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A smooth atlas is a collection of manifold charts covering a topological manifold whose coordinate changes are smooth transition maps. It determines a smooth manifold structure by adjoining every compatible chart.

On a smooth manifold, the sequence of sheaves of abelian groupsis exact. The additive smooth-function sheaf is a fine sheaf, so the connecting map identifies $H^1(X,{\mathcal{C}}_{\mathbb{C}}^{\infty ,\ast })$ with $H^2(X,\mathbb{Z})$. The former classifies smooth complex line bundles and the connecting class is their First Chern class.

A map between smooth manifolds is smooth when its expression in every pair of coordinate charts is a smooth function.

A finite group acting freely by diffeomorphisms on a smooth manifold has a smooth quotient. Choose each coordinate neighbourhood disjoint from all its nontrivial translates and transport its chart through the quotient map. The resulting transition maps are restrictions of the original transition maps composed with group elements.

A smooth surface in ${\mathbb{R}}^3$ is a two-dimensional smooth embedded submanifold. Locally it is parametrized by two coordinates with linearly independent tangent vectors.

The spacetime dimension is the dimension of the smooth manifold used to model spacetime. Ordinary general relativity uses one temporal and three spatial dimensions.

The $n+1$ manifold charts $U_i=\{[X]:X_i\ne 0\}$ have coordinates $X_k/X_i$, with $k\ne i$. Their inverse inserts $1$ as the $i$th homogeneous coordinate. On an overlap, changing the denominator from $X_i$ to $X_j$ divides each remaining coordinate by the nonzero coordinate $X_j/X_i$. These rational smooth transition maps give Real projective space its standard smooth manifold structure.

A symplectic manifold is an even-dimensional smooth manifold $M$ equipped with a closed nondegenerate differential two-form $\omega$.

A symplectic form on a $2n$-dimensional smooth manifold determines the orientation in which the nowhere-zero volume form ${\omega }^n/n!$ is positive. Consequently every symplectic manifold is orientable. In dimension two this forbids a symplectic form on the Möbius strip.

The tangent bundle of a smooth manifold $M$ is the vector bundle whose fiber over $p$ is the tangent space $T_{p}M$.

The tangent space $T_{p}M$ is the vector space of tangent vectors to a smooth manifold $M$ at $p$. For an embedded manifold $M\subseteq {\mathbb{R}}^N$, it isFor a regular parametrized surface $X(u,v)$, it is spanned by $X_u$ and $X_v$.

On a compact connected oriented $n$-dimensional smooth manifold without boundary, integration is a linear isomorphism $H_{\mathrm{dR}}^n(M)\to \mathbb{R}$. Thus two top-degree differential forms have the same de Rham cohomology class exactly when their integrals agree.

For a compact connected oriented boundaryless $n$-dimensional smooth manifold, choose a Riemannian metric. Harmonic functions are constant because $⟨f,\Delta f⟩=\Vert df{\Vert }^2$, and Hodge star commutes with the Hodge Laplacian. Thus the Hodge star operator identifies the constant functions with the harmonic differential forms of top degree, exactly $\mathbb{R}{\omega }_g$. The Hodge decomposition theorem identifies these with the top de Rham cohomology. The metric volume form has nonzero class by Stokes theorem. The harmonic-representative theorem is stated in Denis Auroux's Hodge theory lecture.

If $D=\mathrm{diag}(a_1,\dots ,a_{n+1})$ is invertible, $x\mapsto Dx$ sends the ellipsoid ${\sum }_i{a}_i^2{x}_i^2=1$ to the unit sphere. On $u=(v,s)\in S^n$, the two stereographic projections $v/(1-s)$ and $v/(1+s)$ omit opposite poles and have inverses $(2w/(1+|w{|}^2),\pm (|w{|}^2-1)/(1+|w{|}^2))$, with the sign chosen for the appropriate pole. Their overlap transition is $w\mapsto w/|w{|}^2$, smooth and involutive on ${\mathbb{R}}^n\setminus \{0\}$. Composing their inverses with $D^{-1}$ gives a two-chart smooth atlas of the ellipsoid. Its subspace topology is Hausdorff and has a second-countable space topology, so it is a smooth manifold. For $n=0$ the overlap is empty and each chart is a point.

A smooth map between smooth manifolds with zero differential is locally constant: each target coordinate function has zero derivatives on a sufficiently small source coordinate ball. Integration along straight segments proves constancy there. Its fibers are open and closed, so a connected source makes the map constant globally.