[Tu Differential Geometry] 7. Vector Bundles

This article is one of Manifold, Differential Geometry, Fibre Bundle.

The set $\mathfrak{X}(M)$ of all $C^\infty$ vector fields on a manifold $M$ has the structure of a real vector space, which is the same as a module over the field $\mathbb{R}$ of real numbers. Let $\mathcal{F}=C^\infty(M)$ again be the ring of $C^\infty$ functions on $M$. Since we can multiply a vector field by a $C^\infty$ function. the vector space $\mathfrak{X}(M)$ is also a module over $\mathcal{F}$. Thus the set $\mathfrak{X}(M)$ has two module structure, over $\mathbb{R}$ and over $\mathfrak{F}$. In spaeking of a linear map: $\mathfrak{X}(M)\rightarrow \mathfrak{X}(M)$ one should be careful to specify whether it is $\mathbb{R}$-linear or $\mathcal{F}$-linear. An $\mathcal{F}$-linear map is of course $\mathbb{R}$-linear, but the converse is not true.

The $\mathcal{F}$-linearity of the torsion $T(X,Y)$ and the curvature $R(X,Y)Z$ from the preceding section has an important consequence, namely that these two constructions make sense pointwise. For example, if $X_p$, $Y_p$, and $Z_p$ are tangent vectors to a manifold $M$ at $p$, then one can define $R(X_p,Y_p)Z_p$ to be $(R(X,Y)Z)_p\in T_pM$ for any vector fields $X$, $Y$, and $Z$ on $M$ that extend $X_p$, $Y_p$, and $Z_p$, respectively. While it is possible to explain this fact strictly whithin the framework of vector fields, it is most natural to study it in the context of vector bundles. For this reason, we make a digression on vector bundles in this section.

We will try to understand $\mathcal{F}$-linear maps from the point of view of vector bundles. The main result (Theorem 7.26) asserts the existence of a one-to-one correspondence between $\mathcal{F}$-linear maps $\alpha:\Gamma(E)\rightarrow \Gamma(F)$ of sections of vector bundles and bundle maps $\varphi:E\rightarrow F$.

7.1 Definition of a Vector Bundle

Given an open subset $U$ of a manifold $M$, one can think of $U\times \mathbb{R}^r$ as a family of vector spaces $\mathbb{R}^r$ parametrized by the points in $U$. A vector bundle, intuitively speaking, is a family of vector spaces that locally "looks" liker $U\times \mathbb{R}^r$.

Definition 7.1. A $C^\infty$ surjection $\pi:E\rightarrow M$ is a $C^\infty$ vector bundle of rank $r$ if 

(i) For every $p\in M$, the set $E_p:=\pi^{-1}(p)$ is areal vector space of dimension $r$;

(ii) every point $p\in M$ has an open neighborhood $U$ such that there is a fiber-preserving diffeomorphism $$\phi_U:\pi^{-1}(U)\rightarrow U\times \mathbb{R}^r$$ that restricts to a linear isomorphism $E_p\rightarrow \{p\}\times \mathbb{R}^r$ on each fiber.

The space $E$ is called the total space, the space $M$ the base space, and the space $E_p$ the fiber above $p$ of the vector bundle. We often say that $E$ is a vector bundle over $M$. A vector bundle of rank $1$ is also called a line bundle.

Condition (i) says that $\pi:E\rightarrow M$ is a family of vector spaces, while condition (ii) formalizes the fact that this family is locally looks like $\mathbb{R}^n$. We call the open set $U$ in (ii) a trivializing open subset for the vector bundle, and $\phi_U$ a trivialization of $\pi^{-1}(U)$. A trivializing open cover for the vector bundle is an open cover $\{U_\alpha\}$ of $M$ consisting of trivializing open sets $U_\alpha$ together with trivializations $\phi_\alpha:\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times \mathbb{R}^r$.

Definition 7.5. Let $\pi_E:E\rightarrow M$ and $\pi_F:F\rightarrow N$ be $C^\infty$ vector bundles. A $C^\infty$ bundle map from $E$ to $F$ is a pair of $C^\infty$ maps ($\varphi:E\rightarrow F$, $\varphi_-:M\rightarrow N$) such that 

(i) The diagram

commutes,

(ii) $\varphi$ restricts to a linear map $\varphi_p:E_p\rightarrow F_{\varphi_-(p)}$ of fibers for each $p\in M$.

Abusing language, we often call the map $\varphi:E\rightarrow F$ alone the bundle map.

An important special case of a bundle map occurs when $E$ and $F$ are vector bundles over the same manifold $M$ and the base map $\varphi_-$ is the identity map $1_M$, In this case we call the bundle map ($\varphi:E\rightarrow F,1_M$) a bundle map over $M$.

If there is a bundel map $\psi:F\rightarrow E$ over $M$ such that $\psi\circ \varphi=1_E$ and $\varphi\circ \psi=1_F$, then $\varphi$ is called a bundle isomorphism over $M$, and the vector bundles $E$ and $F$ are said to be isomorphic over $M$.

Definition 7.6. A vector bundle $\pi:E\rightarrow M$ is said to be trivial if it is isomorphic to a product bundle $M\times \mathbb{R}^r\rightarrow M$ over $M$.

7.2 The Vector Space of Sections

A section of a bector bundle $\pi:E\rightarrow M$ over an open set $U$ is a function $s:U\rightarrow E$ such that $\pi\circ s=1_U$, the identity map on $U$. For each $p\in U$, the section $s$ picks out one element of the fiber $E_p$. The set of all $C^\infty$ sections of $E$ over $U$ is denoted by $\Gamma(U,E)$. If $U$ is the manifold $M$, we also write $\Gamma(E)$ instead of $\Gamma(M,E)$.

The set $\Gamma(U,E)$ of $C^\infty$ sections of $E$ over $U$ is clearly a vector space over $\mathbb{R}$. It is in fact a module over the ring $C^\infty(U)$ of $C^\infty$ functions, for if $f$ is a $C^\infty$ function over $U$ and $s$ is a $C^\infty$ section of $E$ over $U$, then the definition $$(fs)(p):= f(p)s(p)\in E_p,\quad p\in U,$$ makes $fs$ into a $C^\infty$ section of $E$ over $U$.

Definition 7.12. A bundle map $\varphi:E\rightarrow F$ over a manifold $M$ (meaning that the base map is the identity $1_M$) induces a map on the space of sections: $$\varphi_\#: \Gamma(E)\rightarrow \Gamma(F),\\ \varphi_\#(s)=\varphi\circ s.$$ This induced map $\varphi_\#$ is $\mathcal{F}$-linear because if $f\in \mathcal{F}$, then $$\left( \varphi_\#(fs)\right) (p)=\left( \varphi\circ (fs)\right) (p)=\varphi \left( f(p)s(p)\right)=f(p)\varphi \left(s(p)\right)=\left( f\varphi_\#(s)\right) (p).$$ 

Our goal in the rest of this chapter is to prove that conversely, every $\mathcal{F}$-linear map $\alpha:\Gamma(E)\rightarrow \Gamma(F)$ comes from a bundle map $\varphi:E\rightarrow F$, i.e., $\alpha=\varphi_\#$.

7.3 Extending a Local Section to a Global Section

Proposition 7.13 (Existence of Global Section). Let $E\rightarrow M$ be a $C^\infty$ vector bundle, $s$ a $C^\infty$ section of $E$ over some open set $U$ in $M$, and $p$ a point in $U$. Then there exists a $C^\infty$ global section $\bar{s}\in \Gamma(M,E)$ that agrees with $s$ over some neighborhood of $p$.

Proof. Choose a $C^\infty$ bump function $f$ on $M$ such that $f\equiv 1$ on a neighborhood $W$ of $p$ contained in $U$ and $\mbox{supp} f\subset U$ (Figure 7.3). Define $\bar{s}:M\rightarrow E$ by $$\bar{s}(q)=\begin{cases}f(q)s(q)& \mbox{for }q\in U,\\ 0 & \mbox{for }q\notin U.\end{cases}$$

On $U$ the section $\bar{s}$ is clearly $C^\infty$ for it is the product of a $C^\infty$ function $f$ and a $C^\infty$ section $s$.

If $p\notin U$, then $p\notin \mbox{supp} f$. Since $\mbox{supp} f$ is a closed set, there is a neighborhood $V$ of $p$ contained in its complement $M\backslash \mbox{supp} f$. On $V$ the section $\bar{s}$ is identically zero. Hence, $\bar{s}$ is $C^\infty$ at $p$. This proves that $\bar{s}$ is $C^\infty$ on $M$.

On $W$, since $f\equiv 1$, the section $\bar{s}$ agrees with $s$.

7.4 Local Operators

In this section. $E$ and $F$ are $C^\infty$ vector bundles over a manifold $M$, and $\mathcal{F}$ is the ring $C^\infty(M)$ of $ C^\infty$ functions on $M$.

Definition 7.14. Let $E$ and $F$ be vector bundles over a manifold $M$. An $\mathbb{R}$-linear map $\alpha:\Gamma(E)\rightarrow \Gamma(F)$ is a local operator if whenever a section $s\in \Gamma(E)$ vanishes on an open set $U$ in $M$, then $\alpha(s)\in \Gamma(F)$ also vanishes on $U$. It is a point operator if whenever a section $s\in \Gamma(F)$ vanishes at a point $p$ in $M$, then $\alpha(s)\in \Gamma(F)$ also vanishes at $p$.

Proposition 7.17 ($\mathcal{F}$-linear is Local Operator). Let $E$ and $F$ be $C^\infty$ vector bundles over a manifold $M$, and $\mathcal{F}=C^\infty(M)$. If a map $\alpha:\Gamma(E)\rightarrow \Gamma(F)$ is $\mathcal{F}$-linear, then it is a local operator.

Proof. Suppose the section $s\in \Gamma(E)$ vanishes on the open set $U$. Let $p\in U$ and let $f$ be a $C^\infty$ bump function such that $f(p)=1$ and $\mbox{supp} f\subset U$ (Figure 7.3). Then $fs\in \Gamma(E)$ and $fs\equiv 0$ on $M$ (FIgure 7.4). So $\alpha(fs)\equiv 0$. By $\mathcal{F}$-linearity, $$0=\alpha(fs)=f\alpha(s).$$ Evaluating at $p$ gives $\alpha(s)(p)=0$. Since $p$ is an arbitrary point of $U$, $\alpha(s)\equiv 0$ on $U$.

7.5 Restriction of a Local Operator to an Open Subset

Theorem 7.20 (Existence of Local Section Map). Let $E$ and $F$ be vector bundles over a manifold $M$. If $\alpha:\Gamma(E)\rightarrow \Gamma(F)$ is a local operator, then for each open subset $U$ of $M$ there is a unique linear map, called the restriction of $\alpha$ of $\alpha$ to $U$, $$\alpha_U:\Gamma(U,E)\rightarrow \Gamma(U,F)$$ such that for any global section $t\in \Gamma(E)$, $$\begin{align}\alpha_U(t|_U)=\alpha(t)|_U.\end{align}$$

Proof. Let $s\in \Gamma(U,E)$ and $p\in U$. By Proposition 7.13, there exists a global section $\bar{s}$ of $E$ that agrees with $s$ in some neighborhood $W$ of $p$ in $U$. We define $\alpha_U(s)(p)$ using (1): $$\alpha_U(s)(p)=\alpha(\bar{s})(p).$$ Suppose $\tilde{s}\in \Gamma(E)$ is another global section that agrees with $s$ in the neighborhood $W$ of $p$. Then $\bar{s}=\tilde{s}$ in $W$. Since $\alpha$ is a local operator, $\alpha(\bar{s})=\alpha(\tilde{s})$ on $W$. Hence, $\alpha(\bar{s})(p)=\alpha(\tilde{s})(p)$. This shows that $\alpha_U(s)(p)$ is independent of the choice of $\bar{s}$, so $\alpha_U$ is well defined and unique. Fix $p\in U$. If $s\in \Gamma(U,E)$ and $\bar{s}\in \Gamma(M,E)$ agree on a neighborhood $W$ of $p$, then $\alpha_U(s)=\alpha(\bar{s})$ on $W$. Hence, $\alpha_U(s)$ is $C^\infty$ as a section of $F$.

If $t\in \Gamma(M,E)$ is a global section, then it is a global extension of its restriction $t|_U$. Hence, $$\alpha_U(t|_U)(p)=\alpha(t)(p)$$ for all $p\in U$. This proves that $\alpha_U(t|_U)=\alpha(t)|_U$.

Proposition 7.21 ($\mathcal{F}$-linear become Local). Let $E$ and $F$ be $C^\infty$ vector bundles over a manifold $M$, let $U$ be an open subset of $M$, let $\mathcal{F}(U)=C^\infty(U)$, the ring of $C^\infty$ functions on $U$. If $\alpha:\Gamma(E)\rightarrow \Gamma(F)$ is $\mathcal{F}$-linear, then the restriction $\alpha_U:\Gamma(U,E)\rightarrow \Gamma(U,F)$ is $\mathcal{F}(U)$-linear.

Proof. Let $s\in \Gamma(U,E)$ and $f\in \mathcal{F}(U)$. Fix $p\in U$ and let $\bar{s}$ and $\bar{f}$ be global extensions of $s$ and $f$ that agree with $s$ and $f$, respectivelty, on a neighborhood of $p$ (Proposition 7.13). Then $$\alpha_U(fs)(p)=\alpha(\bar{f}\bar{s})(p)=\bar{f}(p)\alpha(\bar{s})(p)=f(p)\alpha_U(s)(p).$$ (Use definition of $\alpha_U$, $\mathcal{F}$-linearity of $\alpha$)

Since $p$ is an arbitrary point of $U$, $$\alpha_U(fs)=f\alpha_U(s),$$ proving that $\alpha_U$ is $\mathcal{F}(U)$-linear.

7.6 Frames

A frame for a vector bundle $E$ of rank $r$ over an open set $U$ is a collection of sections $e_1,\cdots,e_r$ of $E$ over $U$ such that at each point $p\in U$, the element $e_1(p),\cdots,e_r(p)$ form a basis for the fiber $E_p$.

Proposition 7.22 (Frame on Trivial Bundle). A $C^\infty$ vector bundle $\pi:E\rightarrow M$ is trivial if and only if it has a $C^\infty$ frame.

Proof. Suppose $E$ is trivial, with $C^\infty$ trivialization $$\phi:E\rightarrow M\times \mathbb{R}^r.$$ Let $v_1,\cdots, v_r$ be the standard basis for $\mathbb{R}^r$. Then the elements $(p,v_i)$, $i=1,\cdots,r$, form a basis for $\{p\}\times \mathbb{R}^r$ for each $p\in M$, and so the sections of $E$ $$e_i(p)=\phi^{-1}(p,v_i),\quad i=1,\cdots,r,$$ provide a basis for $E_p$ at each point $p\in M$.

Conversely, suppose $e_1,\cdots,e_r$ is a frame for $E\rightarrow M$. Then every point $e\in E$ is a linear combination $e=\sum a^ie_i$. The map $$\phi(e)=(\pi(e),a^1,\cdots,a^r):E\rightarrow M\times \mathbb{R}^r$$ is a bundle map with inverse $$\psi:M\times \mathbb{R}^r\rightarrow E,\\ \psi(p,a^1,\cdots,a^r)=\sum a^i(p)e_i(p).$$

It follows from this proposition that over any trivializing open set $U$ of a vector bundle $E$m there is always a frame.

7.7 $\mathcal{F}$-Linearity and Bundle Maps

Throughout this subsection, $E$ and $F$ are $C^\infty$ vector bundles over a manifold $M$, and $\mathcal{F}=C^\infty(M)$ is the ring of $C^\infty$ real-valued functions on $M$. We will show that an $\mathcal{F}$-linear map $\alpha:\Gamma(E)\rightarrow \Gamma(F)$ can be defined pointwise and therefore corresponds uniquely to a bundle map $E\rightarrow F$.

Lemma 7.23 ($\mathcal{F}$-linear Section Map is Point Operator). An $\mathcal{F}$-linear map $\alpha:\Gamma(E)\rightarrow \Gamma(F)$ is a point operator.

Proof. We need to show that if $s\in \Gamma(E)$ vanishes at a point $p$ in $M$, then $\alpha(s)\in \Gamma(F)$ also vanishes at $p$. Let $U$ be an open neighborhood of $p$ over which $E$ is trivial. Thus, over $U$ it is possible to find a frame $e_1,\cdots,e_r$ for $E$. We write $$s|_U=\sum a^ie_i,\quad a^i\in C^\infty(U)=\mathcal{F}(U).$$ Because $s$ vanishes at $p$, all $a^i(p)=0$. Since $\alpha$ is $\mathcal{F}$-linear, it is a local operator (Proposition 7.17) and by Theorem 7.20 its restriction $\alpha_U:\Gamma(U,E)\rightarrow \Gamma(U,F)$ is defined. Then $$\alpha(s)(p)=\alpha_U(s|_U)(p)=\alpha_U\left( \sum a^ie_i \right) (p)=\left( \sum a^i\alpha_U(e_i)\right) (p) = \sum a^i(p)\alpha_U(e_i)(p)=0.$$

Lemma 7.24 (Fibre-Preserving Map and Section Map). Let $E$ and $F$ be $C^\infty$ vector bundles over a manifold $M$. A fiber-preserving map $\varphi:E\rightarrow F$ that is linear on each fiber is $C^\infty$ if and only if $\varphi_\#$ takes $C^\infty$ sections of $E$ to $C^\infty$ sections of $F$.

Proof. ($\Rightarrow$) If $\varphi$ is $C^\infty$, then $\varphi_\#(s)=\varphi\circ s$ clearly takes a $C^\infty$ section $s$ of $E$ to a $C^\infty$ section of $F$.

($\Leftarrow$) Fix $p\in M$ and let $(U,x^1,\cdots,x^n)$ be a chart about $p$ over which $E$ and $F$ are both trivial. Let $e_1,\cdots,e_r\in \Gamma(E)$ be a frame for $E$ over $U$. Likewise, let $f_1,\cdots,f_m\in \Gamma(F)$ be a frame for $F$ over $U$. A point of $E|_U$ can be written as a unique linear combination $\sum a^je_j$. Suppose $$\varphi\circ e_j=\sum_i b^i_jf_i.$$ In this expression the $b^i_j$'s are $C^\infty$ functions on $U$, because by hypothesis $\varphi\circ e_j=\varphi_\#(e_j)$ is a $C^\infty$ section of $F$. Then $$\varphi\circ \left( \sum_j a^j e_j\right) = \sum_{i,j}a^jb^i_jf_i.$$ One can take local  coordinates on $E|_U$ to be $(x^1,\cdots,x^n,a^1,\cdots,a^r)$. In terms of these local coordinates, $$\varphi(x^1,\cdots,x^n,a^1,\cdots,a^r)=\left( x^1,\cdots,x^n,\sum_j a^jb^1_j,\cdots,\sum_j a^jb^m_j\right),$$ which is a $C^\infty$ map.

Proposition 7.25 (Existence and Uniqueness of Linear Bundle Map). If $\alpha: \Gamma(E)\rightarrow \Gamma(F)$ is $\mathcal{F}$-linear, then for each $p\in M$, there is a unique linear map $\varphi_p:E_p\rightarrow F_p$ such that for all $s\in \Gamma(E)$, $$\varphi_p\left( s(p)\right)=\alpha(s)(p).$$

Proof. Given $e\in E_p$, to define $\varphi_p(e)$, choose any section $s\in \Gamma(E)$ such that $s(p)=e$ and define $$\varphi_p(e)=\alpha(s)(p)\in F_p.$$ This definition is independent of the choice of the section $s$, because if $s'$ is another section of $E$ with $s'(p)=e$, then $(s-s')(p)=0$ and so by Lemma 7.23, we have $\alpha(s-s')(p)=0$, i.e., $$\alpha(s)(p)=\alpha(s')(p).$$

Let us show that $\varphi_p:E_p\rightarrow F_p$ is linear. Suppose $e_1,e_2\in E_p$ and $a_1,a_2\in \mathbb{R}$. Let $s_1,s_2$ be global sections of $E$ such that $s_i(p)=e_i$. Then $(a_1s_1+a_2s_2)(p)=a_1e_1+a_2e_2$, so $$\varphi_p(a_1e_1+a_2e_2)=\alpha(a_1s_2+a_2s_2)(p)=a_1\alpha(s_1)(p)+a_2\alpha(s_2)(p)=a_1\varphi_p(e_1)+a_2\varphi_p(e_2).$$

Theorem 7.26 (Bundle Map and $\mathcal{F}$-linear Section Map). There is a one-to-one correspondence $$\{\mbox{bundle maps }\varphi:E\rightarrow F\}\leftrightarrow \{ \mathcal{F}\mbox{-linear maps }\alpha:\Gamma(E)\rightarrow \Gamma(F)\},\\ \varphi\mapsto \varphi_\#.$$

Proof. We first show surjectivity. Suppose $\alpha: \Gamma(E)\rightarrow \Gamma(F)$ is $\mathcal{F}$-linear. By the preceding proposition, for each $p\in M$ there is a linear map $\varphi_p:E_p\rightarrow F_p$ such that for any $s\in \Gamma(E)$, $$\varphi_p(s(p))=\alpha(s)(p).$$ Define $\varphi:E\rightarrow F$ by $\varphi(e)=\varphi_p(e)$ if $e\in E_p$.

For any $s\in \Gamma(E)$ and for every $p\in M$, $$\left( \varphi_\#(s)\right)(p)=\varphi \left( s(p)\right)=\alpha(s)(p),$$ which shows that $\alpha=\varphi_\#$. Since $\varphi_\#$ takes $C^\infty$ sections of $E$ to $C^\infty$ sections of $F$, by Lemma 7.24 the map $\varphi:E\rightarrow F$ is $C^\infty$. Thus, $\varphi$ is a bundle map.

Next we prove the injectivity of the correspondence. Suppose $\varphi,\psi:E\rightarrow F$ are two bundel maps such that $\varphi_\#=\psi_\#:\Gamma(E)\rightarrow \Gamma(F)$. For any $e\in E_p$, choose a section $s\in \Gamma(E)$ such that $s(p)=e$. Then $$\varphi(e)=\varphi\left( s(p)\right)=\left( \varphi_\#(s)\right) (p)=\left( \psi_\#(s)\right) (p)=(\psi\circ s)(p)=\psi(e).$$ Hence, $\varphi=\psi$.

Corollary 7.27 (Defining of $1$-Form). An $\mathcal{F}$-linear map $\omega:\mathfrak{X}(M)\rightarrow C^\infty(M)$ is a $C^\infty$ $1$-form on $M$.

Proof. By proposition 7.25, one can define for each $p\in M$, a linear map $\omega_p:T_pM\rightarrow \mathbb{R}$ such that for all $X\in \mathfrak{X}(M)$, $$\omega_p(X_p)=\omega(X)(p).$$ This shows that $\omega$ is a $1$-form on $M$.

For every $C^\infty$ vector field $X$ on $M$, $\omega(X)$ is a $C^\infty$ function on $M$. This shows that as a $1$-form, $\omega$ is $C^\infty$.

7.8 Multilinear Maps over Smooth Functions

By Proposition 7.25, if $\alpha:\Gamma(E)\rightarrow \Gamma(F)$ is an $\mathcal{F}$-linear map of sections of vector bundles over $M$, then at each $p\in M$, it is possible to define a linear map $\varphi_p:E_p\rightarrow F_p$ such that for any $s\in \Gamma(E)$, $$\varphi_p(s(p))=\alpha(s)(p).$$ This can be generalized to $\mathcal{F}$-multilinear maps.

Proposition 7.28 (Multi-Section Map). Let $E,E',F$ be vector bundles over a manifold $M$. If $$\alpha:\Gamma(E)\times \Gamma(E')\rightarrow \Gamma(F)$$ is $\mathcal{F}$-bilinear, then for each $p\in M$ there is a unique $\mathbb{R}$-bilinear map $$\varphi_p:E_p\times E'_p\rightarrow F_p$$ such that for all $s\in \Gamma(E)$ and $s'\in \Gamma(E')$, $$\varphi_p(s(p),s'(p))=(\alpha(s,s'))(p).$$

Since the proof is similar to that of Proposition 7.25.

Of course, Proposition 7.28 generalizes to $\mathcal{F}$-linear maps with $k$ argyments. Just as in Corollary 7.27, we conclude that if an alternating map $$\omega:\mathfrak{X}(M)\times \cdots\times \mathfrak{X}(M)(k\mbox{ times})\rightarrow C^\infty(M)$$ is $\mathcal{F}$-linear in each argument, then $\omega$ induces a $k$-form $\tilde{\omega}$ on $M$ such that for $X_1,\cdots,X_k\in \mathfrak{X}(M)$, $$\tilde{\omega}_p(X_{1,p},\cdots,X_{k,p})=(\omega(X_1,\cdots,X_k))(p).$$ It is customary to write the $k$-form $\tilde{\omega}$ as $\omega$.