I'm reading Moroianu's Lectures on Kähler Geometry and had a question on how he introduced the inner product of $k$-forms. He assumes a Riemannian manifold $(M^n, g)$ and proceeds:

There is a natural embedding $\varphi$ of $\Lambda^kM$ in $(T^*M)^{\otimes k}$ given by $$ \varphi(\omega)(X_1, \ldots, X_k) \;\; =\;\; \omega(X_1, \ldots, X_k), $$ which in the local basis reads $$ \varphi(e_1\wedge \ldots \wedge e_k) \;\; = \;\; \sum_{\sigma \in \mathfrak{S}_k} \varepsilon(\sigma) e_{\sigma_1} \otimes \ldots \otimes e_{\sigma_k}. $$ The Riemannian product $g$ induces a Riemannian product on all tensor bundles. We consider the following weighted scalar product on $\Lambda^kM$: $$ \langle \omega, \tau \rangle \;\; =\;\; \frac{1}{k!} g(\varphi(\omega), \varphi(\tau)), $$ $\ldots $

I'm not clear in what sense this defines an inner product, since $g$ is a symmetric 2-tensor on $TM$, I would anticipate that $\varphi(\omega)$ would be a tangent vector on $M$ in order for it to be inserted into the Riemannian metric. $\varphi$ appears to just be inclusion in the tensor bundle, so it doesn't (as far as I can tell) change its input argument to anything that drastically different.

This is all in contrast to how I've seen this inner product defined in other contexts. Take Bleecker's Gauge Theory and Variational Principles where he unambiguously defines $$ \widetilde{g}(\alpha, \beta) \;\; =\;\; \frac{1}{k!}\sum g^{i_1j_1}g^{i_2j_2}\ldots g^{i_kj_k} \alpha_{i_1\ldots i_k} \beta_{j_1\ldots j_k} $$ where the $g^{ij}$ represent the components of the inverse of $g$ in a pre-chosen local frame, and similarly the components of $\alpha,\beta \in \Lambda^kM$.

I'm curious what is the relationship between Moroianu's definition and Bleecker's, and also to understand whether Moroianu is implying some sort of shorthand via the embedding $\varphi$.


1 Answer 1


As a preliminary, the tensor product is functorial, in the sense that it has a natural extension to linear maps. That is, if if we have linear maps between vector spaces $L_1:U_i\to V_1$ and $L_2:U_2\to V_2$, we may define $L_1\otimes L_2:U_1\otimes U_2\to V_1\otimes V_2$ as the unique linear extension of the expression $(L_1\otimes L_2)(u_1\otimes u_2)=L_1(u_1)\otimes L_2(u_2)$, and this product of maps several important properties, namely that the tensor product of isomorphisms is an isomorphism.

It's more convenient here to think of an inner product $g$ on a finite dimensional vector space $V$ as an isomorphism with the dual space $g:V\to V^*$ with inverse $\tilde{g}:V^*\to V$. The functorality of tensor product ensures that there is a unique extension to controvariant tensor spaces that is compatible with tensor products, namely $g^{\otimes k}:V^{\otimes{k}}\to(V^*)^{\otimes{k}}$, which is also an isomorphism and thus an inner product. Taking appropriate tensor products of $g$ and $\tilde g$ allows one to construct similar isomorphisms on tensors of any signature. Going back to bilinear forms, this gives the proprty $g(v_1\otimes u_1,v_2\otimes u_2)=g(u_1,u_2)g(v_1,v_2)$.

It seems from the snippets you provide that Moroianu simply states this as a known result (usung $g$ for any such inner product, which I will also do), while Bleeker states the coordinate form, which can be derived without too much fuss. Letting $A$, $B$ be tensors of signature $(0,k)$, $$\begin{align} g(A,B)&=g\left(A_{i_1\dots i_k}e^{i_1}\otimes\dots\otimes e^{i_k},B_{j_1\dots j_k}e^{j_1}\otimes\dots\otimes e^{j_k}\right) \\ &=A_{i_1\dots i_k}B_{j_1\dots j_k}g\left(e^{i_1}\otimes\dots\otimes e^{i_k},e^{j_1}\otimes\dots\otimes e^{j_k}\right) \\ &=A_{i_1\dots i_k}B_{j_1\dots j_k}g(e^{i_1},e^{j_1})\dots g(e^{i_k},e^{j_k}) \\ &=g^{i_1 j_1}\dots g^{i_k j_k}A_{i_1\dots i_k}B_{j_1\dots j_k} \end{align}$$ (the generalization to arbitrary signature is the same computation) The wedge product spaces are isomorphic to the spaces of alternating tensors, which are vector subspaces of the tensor spaces, so they inherit the inner product as a restriction of domain.

Another way of doing things is to construct the wedge product spaces independently. The wedge product is also functorial, so we can carry out exactly the same construction replacing $\otimes$ with $\wedge$, and obtain a similar family of inner products These will correspond up to a choice of normalization; in particular, the authors you cite chosen conventions such that the two differ by a factor of $k!$.


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