Covariant Derivatives of Tensor Fields

A previous post introduced connections on manifolds and covariant derivatives of vector fields. This post discusses covariant derivatives of tensor fields.

Connections in Vector Bundles

The notion of an affine connection on the tangent bundle can be readily generalized to the notion of a connection on an arbitrary vector bundle.

Definition (Connection in a Vector Bundle). A connection in a smooth vector bundle \(\pi : E \rightarrow M\) over a smooth manifold \(M\) is a map

\[\nabla : \mathfrak{X}(M) \times \Gamma(E) \rightarrow \Gamma(E),\]

written

\[(X,Y) \mapsto \nabla_XY,\]

where \(\Gamma(E)\) is the space of smooth sections of \(E\), such that the following properties are satisfied for all \(X_1,X_2 \in \mathfrak{X}(M)\), \(Y_1,Y_2 \in \Gamma(E)\),\(f \in C^{\infty}(M)\), and \(a \in \mathbb{R}\):

• \(\nabla_XY\) is linear over \(C^{\infty}(M)\) in \(X\):
  • \(\nabla_{fX}Y = f\nabla_XY\).
  • \(\nabla_{X_1 + X_2}Y = \nabla_{X_1}Y + \nabla_{X_2}Y\).
• \(\nabla_XY\) is linear over \(\mathbb{R}\) in \(Y\):
  • \(\nabla_X(aY) = a\nabla_XY\).
  • \(\nabla_X(Y_1 + Y_2) = \nabla_XY_1 + \nabla_XY_2.\)
• \(\nabla\) satisfies the following product rule:

\[\nabla_X(fY) = f\nabla_XY + (\nabla_Xf)Y,\]

where

\[\nabla_Xf = Xf.\]

\(\nabla_XY\) is called the covariant derivative of \(Y\) in the direction of \(X\).

Note that this definition requires \(X\) to be a vector field, but generalizes the notion of a connection on the tangent bundle by permitting \(Y\) to be a smooth section of an arbitrary vector bundle.

Connections in Tensor Bundles

Since each tensor bundle \(T^{(k,l)}TM\) is a vector bundle, we can define connections in tensor bundles. In fact, a connection on a tangent bundle induces a unique connection (satisfying specific conditions) on every tensor bundle.

First, we will define the covariant derivative of functions (i.e. \((0,0)\)-tensor fields or \(0\)-forms) as \(\nabla_Xf = Xf\).

Definition (Covariant Derivative of a Function). Given a connection \(\nabla\) on the tangent bundle of a smooth manifold \(M\), the covariant derivative of a function \(f \in C^{\infty}(M)\) in the direction of a vector field \(X\) is the function \(\nabla_Xf \in C^{\infty}(M)\) defined as

\[\nabla_Xf = Xf.\]

Next, we will define the covariant derivative of \(1\)-forms (smooth covector fields). Note that the covariant derivative \(\nabla_X\omega\) of a covector field \(\omega\) along a vector field \(X\) is a map \(\nabla_X\omega : \mathfrak{X}(M) \rightarrow C^{\infty}(M)\) which characterizes another covector field. To arrive at a definition, we require that the covariant derivative be compatible with the natural pairing of covector fields and vector fields in the sense that the covariant derivative satisfies the following product rule relative to the natural pairing of covector fields and vector fields:

\[\nabla_X\langle \omega,Y\rangle = \langle\nabla_X\omega,Y\rangle + \langle\omega,\nabla_XY\rangle.\]

Recall that the natural pairing of a covector field \(\omega\) and a vector field \(Y\) is the smooth function defined as

\[\langle \omega,Y \rangle = \omega(Y).\]

The compatibility condition then means that

\[X\langle\omega,Y\rangle = \nabla_X\omega(Y) + \omega(\nabla_XY),\]

and hence

\[X(\omega(Y)) = \nabla_X\omega(Y) + \omega(\nabla_XY),\]

and

\[(\nabla_X\omega)(Y) = X(\omega(Y)) - \omega(\nabla_XY).\]

We thus stipulate this as a definition.

Definition (Covariant Derivative of a Covector Field). Given a connection \(\nabla\) on the tangent bundle of a smooth manifold \(M\), the covariant derivative of a covector field \(\omega \in \mathfrak{X}^*(M)\) in the direction of a vector field \(X \in \mathfrak{X}(M)\) is the map \(\nabla_X\omega : \mathfrak{X}(M) \rightarrow C^{\infty}(M)\) defined as follows for all vector fields \(Y \in \mathfrak{X}(M)\):

\[(\nabla_X\omega)(Y) = X(\omega(Y)) - \omega(\nabla_XY).\]

This map characterizes a covector field by the Characterization Lemma. Consider the following:

\begin{align}(\nabla_X\omega)(fY) &= X(\omega(fY)) - \omega(\nabla_X(fY)) \\&= X(f\omega(Y)) - \omega(f\nabla_XY+(Xf)Y) \\&= fX(\omega(Y)) + (Xf)\omega(Y) - \omega(f\nabla_XY) - \omega((Xf)Y) \\&= fX(\omega(Y)) + (Xf)\omega(Y) - f\omega(\nabla_XY) - (Xf)\omega(Y) \\&= fX(\omega(Y)) - f\omega(\nabla_XY) \\&= f(X(\omega(Y)) - \omega(\nabla_XY)) \\&= f\nabla_X\omega.\end{align}

The operation is thus linear over \(C^{\infty}(M)\) in \(Y\) and so characterizes a covector field.

Furthermore, it satisfies the requisite properties of a connection. It is linear over \(C^{\infty}(M)\) in \(X\) since

\begin{align}(\nabla_{fX}\omega)(Y) &= fX(\omega(Y)) - \omega(\nabla_{fX}Y) \\&= fX(\omega(Y)) - \omega(f\nabla_XY) \\&= fX(\omega(Y)) - f\omega(\nabla_XY) \\&= f(X(\omega(Y)) - \omega(\nabla_XY)) \\&= f(\nabla_X\omega)(Y).\end{align}

and

\begin{align}(\nabla_{X_1+X_2}\omega)(Y) &= (X_1+X_2)(\omega(Y)) - \omega(\nabla_{X_1+X_2}Y) \\&= X_1(\omega(Y)) + X_2(\omega(Y)) - \omega(\nabla_{X_1}Y + \nabla_{X_2}Y) \\&= X_1(\omega(Y)) + X_2(\omega(Y)) - \omega(\nabla_{X_1}Y) - \omega(\nabla_{X_2}Y)) \\&= \nabla_{X_1}Y + \nabla_{X_2}Y.\end{align}

It is linear over \(\mathbb{R}\) in \(Y\) since

\begin{align}\nabla_X(aY) &= X(\omega(aY)) - \omega(\nabla_X(aY)) \\&= X(a\omega(Y)) - \omega(a\nabla_X(Y)) \\&= aX(\omega(Y)) - a\omega(\nabla_X(Y)) \\&= a(X(\omega(Y)) - \omega(\nabla_X(Y))) \\&= a\nabla_XY\end{align}

and

\begin{align}\nabla_X(Y_1+Y_2) &= X(\omega(Y_1+Y_2)) - \omega(\nabla_X(Y_1+Y_2)) \\&= X(\omega(Y_1)+\omega(Y_2)) - \omega(\nabla_XY_1+\nabla_XY_2) \\&= X(\omega(Y_1))+X(\omega(Y_2)) - \omega(\nabla_XY_1)-\omega(\nabla_XY_2) \\&= \nabla_XY_1 + \nabla_XY_2. \end{align}

It satisfies the product rule since

\begin{align}(\nabla_X\omega)(fY) &= X(\omega(fY)) - \omega(\nabla_X(fY)) \\&= X(f\omega(Y)) - \omega(f\nabla_XY + (Xf)Y) \\&= fX(\omega(Y)) - \omega(f\nabla_XY) + \omega((Xf)Y) \\&= fX(\omega(Y)) - f\omega(\nabla_XY) + (Xf)\omega(Y) \\&= f(X(\omega(Y)) - \omega(\nabla_XY)) + (Xf)\omega(Y) \\&= f(\nabla_X\omega)(Y) + (Xf)\omega(Y) \\&= (f(\nabla_X\omega) + (Xf)\omega)(Y).\end{align}

Finally, we can extend the definition of the covariant derivative of a covector field to a definition of the covariant derivative of an arbitrary tensor field by applying the compatibility condition argument-wise, as in the following definition.

Definition (Covariant Derivative of a Tensor Field). Given a connection \(\nabla\) on the tangent bundle of a smooth manifold \(M\), the covariant derivative of a tensor field \(F \in \Gamma(T^{(k,l)}TM)\) in the direction of a vector field \(X \in \mathfrak{X}(M)\) is the map

\[\nabla_XF : \underbrace{\mathfrak{X}^*(M) \times \dots \mathfrak{X}^*(M)}_{k~\text{times}} \times \underbrace{\mathfrak{X}(M) \times \dots \mathfrak{X}(M)}_{l~\text{times}} \rightarrow C^{\infty}(M)\]

defined as follows for any covector fields \(\omega^1,\dots,\omega^k\) and vector fields \(Y_1,\dots,Y_l\):

\begin{align}(\nabla_XF)(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l) &= X(F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l)) \\&- \sum_{i=1}^kF(\omega^1,\dots,\nabla_X\omega^i,\dots,\omega^k,Y_1,\dots,Y_l) \\&- \sum_{j=1}^lF(\omega^1,\dots,\omega^k,Y_1,\dots,\nabla_XY_j,\dots,Y_l).\end{align}

Note that this map does indeed characterize a tensor field by the Characterization Lemma since it is multilinear over \(C^{\infty}(M)\):

\begin{align}(\nabla_XF)(\omega^1,\dots, f\omega^n,\dots,\omega^k,Y_1,\dots,Y_l) &= X(F(\omega^1,\dots,f\omega^n,\dots,\omega^k,Y_1,\dots,Y_l)) \\&- \sum_{i \ne n}F(\omega^1,\dots,\nabla_X\omega^i,\dots,f\omega^n,\dots,\omega^k,Y_1,\dots,Y_l) \\&- F(\omega^1,\dots,\nabla_X(f\omega^n),\dots,\omega^k,Y_1,\dots,Y_l) \\&- \sum_jF(\omega^1,\dots,f\omega^n,\dots,\omega^k,Y_1,\dots,\nabla_XY_j,\dots,Y_l) \\&= X(fF(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l)) \\&- \sum_{i \ne n}fF(\omega^1,\dots,\nabla_X\omega^i,\dots,\omega^k,Y_1,\dots,Y_l) \\&- F(\omega^1,\dots,f\nabla_X(\omega^n)+(Xf)\omega^n,\dots,\omega^k,Y_1,\dots,Y_l) \\&- \sum_jfF(\omega^1,\dots,\omega^k,Y_1,\dots,\nabla_XY_j,\dots,Y_l) \\&= fX(F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l)) \\&+ (Xf)F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l) \\&- f\sum_{i \ne n}F(\omega^1,\dots,\nabla_X\omega^i,\dots,\omega^k,Y_1,\dots,Y_l) \\&- F(\omega^1,\dots,f\nabla_X(\omega^n),\dots,\omega^k,Y_1,\dots,Y_l) \\&- F(\omega^1,\dots,(Xf)\omega^n,\dots,\omega^k,Y_1,\dots,Y_l) \\&- f\sum_jF(\omega^1,\dots,\omega^k,Y_1,\dots,\nabla_XY_j,\dots,Y_l) \\&= fX(F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l)) \\&- f\sum_{i=1}^kF(\omega^1,\dots,\nabla_X\omega^i,\dots,\omega^k,Y_1,\dots,Y_l) \\&- f\sum_{j=1}^lF(\omega^1,\dots,\omega^k,Y_1,\dots,\nabla_XY_j,\dots,Y_l) \\&=f(\nabla_XF)(\omega^1,\dots, f\omega^n,\dots,\omega^k,Y_1,\dots,Y_l) .\end{align}

A similar calculation involving \(fY_n\) yields the same result. The expression is clearly multilinear under the addition of arguments.

Moreover, this does indeed define a covariant derivative in the respective tensor bundle, since linearity over \(C^{\infty}(M)\) in \(X\) and linearity over \(\mathbb{R}\) in \(Y\) are both apparent and the product rule is immediate since \(X\) satisfies the product rule.

We can now define the total covariant derivative in analogy to the total derivative of a function.

Definition (Total Covariant Derivative). Given a connection \(\nabla\) in \(TM\) on a smooth manifold \(M\), the total covariant derivative of a smooth tensor field \(F \in \Gamma(T^{(k,l)}TM)\) is the map

\[\nabla F : \underbrace{\mathfrak{X}^*(M) \times \dots \times \mathfrak{X}^*(M)}_{k ~ \text{times}} \times \underbrace{\mathfrak{X}(M) \times \dots \times \mathfrak{X}(M)}_{l+1 ~ \text{times}} \rightarrow C^{\infty}(M)\]

defined as follows for any vector field \(X\), covector fields \(\omega^1,\dots,\omega^k\), and vector fields \(Y_1,\dots,Y_l\):

\[(\nabla F)(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l, X) = (\nabla_XF)(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l).\]

Covariant Derivative in Coordinates

Next, we will consider the coordinate expression of the covariant derivative of a tensor field. Suppose \(M\) is a smooth manifold, \(\nabla\) is a connection in \(TM\), \((E_i)\) is a local frame for \(M\) with dual coframe \((\varepsilon^j)\), \(X\) is a smooth vector field on \(M\).

Then, for any covector field \(\omega\):

\begin{align}\nabla_X\omega &= (\nabla_X\omega)(E_k)\varepsilon^k \\&= (X(\omega(E_k)) - \omega(\nabla_XE_k))\varepsilon^k \\&= (X(\omega_i\varepsilon^i(E_k)) - \omega(X(E_k^i)E_i + X^jE_k^l\Gamma_{jl}^iE_i))\varepsilon^k \\&= (X(\omega_i\delta^i_k) - \omega(X(\delta_k^i)E_i + X^j\delta_k^l\Gamma_{jl}^iE_i))\varepsilon^k \\&= (X(\omega_k) - \omega(X^j\Gamma_{jk}^iE_i))\varepsilon^k \\&= (X(\omega_k) - X^j\Gamma_{jk}^i\omega(E_i))\varepsilon^k \\&= (X(\omega_k) - X^j\omega_i\Gamma_{jk}^i)\varepsilon^k. \end{align}

Then, for any tensor field \(F \in \Gamma(T^{(k,l)}(TM))\):

\begin{align}(\nabla_XF)(\varepsilon^{i_1},\dots,\varepsilon^{i_k},E_{j_1},\dots,E_{j_l}) &= X(F((\varepsilon^{i_1},\dots,\varepsilon^{i_k},E_{j_1},\dots,E_{j_l}))) \\&- \sum_{s=1}^kF(\varepsilon^{i_1},\dots,\nabla_X\varepsilon^{i_s},\dots,\varepsilon^{i_k},E_{j_1},\dots,E_{j_l}) \\&- \sum_{j=1}^lF(\varepsilon^{i_1},\dots,\varepsilon^{i_k},E_{j_1},\dots,\nabla_XE_{j_s},\dots,E_{j_l}) \\&= X(F^{i_1 \dots i_k}_{j_1 \dots j_l}) \\&- \sum_{s=1}^kF(\varepsilon^{i_1},\dots,(X(\varepsilon^{i_s}_p) - X^m\varepsilon^{i_s}_{i_s}\Gamma_{mp}^{i_s})\varepsilon^p,\dots,\varepsilon^{i_k},E_{j_1},\dots,E_{j_l}) \\&- \sum_{s=1}^lF(\varepsilon^{i_1},\dots,\varepsilon^{i_k},E_{j_1},\dots,(X(E_{j_s}^p) + X^mE_{j_s}^q\Gamma^p_{mq})E_p,\dots,E_{j_l}) \\&= X(F^{i_1 \dots i_k}_{j_1 \dots j_l}) \\&- \sum_{s=1}^kF(\varepsilon^{i_1},\dots,(X(\delta^{i_s}_p) - X^m\delta^{i_s}_{i_s}\Gamma_{mp}^{i_s})\varepsilon^p,\dots,\varepsilon^{i_k},E_{j_1},\dots,E_{j_l}) \\&- \sum_{s=1}^lF(\varepsilon^{i_1},\dots,\varepsilon^{i_k},E_{j_1},\dots,(X(\delta_{j_s}^p) + X^m\delta_{j_s}^q\Gamma^p_{mq})E_p,\dots,E_{j_l}) \\&= X(F^{i_1 \dots i_k}_{j_1 \dots j_l}) \\&- \sum_{s=1}^kF(\varepsilon^{i_1},\dots, - X^m\Gamma_{mp}^{i_s}\varepsilon^p,\dots,\varepsilon^{i_k},E_{j_1},\dots,E_{j_l}) \\&- \sum_{s=1}^lF(\varepsilon^{i_1},\dots,\varepsilon^{i_k},E_{j_1},\dots,X^m\Gamma^p_{mj_s}E_p,\dots,E_{j_l}) \\&= X(F^{i_1 \dots i_k}_{j_1 \dots j_l}) \\&+X^m\Gamma_{mp}^{i_s} \sum_{s=1}^kF(\varepsilon^{i_1},\dots, \varepsilon^p,\dots,\varepsilon^{i_k},E_{j_1},\dots,E_{j_l}) \\&- X^m\Gamma^p_{mj_s}\sum_{s=1}^lF(\varepsilon^{i_1},\dots,\varepsilon^{i_k},E_{j_1},\dots,E_p,\dots,E_{j_l}) \\&= X(F^{i_1 \dots i_k}_{j_1 \dots j_l}) \\&+ \sum_{s=1}^k X^m F^{i_1 \dots p \dots i_k}_{j_1 \dots j_l} \Gamma_{mp}^{i_s} \\&- \sum_{s=1}^l X^m F^{i_1 \dots i_k}_{j_1 \dots p \dots j_l} \Gamma^p_{mj_s}.\end{align}

It thus follows that

\[\nabla_XF = \left(X(F^{i_1 \dots i_k}_{j_1 \dots j_l}) + \sum_{s=1}^k X^m F^{i_1 \dots p \dots i_k}_{j_1 \dots j_l} \Gamma_{mp}^{i_s} - \sum_{s=1}^l X^m F^{i_1 \dots i_k}_{j_1 \dots p \dots j_l} \Gamma^p_{mj_s}\right) \cdot \varepsilon^{i_1} \otimes \dots \otimes \varepsilon^{i_k} \otimes E_{j_1} \otimes \dots \otimes E_{j_l}.\]

Note that this notation is meant to indicate that \(p\) occurs in the \(s\)-th position of the list of indices.

Using this expression, we can also compute an expression for the total covariant derivative of a tensor field in coordinates.

\begin{align}(\nabla F)(\varepsilon^{i_1},\dots,\varepsilon^{i_1},E_{j_1},\dots,E_{j_l},E_m) &= (\nabla_XF)(\varepsilon^{i_1},\dots,\varepsilon^{i_1},E_{j_1},\dots,E_{j_l},E_m) \\&= \left(E_m(F^{i_1 \dots i_k}_{j_1 \dots j_l}) + \sum_{s=1}^k E_m^m F^{i_1 \dots p \dots i_k}_{j_1 \dots j_l} \Gamma_{mp}^{i_s} - \sum_{s=1}^l E_m^m F^{i_1 \dots i_k}_{j_1 \dots p \dots j_l} \Gamma^p_{mj_s}\right) \\&= \left(E_m(F^{i_1 \dots i_k}_{j_1 \dots j_l}) + \sum_{s=1}^k \delta_m^m F^{i_1 \dots p \dots i_k}_{j_1 \dots j_l} \Gamma_{mp}^{i_s} - \sum_{s=1}^l \delta_m^m F^{i_1 \dots i_k}_{j_1 \dots p \dots j_l} \Gamma^p_{mj_s}\right) \\&=\left(E_m(F^{i_1 \dots i_k}_{j_1 \dots j_l}) + \sum_{s=1}^k F^{i_1 \dots p \dots i_k}_{j_1 \dots j_l} \Gamma_{mp}^{i_s} - \sum_{s=1}^l F^{i_1 \dots i_k}_{j_1 \dots p \dots j_l} \Gamma^p_{mj_s}\right).\end{align}

Properties of The Covariant Derivative

The covariant derivative of a tensor field satisfies a few important properties with respect to operations on tensors such as the trace and tensor product.

Tensor Product

Consider the following for tensor fields \(F \in \Gamma(T^{(k,l)}(TM))\) and \(G \in \Gamma(T^{(p,q)}(TM))\):

\begin{align}\nabla_X(F \otimes G)(\omega^1,\dots,\omega^{k+p},Y_1,\dots,Y_{l+q}) &= X((F \otimes G)(\omega^1,\dots,\omega^{k+p},Y_1,\dots,Y_{l+q})) \\&- \sum_{i=1}^{k+p}(F \otimes G)(\omega^1,\dots,\nabla_X\omega^i,\dots,\omega^{k+p},Y_1,\dots,Y_{l+q}) \\&- \sum_{j=1}^{l+q}(F \otimes G)(\omega^1,\dots,\omega^{k+p},Y_1,\dots,\nabla_XY_j,\dots,Y_{l+q}) \\&= X(F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l)G(\omega^{k+1},\dots,\omega^{k+p},Y_{l+1},\dots,Y_{l+q})) \\&- \sum_{i=1}^kF(\omega^1,\dots,\nabla_X\omega^i,\dots,\omega^k,Y_1,\dots,Y_l)G(\omega^{k+1},\dots,\omega^{k+p},Y_{l+1},\dots,Y_{l+q}) \\&- \sum_{i=k+1}^{k+p}F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l)G(\omega^{k+1},\dots,\nabla_X\omega^i,\dots,\omega^{k+p},Y_{l+1},\dots,Y_{l+q}) \\&- \sum_{j=1}^lF(\omega^1,\dots,\omega^k,Y_1,\dots,\nabla_XY_j,\dots,Y_l) G(\omega^{k+1},\dots,\omega^{k+p},Y_{l+1},\dots,Y_{l+q}) \\&- \sum_{j=l+1}^{l+q}F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_{l+q}) G(\omega^{k+1},\dots,\omega^{k+p},Y_{l+1},\dots,\nabla_XY_j,\dots,Y_{l+q})\\&= X(F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l))G(\omega^{k+1},\dots,\omega^{k+p},Y_{l+1},\dots,Y_{l+q}) \\&+ F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l)X(G(\omega^{k+1},\dots,\omega^{k+p},Y_{l+1},\dots,Y_{l+q})) \\&- \left(\sum_{i=1}^kF(\omega^1,\dots,\nabla_X\omega^i,\dots,\omega^k,Y_1,\dots,Y_l)\right)G(\omega^{k+1},\dots,\omega^{k+p},Y_{l+1},\dots,Y_{l+q}) \\&- F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l)\left(\sum_{i=k+1}^{k+p}G(\omega^{k+1},\dots,\nabla_X\omega^i,\dots,\omega^{k+p},Y_{l+1},\dots,Y_{l+q}) \right)\\&- \left(\sum_{j=1}^lF(\omega^1,\dots,\omega^k,Y_1,\dots,\nabla_XY_j,\dots,Y_l)\right) G(\omega^{k+1},\dots,\omega^{k+p},Y_{l+1},\dots,Y_{l+q}) \\&- F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_{l+q}) \left(\sum_{j=l+1}^{l+q}G(\omega^{k+1},\dots,\omega^{k+p},Y_{l+1},\dots,\nabla_XY_j,\dots,Y_{l+q}) \right) \\&= \left((\nabla_XF) \otimes G + F \otimes (\nabla_XG)\right)(\omega^1,\dots,\omega^{k+p},Y_1,\dots,Y_{l+q}).\end{align}

It thus follows that

\[\nabla_X(F \otimes G) = (\nabla_XF) \otimes G + F \otimes (\nabla_XG).\]

Trace

Let \(F\) be a \((k+1,l+1)\)-tensor field, and consider its trace (contraction) at indices \(i\) and \(j\). Its trace on these indices is the tensor field \(\mathrm{tr}_{ij}~F\) defined as follows relative to a given coordinate frame \((E_i)\) with dual coframe \((\varepsilon^j)\) (where a sum over \(m\) is implied):

\[(\mathrm{tr}_{ij}~F)(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l) = F(\omega^1,\dots,\underbrace{\varepsilon^m}_{i\text{-th covector}},\omega^k,Y_1,\dots,\underbrace{E_m}_{j\text{-th vector}},\dots,Y_l).\]

It then follows that

\begin{align}\nabla_X(\mathrm{tr}_{ij}~F)(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l) &= X((\mathrm{tr}_{ij}~F)(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l)) \\&- \sum_{s=1}^k(\mathrm{tr}_{ij}~F)(\omega^1,\dots,\nabla_X\omega^s,\dots,\omega^k,Y_1,\dots,Y_l) \\&- \sum_{s=1}^l(\mathrm{tr}_{ij}~F)(\omega^1,\dots,\omega^k,Y_1,\dots,\nabla_XY_s,\dots,Y_l) \\&= X(F(\omega^1,\dots,\underbrace{\varepsilon^m}_{i\text{-th covector}},\dots,\omega^k,Y_1,\dots,\underbrace{E_m}_{j\text{-th vector}},\dots,Y_l)) \\&- \sum_{s=1}^kF(\omega^1,\dots,\underbrace{\varepsilon^m}_{i\text{-th covector}},\dots,\nabla_X\omega^s,\dots, \omega^k,Y_1,\dots,\underbrace{E_m}_{j\text{-th vector}},\dots,Y_l) \\&- \sum_{s=1}^lF(\omega^1,\dots,\underbrace{\varepsilon^m}_{i\text{-th covector}},\dots,\omega^k,Y_1,\dots,\underbrace{E_m}_{j\text{-th vector}},\dots,\nabla_XY_s,\dots,Y_l) \\&= (\nabla_XF)(\omega^1,\dots,\underbrace{\varepsilon^m}_{i\text{-th covector}},\omega^k,Y_1,\dots,\underbrace{E_m}_{j\text{-th vector}},\dots,Y_l) \\&= \mathrm{tr}_{ij}(\nabla_XF)(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l).\end{align}

Thus,

\[\nabla_X(\mathrm{tr}_{ij}~F) = \mathrm{tr}_{ij}(\nabla_XF).\]

Uniqueness of the Covariant Derivative

Note that, for a covector field \(\omega\) and a vector field \(Y\), \(\omega \otimes Y\) can be written in coordinates as \(\omega \otimes Y = \omega_iY^j (\varepsilon^i\otimes E_j)\) in terms of a local frame \((E_j)\) with dual coframe \((\varepsilon^i)\). This is thus a \((1,1)\) tensor. The trace of such a tensor is computed as

\begin{align}\mathrm{tr}(\omega \otimes Y) &= (\omega \otimes Y)(E_i, \varepsilon_i) \\&= \omega(E_i)Y(\varepsilon^i) \\&= \omega_i\varepsilon^i(Y) \\&= \omega_iY^i.\end{align}

Note also that

\begin{align}\langle \omega,Y\rangle &= \omega(Y) \\&= \omega_i\varepsilon^i(Y) \\&= \omega_iY^i.\end{align}

Thus, since these two tensor fields always have the same expression in coordinates, they are equal, i.e.

\[\langle \omega,Y\rangle = \mathrm{tr}(\omega \otimes Y).\]

Note further that this fact combined with the properties established previously indicate the following:

\begin{align}\nabla_X\langle \omega,Y \rangle &= \nabla_X(\mathrm{tr}(\omega \otimes Y)) \\&= \mathrm{tr}(\nabla_X(\omega \otimes Y)) \\&= \mathrm{tr}(\nabla_X\omega \otimes Y + \omega \otimes \nabla_XY) \\&= \mathrm{tr}(\nabla_X\omega \otimes Y) + \mathrm{tr}(\omega \otimes \nabla_XY) \\&= \langle \nabla_X\omega \rangle + \langle \omega, \nabla_XY \rangle.\end{align}

Thus, any covector field satisfying the relevant properties (with respect to tensor products and traces) necessarily determines the same covariant derivative.

This result can be extended by induction to higher rank tensors. Observe the following for a \((k,l)\)-tensor \(F\):

\begin{align}\mathrm{tr_{1,l+1}}(F \otimes \omega^1)(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l) &= (F \otimes \omega^1)(\varepsilon^m,\dots,\omega^k,Y_1,\dots,Y_l, E_m) \\&= F(\varepsilon^m,\dots,\omega^k,Y_1,\dots,Y_l)\omega^1(E_m) \\&= \omega^1_m \cdot F(\varepsilon^m,\dots,\omega^k,Y_1,\dots,Y_l) \\&= F(\omega^1_m \cdot \varepsilon^m,\dots,\omega^k,Y_1,\dots,Y_l) \\&= F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l).\end{align}

It then follows that

\[F(\omega^1,\dots,\omega^k,Y_1,\dots,Y_l) = \underbrace{\mathrm{tr} \circ \dots \circ \mathrm{tr}}_{k+l ~\text{times}}(F \otimes \omega^1 \otimes \dots \otimes \omega^k \otimes Y_1 \otimes \dots \otimes Y_l).\]

The argument for covectors can then be extended by induction to demonstrate that the definition of \(\nabla_XF\) must coincide for any tensor field satisfying the relevant properties regarding tensor products and traces. Thus, these properties uniquely determine the covariant derivative.