b_k}^{a_1a_2\ldots c}. \end{aligned}$$ Relative to a smooth family of local frames $\{e_{i}(\tau)\}_{i=1}^{n}$ evolving so as to remain orthonormal relative to $g(\tau$), the components of $D_{\tau}V$ express the total derivatives $$D_{\tau}V_{b_1b_2 \ldots b_k}^{a_1a_2\ldots a_l} = {{\frac{\partial }{\partial\tau}}}\left(V(e_{b_1}, e_{b_2}, \ldots, e_{b_k}, e_{a_1}^*, e_{a_2}^*, \ldots, e_{a_l}^*)\right).$$ Alternatively, $D_\tau$ may be regarded as a vector on the product of the frame bundle with $[0, T]$ tangent to the bundle of $g(\tau)$-orthonormal frames. See [@HamiltonHarnack] or Appendix F of [@RFV2P2] for details.

Equations and are then equivalent to the assertions that $D_{\tau} g = 0$ and $D_{\tau} J =0$, and imply that $$J^2  = -\operatorname{Id}, \quad g(\cdot, \cdot) = g(J\cdot, J\cdot),$$ that is, that $J$ remains an almost complex structure and $g$ remains Hermitian with respect to $J$ on $M\times[0, T]$,

A time-dependent splitting of ${\wedge^2 T^*M}$.
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We will now examine the relationships between several families of endomorphisms of ${\wedge^2 T^*M}$ induced by $J$. Below, we will write $$\theta\wedge \sigma = \theta\otimes\sigma - \sigma\otimes \theta,$$ for $\theta$, $\sigma\in T^*M$, and use the metric $$\label{eq:metric}
 \langle \theta\wedge\sigma, \phi\wedge \psi\rangle = \det\left(\begin{array}{cc}
                                                               \langle \theta, \phi \rangle & \langle \theta, \psi\rangle \\
                                   \langle \sigma, \phi\rangle & \langle \sigma, \psi\rangle
                                                              \end{array}\right)$$ induced on ${\wedge^2 T^*M}$ by $g(\tau)$.

Now 