Action of almost complex structure on tensors/forms

Question

I am currently trying to lear about almost complex structures and how they are extended to tensor fields especially differential forms.

I have seen some variations but am confused about certain aspects, e.g. one given definition was:

Let $(M, J)$ be a manifold of real dimension 2m with almost complex structure $J$.
We can extend the almost complex structure to $k$-forms in the following way:
$(J\alpha)(X_1, ..., X_k)\colon = \alpha(J X_1, ..., J X_k)$.

I, however, do not see, how this yields an almost complex structure on the space of e.g. two forms, because, by my "naïve" calculation for a two form, one would get the following "contradiction":
\begin{align*} -\alpha(X, Y) &= (J^2 \alpha)(X, Y) = (J(J\alpha))(X, Y) = (J\alpha)(J X, J Y) = \alpha(J^2 X, J^2 Y)\\ &= \alpha(-X, -Y) = \alpha (X, Y). \end{align*}

Now obviously this works just fine for one-forms but not for two-forms or at least not in the naïve way.

Another definition I stumbled upon was the following:

$(J\alpha)(X_1, ...,X_k) \colon = \sum\limits_{n=1}^k \alpha(X_1,..., J X_n, ...,X_k)$.

Here, I also fail to see, how this reproduces the $J^2 \alpha = -\alpha$ property an almost complex structure should have.

So basically my questions are, whether I am wrong somewhere and this works out or, if I am correct in my assessments, how one can extend an almost complex structure correctly to differential forms and even tensors.

Thanks in advance and cheers!

Edit: Fixed some layout of equations.

Answer 1

Thanks to Ivo Terek's comment, I have come up with a satisfying answer after giving this some more thought.

Edit 3: Didier and Travis Willse respectively have made a smart suggestion and alerted me to and error in my thinking. For one, there is a nicer way to define the action of the almost complex strucutre on one forms. For the other, my proof for the equivalence of the two expressions of the almost complex structure is incorrect insofar that it only works on completely symmetric or antisymmetric tensor fields or tensor fields that are combination of the former two types because tensors of type $(r, s)$ for which $r + s > 2$ in general cannot be decomposed into just a completely symmetric and completely antisymmetric part but rather have more complicated symmetries (although the equivalence of the two expressions does still hold for the (Riemann-)curvature tensor, as long as it exhibits the same symmetries as the Levi-Civita connexion's (Riemann-)curvature tensor). I will be marking the changes from this third edit in blue so that it is obvious to the reader.

Let $(M, J)$ be a manifold of real dimension $2m$ with almost complex structure $J$.

The extension of $J$ to one-forms is straightforward:
$(J\alpha)(X)\colon= \color{blue}{\mathbf{-}}\alpha(JX)$, where $\alpha \in \Omega^1(M)$ and $X \in \Gamma TM$, $\style{font-family:inherit;}{\color{blue}{\text{such that $(J\alpha)(JX) = \alpha(X)$.}}}$ $\style{font-family:inherit;}{\color{blue}{\text{One cou just as well choose $(J\alpha)(X)\colon= \alpha(JX)$; this, however, does not have the "nice"}}}$ $\style{font-family:inherit;}{\color{blue}{\text{or "expected" property that $(J\alpha)(JX) = \alpha(X)$ but rather that $(J\alpha)(JX) = -\alpha(X)$.}}}$

$\style{font-family:inherit;}{\color{blue}{\text{We can extend $J$ to act on arbitrary $(r, s)$-tensor-fields in the following way:}}}$ $\color{blue}{(J_kT)(\alpha_1, ..., \alpha_{r+s})\colon = -T(\alpha_1, ..., J\alpha_k, ..., \alpha_{r+s}),}$ $\style{font-family:inherit;}{\color{blue}{\text{where each $\alpha_n$ is an element of either $\Gamma TM$ or $\Omega^1(M)$, and $T\in \Gamma TM_r^s$.}}}$ $\style{font-family:inherit;}{\color{blue}{\text{As far as I can tell this yields no unique extension but rather presents one with}}}$ $\style{font-family:inherit;}{\color{blue}{\text{$r+s$ extensions from which to choose.}}}$

We can extend $J$ to act on arbitrary $\style{font-family:inherit;}{\color{blue}{\text{completely symmetric or completely antisymmetric}}}$
$(r, s)$-tensor-fields $\style{font-family:inherit;}{\color{blue}{\text{or those that can be decomposed into a completely symmetric}}}$ $\style{font-family:inherit;}{\color{blue}{\text{and completely antisymmetric part}}}$ in the following way: $(JT)(\alpha_1, ..., \alpha_{r+s})\colon = \color{blue}{\mathbf{-}}T(J\alpha_1, ...,\alpha_{r+s})\equiv \color{blue}{\mathbf{-}}\frac{1}{r+s}\sum\limits_{k=1}^{r+s}T(\alpha_1,..., J\alpha_k,..., \alpha_{r+s})$,
where each $\alpha_n$ is an element of either $\Gamma TM$ or $\Omega^1(M)$, and $T\in \Gamma TM_r^s$. $\style{font-family:inherit;}{\color{blue}{\text{On these kinds of tensor fields the different choices of extending $J$ are thus equivalent}}}$ $\style{font-family:inherit;}{\color{blue}{\text{yielding a unique (up to a sign) extension.}}}$

Since tensors are multilinear, we directly see that this indeed satisifies the $J^2T =-T$ property we want from an almost complex structure. What remains to be shown is the last equality, which we will do in two steps:

First, we remark, that we can always decompose tensors into a symmetric and an antisymmetric part: $T= S + A$, where $S$ denotes the symmetric part of $T$ and $A$ denotes the anitsymmetric part of $T$. We , therefore, prove the last equality separately for a completely antisymmetric tensor and for a completely symmetric tensor:

Case I: $T$ is completely antisymmetric:

\begin{align*} (JT)(\alpha_1, ..., \alpha_{r+s}) &= \color{blue}{\mathbf{-}}T(J\alpha_1,..., \alpha_{r+s})\\ &= \color{blue}{\mathbf{-}}T(J\alpha_1,...,\alpha_{k-1}, \alpha_{k}, \alpha_{k+1},..., \alpha_{r+s})\\ &= \color{blue}{\mathbf{-}}T(J\alpha_1,...,\alpha_{k-1}, -J^2\alpha_{k}, \alpha_{k+1},..., \alpha_{r+s})\\ &= \color{blue}{\mathbf{-}}(-1)^{k-1}T(-J^2\alpha_{k}, J\alpha_1,..., \alpha_{r+s})\\ &= \color{blue}{\mathbf{-}}(-1)^{k-1}(-1)T(J^2\alpha_{k}, J\alpha_1,..., \alpha_{r+s})\\ &= \color{blue}{\mathbf{-}}(-1)^{k-1}(-1)(JT)(J\alpha_k, J\alpha_1,..., \alpha_{r+s})\\ &= \color{blue}{\mathbf{-}}(-1)^{k-1}(-1)^{k-1}(-1)(JT)(J\alpha_1, ...,\alpha_{k-1}, J\alpha_{k}, \alpha_{k+1},..., \alpha_{r+s})\\ &=\color{blue}{\mathbf{-}}(-1)^{2(k-1)} (-1) T(J^2\alpha_1,..., J\alpha_k,..., \alpha_{r+s})\\ &=\color{blue}{\mathbf{-}}(-1)^{2(k-1)} (-1) T(-\alpha_1,..., J\alpha_k,..., \alpha_{r+s})\\ &=\color{blue}{\mathbf{-}}(-1)^{2(k-1)} (-1)^2 T(\alpha_1,..., J\alpha_k,..., \alpha_{r+s})\\ &=\color{blue}{\mathbf{-}}T(\alpha_1,..., J\alpha_k,..., \alpha_{r+s}), \end{align*} where we have taken care to make clear which factors of $(-1)$ come from which operation: the factor $(-1)^{2(k-1)}$ comes from the antisymmetry of the tensor and the factor $(-1)^2$ comes from switching around the almost complex structure.

Case II: $T$ is completely symmetric:

This follows completely analogously to the calculation above, but we do not receive a factor of $(-1)^{2(k-1)}$ since the tensor is completely symmetric now.

Combining these two results yields:
$(JT)(\alpha_1,..., \alpha_{r+s}) = \color{blue}{\mathbf{-}}T(J\alpha_1,..., \alpha_{r+s}) \equiv \color{blue}{\mathbf{-}}T(\alpha_1,..., J\alpha_k,..., \alpha_{r+s})\;\forall\, k$, i.e. the almost complex structure can freely "hop" between arguments. This, in turn, yields:
$(JT)(\alpha_1,..., \alpha_{r+s}) = \color{blue}{\mathbf{-}}T(J\alpha_1,..., \alpha_{r+s}) \equiv \color{blue}{\mathbf{-}}\frac{1}{r+s}\sum\limits_{k=1}^{r+s}T(\alpha_1,..., J\alpha_k,..., \alpha_{r+s})$, thereby proving our claim.
q.e.d.

So, it turns out that the second suggestion on how to induce an almost complex structure on forms wasn't too far off, only by a constant multiple to be precise.

$\style{font-family:inherit;}{\color{blue}{\text{If one has a tensor-field with a block (anti-)symmetry and blocks that are within}}}$ $\style{font-family:inherit;}{\color{blue}{\text{themselves (anti-)symmetric, this second epression for extending $J$}}}$ $\style{font-family:inherit;}{\color{blue}{\text{applies to it, too. See, e.g. the Levi-Civita connexion's (Riemann-)curvature tensor}}}$ $\style{font-family:inherit;}{\color{blue}{\text{for which $R(X, Y, Z, U) = -R(Y, X, Z, U) = - R(X, Y, U, Z) = R(Z, U, X, Y)$.}}}$ $\style{font-family:inherit;}{\color{blue}{\text{Since many tensor-fields of interest have extensive symmetries, one often is capable}}}$ $\style{font-family:inherit;}{\color{blue}{\text{of uniquely extending $J$ to the class of tensor-fields with these symmetries in the way }}}$ $\style{font-family:inherit;}{\color{blue}{\text{we have shown.}}}$

$\style{font-family:inherit;}{\color{blue}{\text{Conclusion: We have not managed to find a unique (up to a sign) extension of $J$}}}$ $\style{font-family:inherit;}{\color{blue}{\text{to arbitrary $(r, s)$-tensor-fields. We have, however, found a unique extension for $k$-forms}}}$ $\style{font-family:inherit;}{\color{blue}{\text{and some other kinds of tensor-fields.}}}$
$\style{font-family:inherit;}{\color{blue}{\text{If anyone has a way of proving the uniqueness of this extension or the equivalence of the }}}$ $\style{font-family:inherit;}{\color{blue}{\text{different extensions we have found by some clever argument about decomposing arbitrary}}}$ $\style{font-family:inherit;}{\color{blue}{\text{tensor-fields and using their components' symmetries, please let me/us know!}}}$

Edit: Reformulated something to be more clear.
Edit 2: I have been sloppy by omitting that we can only even try to extend the almost complex structure to tensors/tensor bundles after we have complexified them. Otherwise one runs into problems because, e.g. the space of all two forms at a point on an almost complex manifold of real dimension six is itself of real dimension 15 which is not even and thus cannot carry any complex structure whatsoever. Complexifying beforehand solves this because now the space of all two forms at a point is of complex dimension 15 which means it is of real dimension 30 which is an even number thus not immediately crushing our hopes of defining an almost complex structure on it.