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Two closed symplectic forms agreeing as ambient bilinear forms along a compact embedded submanifold are equivalent near it by a diffeomorphism fixing it pointwise. Shrink until their affine interpolation is nondegenerate. A relative homotopy primitive $\eta$ for their difference vanishes as an ambient covector on the submanifold. Solving ${\iota }_{Z_t}{\omega }_t=-\eta$ gives a vector field vanishing there; its local flow fixes the submanifold and makes ${\omega }_t$ constant under pullback.

A diffeomorphism of the entire cotangent bundle preserving the Liouville one-form is the cotangent lift of a diffeomorphism of the base. It preserves the zero section and the Liouville vector field, hence commutes with the cotangent fiber-dilation flow. The limit as the fiber contracts forces it to cover its restriction to the zero section. Evaluation of the preserved one-form then forces the inverse-transpose derivative action on each covector.

A smooth isotopy is a smooth family of smooth embeddings or diffeomorphisms parametrized by an interval and beginning at the given map.

Let $Y$ be a embedded submanifold of a complex manifold. If its tangent spaces are invariant under the almost complex structure induced by a complex atlas, choose ambient holomorphic coordinates making its tangent space at one point ${\mathbb{C}}^k\times 0$. The first $k$ coordinates restrict to a local real diffeomorphism by the inverse function theorem, so $Y$ is locally the graph of a smooth function $F$. Invariance of the tangent graph gives $dF(iv)=idF(v)$, the Cauchy-Riemann equations. Thus $F$ is holomorphic and these graph coordinates give the induced complex submanifold structure. This proves the result directly, without needing the general integrability theorem as an unproved step.

If $x=\chi (x^{\prime })$ is an angular diffeomorphism and $\eta \cdot dx$ is a closed differential one-form, the map $y=\eta (\chi (x^{\prime }))+D\chi (x^{\prime }{)}^{-T}{y}^{\prime }$ is a symplectomorphism. It combines a cotangent lift of a diffeomorphism with translation by a closed differential one-form. In particular $\eta =du+a$ is allowed on a torus even when the constant one-form $a\cdot dx$ is not exact.

A symplectomorphism $\varphi :(M,\omega )\to (M^{\prime },{\omega }^{\prime })$ is a diffeomorphism satisfying ${\varphi }^{\ast }{\omega }^{\prime }=\omega$.

Every closed topological manifold with the homotopy type of $S^d$ is homeomorphic to $S^d$. This is a classification theorem about homeomorphisms, and does not assert a diffeomorphism between prescribed smooth structures. It is the precise final step needed after proving that a cone link is a simply connected integral homology sphere in the manifold criterion for a suspension.

For a sufficiently small analytic periodic vector field $w$, a near-identity angular diffeomorphism $\chi$ and a constant vector $a$ solve $D\chi \omega =\omega +w\circ \chi +a$. They satisfy $\chi -\mathrm{id}=O(w)$, $a=O(w)$; if $⟨w⟩=0$, then $a=O(w^2)$. For the defect $e={\mathcal{D}}_{\omega }\chi -\omega -w\circ \chi -a$, set $A=D\chi$, choose $\Delta a=⟨A^{-1}{⟩}^{-1}⟨A^{-1}e⟩$, and solve ${\mathcal{D}}_{\omega }v=A^{-1}(\Delta a-e)$ with zero mean. Updating $\chi$ by $Av$ leaves a quadratic defect $(De)v-[w(\chi +Av)-w(\chi )-Dw(\chi )Av]$. The torus small-divisor estimate and geometrically decreasing strip losses yield a convergent analytic iteration.

A smooth embedded submanifold has a neighborhood diffeomorphic to a neighborhood of the zero section of its normal bundle, with the zero section identified with the submanifold. One may prescribe the normal-bundle splitting of the derivative along that section. Choose a Riemannian metric making the splitting orthogonal and use its normal exponential map: its derivative there is the required isomorphism, so the inverse function theorem supplies local diffeomorphisms. A sufficiently small, possibly varying, fibre radius makes these fit into a tubular neighborhood.

Write each sphere as two disks, $S^2=D_{-}^2\cup D_{+}^2$. In their product, $D_{-}^2\times D_{-}^2$ is a zero-handle; $D_{+}^2\times D_{-}^2$ and $D_{-}^2\times D_{+}^2$ are two-handles. Their attaching circles are the core circles of the complementary solid tori in $\partial (D^2\times D^2)$, hence a Hopf link. Constant normal directions from the disk factors give both components the zero Seifert framing. The remaining $D_{+}^2\times D_{+}^2$ is a four-handle. This product construction establishes the actual diffeomorphism type, rather than relying on its intersection form alone. The resulting intersection matrix is $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$.