# Ex 14.22 Lee's Smooth Manifold: interior multiplication

14.22: pg 362 Lee's Smooth Manifold Let $$X$$ be a smooth vector field on $$M$$.

1. If $$w$$ is a smooth differential form then $$i_Xw$$ is smooth.
$$i_X w:= (X \lrcorner w)_p =X_p \lrcorner w_p.$$

2. $$i_X:\Omega^k(M) \rightarrow \Omega^{k-1}(M)$$ is linear over $$C^\infty(M)$$, corresponding to a smooth bundle homomoprhism $$i_X: \wedge ^kT^*M \rightarrow \wedge^{k-1}T^*M.$$

I would really appreciate nice explanations for both aprts. My solution for 1. is ugly (below). For 2. how does linearity over $$C^\infty(M)$$ imply bundle homomorphism?

My thoughts for 2. are using lemma 14.13 and working in a local coordinate chart.

Lemma 14.13, pg 358: If $$w \in \wedge^k(V^*)$$ and $$\eta \in \wedge^l(V^*)$$, $$i_v (w \wedge \eta) = (i_v w ) \wedge \eta + (-1)^k w \wedge (i_v \eta).$$

which I believe is what the solution wants.

May attempt for 1.

We work locally on a local chart $$(U, (x^i))$$ around $$w$$. We know $$w = \sum w_I dx^I$$, $$X=X^i \frac{\partial}{\partial x^i}$$, where $$w_I, X^i$$ are smooth on $$U$$. Thus, for $$p \in U$$, $$v_2, \ldots, v_k \in T^kM$$, \begin{align*} X_p \lrcorner w_p (v_2, \ldots, v_k) &= w_p(X_p, v_2, \ldots, v_k) \\ &= \sum_I w_I(p) \sum_i X^i (p)dx^I(\frac{\partial }{\partial x^i}, v_2, \ldots, v_k) \end{align*}

In this form, we can see the expression is clearly representable $$\sum_{J} l_Jdx^J$$ for $$J$$ of length $$k-1$$., such that $$l_J$$ is smooth on $$U$$.

EDIT:

Thanks for the comments below. The general theorem show that $$i_X$$ is then given by $$i_X (v) = i_X (\tilde{v})(p)$$ for some $$\tilde{v} \in \Omega^k(M)$$ where $$\tilde{v}(p) = v$$.

• Your solution for 1 is fine. I wouldn't call working with coordinates ugly. In fact, in some cases, there is no other way. As for 2, it is a general (and important) fact that if $E,F\to M$ are vector bundles and $L:\Gamma(E)\to\Gamma(F)$ is linear over $C^\infty(M)$, then $L$ corresponds to a bundle morphism $E\to F$. Regardless, it follows from the definition that interior multiplication corresponds to a bundle morphism. – Amitai Yuval Oct 3 '18 at 10:14
• As @AmitaiYuval said. The second part is true by Lemma 10.29 (Bundle Homomorphism Characterization Lemma) in Lee’s. – Sou Oct 3 '18 at 11:38
• Thanks AmitaiYuval and Sou, I understand now. @Amitai do you want to copy/post your solution so the problem is resolved? – CL. Oct 5 '18 at 9:20