Periods and primitives on a spherical exterior

Set $S_{t,r}=i(S^2)$. If $z:U\to V\subset {\mathbb{R}}^2$ is a sphere chart and $I,J$ are open intervals about $t,r$ with $J\subset (1,\mathrm{\infty })$, the coordinates $(z(x),u-t,v-r)$ on $\mathrm{\Phi }(I\times J\times U)$ are smooth charts of $E$. They identify its intersection with $S_{t,r}$ with $V\times \{0\}\times \{0\}$. Thus $S_{t,r}$ is an embedded smooth surface with the subspace topology. In these charts $i$ is $z\mapsto (z,0,0)$, and its inverse on the image is the restriction of $p$; both are smooth, so $i:S^2\to S_{t,r}$ is a diffeomorphism. Give $S^2$ the outward orientation, in which $(v,w)$ is positive when $\det (x,v,w)>0$, and transport it by $i$ to $S_{t,r}$. This is a smooth orientation since the displayed determinant varies smoothly and never vanishes on a tangent basis. For a two-form $\alpha$ on $E$, its integral over $S_{t,r}$ is the integral of its pullback by the inclusion $S_{t,r}\hookrightarrow E$. The orientation-preserving diffeomorphism $i$ and [F4] identify that integral with ${\int }_{S^2}{i}^{\ast }\alpha$.

We have $p\circ i={\mathrm{id}}_{S^2}$. A smooth homotopy from ${\mathrm{id}}_E$ to $i\circ p$ is obtained in product coordinates by $(u,v,x)\mapsto ((1-s)u+st,(1-s)v+sr,x)$ for $0\le s\le 1$; the second coordinate remains greater than $1$. Hence $i$ and $p$ are smooth homotopy inverses. Pullback by $i$ is therefore an isomorphism in every de Rham degree. In degree one, [F2] and [F1] imply that every closed real one-form on $E$ is exact.

On the outward-oriented unit sphere define $\sigma =x_{1}d{x}_2\wedge d{x}_3-x_{2}d{x}_1\wedge d{x}_3+x_{3}d{x}_1\wedge d{x}_2$, restricted to $S^2$. Its value on a tangent basis $(v,w)$ is $\det (x,v,w)$, so it is a positive smooth top form. It is closed because every three-form on the two-dimensional sphere vanishes. By [F4], ${\int }_{S^2}\sigma >0$ and an exact two-form has zero integral. Thus $[\sigma ]$ is nonzero and, by the one-dimensionality in [F2], generates $H_{\mathrm{d}\mathrm{R}}^2(S^2)$. For any closed real two-form $\beta$ on $S^2$, write $[\beta ]=c[\sigma ]$; then $\beta -c\sigma$ is exact, and linearity and [F4] give $\int \beta =c\int \sigma$. Consequently $\beta$ is exact exactly when its integral vanishes.

Let $\alpha$ be a closed real two-form on $E$. Its pullback $i^{\ast }\alpha$ is closed, and the integral over the specified oriented sphere is ${\int }_{S^2}{i}^{\ast }\alpha$. By step 2.2 this integral vanishes exactly when $i^{\ast }[\alpha ]=0$; the isomorphism in step 2.1 makes that equivalent to $[\alpha ]=0$, hence to exactness of $\alpha$. The argument works for every choice of $t$ and $r>1$, so a vanishing period on any one such sphere suffices.

For complex forms, write the form as its real part plus $i$ times its imaginary part. Exterior differentiation and integration act componentwise, and a complex form is exact precisely when both real components are exact: combine their real primitives into a complex primitive. Applying the real one-form result and the real two-form criterion to both components proves the stated complex cases.