Zoran Skoda covariant derivative

The covariant differentiation is one of the several equivalent forms of the data of a connection on a frame bundle of the tangent bundle on a differentiable manifold.

A vector field (i.e. a section of the tangent bundle $T M\to M$ ) can be considered in some local coordinates on the tangent bundle which are induced by local coordinates on the manifold $(U,\phi)$ , $\phi: U\to \mathbf{R}^n$ , $U^{open}\subset M$ . By abuse of notation, one writes $x^i(p)$ where $p\in U$ , instead of $x^i(\phi(p))$ , where $x^i: \mathbf{R}^n\to \mathbf{R}$ are the coordinate projections. A vector $X$ at $p$ will be then presented as $(x^1(p),\ldots, x^n(p), X^1,\ldots, X^n)$ . It appears that when changing the local coordinates the partial derivatives

\frac{\partial X^i}{\partial x^j}

will not behave as components of a tensor. A similar problem occurs with differentiation of components of other types of tensors with respect to local coordinates, e.g. of differential forms. The rule of covariant differentiation rectifies this problem.

A covariant differentiation $\nabla$ on a manifold $M$ is a rule

\nabla: X\mapsto{\nabla}_X,

which assigns to to every vector field $X$ on a manifold $M$ , an operator $\nabla_X : \Gamma T M\to \Gamma T M$ on the space of smooth vector fields which is

additive $\nabla_X (Y+Z) = \nabla_X(Y)+\nabla_X(Z)$
derivation in the sense $\nabla_X(f Y) = X(f) Y + f\nabla_X(Y)$ for every $f\in C^\infty(M)$ .

such that the rule $X\mapsto \nabla_X$ is linear in $X$ , i.e.

\nabla_{f X + g Y} Z = f\nabla_X(Z) + g\nabla_Y(Z)

Thanks to the derivation rule, the covariant derivative operators are local and hence can be restricted to open neighborhoods. If the local neighborhoods correspond to charts then one can consider the behaviour under changes of local coordinates. Conversely, if a system of locally defined connections $\nabla_U$ in charts $(U,\phi)$ covering the manifold satisfies those transition rules they globally define a covariant differentiation on $M$ .

To find those transition rules, consider two charts $(U,\phi)$ , $(V,\psi)$ with nonempty intersection $U\cap V$ . Denote the local coordinates in $U$ by and $x^i$ and in $V$ by $y^a$ , and $(\nabla_U)_{\partial/\partial x^i}$ by $\nabla_i$ , $(\nabla_V)_{\partial/\partial y^a}$ by $\tilde\nabla_a$ . Define the functions in coordinate charts, called the Christofel symbols $\Gamma$ , by (Einstein’s summation convention understood)

\nabla_i \frac{\partial}{\partial x^j} = \Gamma^k_{i j} \frac{\partial}{\partial x^k}

\tilde\nabla_a\frac{\partial}{\partial y^b} = \Gamma^c_{a b} \frac{\partial}{\partial y^c}

By linearity of the rule $\nabla$ , the collection of Christoffel symbols in some coordinate chart, determine the connection in this chart.

Then by the rules for a covariant derivative one gets

\Gamma_{a b}^c\frac{\partial}{\partial y^c} = \frac{\partial^2 x^i}{\partial y^a \partial y^b}\frac{\partial}{\partial x^i} + \frac{\partial x^i}{\partial y^b}\frac{\partial x^j}{\partial y^a} \Gamma_{i j}^k\frac{\partial}{\partial x^k}

…to be continued.

hence the transformation of the Christoffel symbols is

\Gamma_{a b}^c = \frac{\partial^2 x^i}{\partial y^a \partial y^b}\frac{\partial y^c}{\partial x^i} + \frac{\partial x^i}{\partial y^a}\frac{\partial x^j}{\partial y^b}\frac{\partial y^c}{\partial x_k}\Gamma^k_{i j}