# Show that locally a differential form $\omega$ equals $\Sigma _I\omega _Idx^I$ in which $\omega _I$ are smooth functions

First I give the definition of differential form that I'm using.

Definition: Let $$M\subseteq\mathbb{R}^n$$ be a smooth submanifold and $$p\in\mathbb{N}^\times$$. We say that $$\omega:M\to \cup _{x\in M}\Lambda ^p(T_xM)^*$$ is differential $$p$$-form (which we denote by $$\omega \in \Omega ^p(M)$$) if the following propositions are true:

1. $$\omega (x)\in \Lambda ^p(T_xM)^*$$ for all $$x\in M$$;
2. $$M\to \mathbb{R},\,x\mapsto \omega (x)(X_1(x),\cdots,X_p(x))$$ is smooth for all smooth vector fields $$X_1,\cdots,X_p:M\to\mathbb{R}^n$$.

My question is: Suppose that $$\omega \in \Omega ^p(M)$$. How can I show that locally $$\omega$$ equals $$\Sigma _I\omega _Idx^I$$ in which $$\omega _I$$ are smooth functions?

I already proved that locally $$\omega$$ equals $$\Sigma _I\omega _Idx^I$$ in which $$\omega _I$$ are functions, however I don't know how to prove that those functions are smooth. My problem is with the following fact:

• Suppose that $$A\subseteq M$$ is open in $$M$$. I don't know how to prove that the restriction $$\omega |_A$$ is also a differential $$p$$-form. It's easy to prove that $$\omega (x)\in \Lambda ^p(T_xA)^*$$ for all $$x\in A$$ but I don't know how to prove that $$A\to \mathbb{R},\,x\mapsto \omega (x)(X_1(x),\cdots,X_p(x))$$ is smooth for all smooth vector fields $$X_1,\cdots,X_p:\color{red}{A}\to \mathbb{R}^n$$ because we can't guarantee that given a smooth vector field $$X:A\to \mathbb{R}^n$$ in $$A$$ there's a smooth vector field $$\tilde X:M\to \mathbb{R}^n$$ in $$M$$ such that $$\tilde X|_A=X$$. I tried to solve this problem using partition of unity and coordinate vector fields $$\frac{\partial}{\partial x_i}$$ but I failed.

Please don't assume that $$\cup _{x\in M}\Lambda ^p(T_xM)^*$$ is a smooth manifold because I'm only studying submanifolds of $$\mathbb{R}^n$$ and I'm not considering the general definition of manifold.

You can use smooth bump functions. Say $$A$$ is the coordinate patch you are interested in, then for any $$x\in A$$ consider a small closed disk $$D\subset A$$ centered at $$x$$. There then exists a bump function $$b\colon M\to\mathbb{R}$$ with support in $$A$$ such that $$b|_D\equiv 1$$ is constant. Then consider the global vector fields $$X_i := b\cdot \partial_i$$, and using property (2) is is easy to show that the the functions $$b\cdot\omega_i$$ then are smooth. We conclude that then $$\omega_i$$ is smooth at $$x$$, and since this can be done for any point in $$A$$ we have shown the $$\omega_i$$ are smooth.
The idea is to take vector fields $$X_1,… X_p$$ such that $$\omega(X_1,… X_p) = \omega_I$$. Taking $$A$$ to be a coordinate open set, if $$I=i_1i_2…i_p$$, take $$\frac{\partial}{\partial x_{i_1}},…, \frac{\partial}{\partial x_{i_p}}$$ to be the vector field. And the condition $$2)$$ says that $$\omega_I = \omega(\frac{\partial}{\partial x_{i_1}},…, \frac{\partial}{\partial x_{i_p}})$$ is smooth.
• But generally the domain of the coordinate vector fields $\frac{\partial}{\partial x_{i_1}},…, \frac{\partial}{\partial x_{i_p}}$ isn't the submanifold $M$. To use the condition 2) the vector fields must be defined in the whole submanifold $M$. Feb 21, 2022 at 13:29