}_k \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k \in \Gamma(\mathscr{B}) which obtains

D_l \mathbf{T} = (\partial_l T^{ij}_k) \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k + T^{ij}k (\Gamma{il}^m\mathbf{e}_m) \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k + T^{ij}_k \mathbf{e}i \otimes (\Gamma^m{jl} \mathbf{e}_m) \otimes \mathbf{\hat e}^k + T^{ij}_k \mathbf{e}_i \otimes \mathbf{e}j \otimes (\hat \Gamma^k{ml} \mathbf{\hat e}^m),

with the dual linear connection symbols given by \mathbf{k}(\nabla_{\partial_k} \mathbf{\hat e}^i, \mathbf{e}_j) = \hat \Gamma^j_{ik} with \nabla_{\partial_k} \mathbf{\hat e}^i = \hat \Gamma^j_{ik} \mathbf{\hat e}^j. We then have the following proposition such that we can simplify the above expression.

Proposition 1: let \Gamma^j_{ik} be the linear connection symbols of a covariant derivative and let \hat \Gamma^j_{ik} be the dual linear connection symbols given by \mathbf{k}(\nabla_{\partial_k} \mathbf{\hat e}^i, \mathbf{e}_j) = \hat \Gamma^j_{ik}, then we have that

\hat \Gamma^j_{ik} = - \Gamma^j_{ik},

for all (i,j,k) \in \mathbb{N}^3.

??? note "Proof:"

Will be added later.

With the result of proposition 1 we may write

D_l \mathbf{T} = (\partial_l T^{ij}k + \Gamma{ml}^i T^{mj}k + \Gamma{ml}^j T^{im}k - \Gamma{kl}^m T^{ij}_m) \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k.

Transformation of linear connection symbols

Will be added later.

Parallel transport

Definition 3: let \mathbf{v} \in \Gamma(\mathrm{TM}), then parallel transport of \mathbf{v} occurs along the manifold \mathrm{M} when

D_k \mathbf{v} = \mathbf{0}.

For example, a parameterised vector field \mathbf{v}: x(t) \mapsto \mathbf{v}(x(t)) \in \Gamma(\mathrm{TM}) is transported parallel if

D_t \mathbf{v} = (\partial_k v^i) \dot x^k \partial_i + \Gamma^i_{jk} v^j \partial_i = \mathbf{0},

so (\partial_k v^i) \dot x^k + \Gamma^i_{jk} v^j = 0

Contravariant derivative