Smooth map between manifolds

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

A map between smooth manifolds is smooth when its coordinate expressions in compatible charts have derivatives of every order. The condition is invariant under smooth chart changes and expresses smoothness in charts.

A continuous map from a smooth manifold to ${\mathbb{R}}^d$ can be approximated by a smooth map with any prescribed positive continuous pointwise error bound. In coordinate charts, smoothing by local convolution and then combining with a smooth partition of unity gives the approximation. On a compact subset of an open chart domain, a smooth cutoff allows the resulting straight-line homotopy to be supported in that domain. This permits the Sard theorem to be used after approximation; the original continuous map can be space-filling and need not omit a point.

A smooth map $f:M\to N$ is transverse to an embedded submanifold $S\subset N$ whenThis is a sum of vector subspaces, not necessarily a direct sum. Equivalently the composition of $d{f}_p$ with the quotient to $T_{f(p)}N/T_{f(p)}S$ is surjective. In a slice chart for an embedded submanifold of codimension $k$, the $k$ normal coordinate functions composed with $f$ have surjective derivative on their common zero set. If the preimage is empty the condition holds vacuously.

Smooth maps of finitely many circles into a manifold of dimension at least three can be perturbed within their homotopy classes to disjoint embeddings. The transversality dimension count for two curve branches is $1+1-m<0$, so their images avoid one another after perturbation; one also removes self-intersections and obtains immersions by the jet version of transversality. This allows disjoint representative loops for relators in high-dimensional surgery constructions of prescribed fundamental groups.

If $f:M\to N$ satisfies transversality of a map to a submanifold $S$, its nonempty preimage is an embedded submanifold of the displayed codimension. Choose a slice chart for an embedded submanifold around $f(p)$ in which $S$ is the zero set of the first $k$ coordinates. Their composition with $f$ has surjective derivative on its zero set, so the regular level set theorem gives a local embedded slice of dimension $\dim M-k$. These slices give the subspace topology and a compatible smooth atlas on the preimage. Its tangent space is

A pushforward of a curve composes a parametrized curve in the source with a smooth map between manifolds. Its tangent is the image of the original tangent under the differential of a smooth map. A constant image curve has zero tangent, so a general smooth map need not preserve regularity of curves.

A pullback of a smooth function composes a function on the target with a smooth map between manifolds, producing a function on the source. It needs no inverse map. Acting on the pulled-back function defines the differential of a smooth map intrinsically as a map of derivations.

For a smooth map between manifolds $f:M\to N$, the differential $d{f}_x:T_{x}M\to T_{f(x)}N$ is the linear map obtained by differentiating in charts. Intrinsically it sends the tangent vector of a curve $\gamma$ through $x$ to the tangent vector of $f\circ \gamma$. The chain rule makes it independent of charts and gives $d(g\circ f{)}_x=d{g}_{f(x)}\circ d{f}_x$.

For a real-valued smooth function on a smooth manifold, its differential of a smooth map is a differential one-form: $d{f}_p$ maps a tangent vector to the directional derivative of the function. In local coordinates, $df={\sum }_i({\partial }_{i}f)d{x}^i$. The exterior derivative extends this operation from functions to higher-degree differential forms.

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.

The pushforward of a contravariant tensor applies the differential of a smooth map to each vector factor of a type $(r,0)$ tensor. It is defined pointwise at a specified source point. A field generally gives a field along the map, not necessarily a unique tensor field on the image when several points have the same image. A diffeomorphism allows the usual transport of arbitrary mixed tensors through its inverse as well.

The pullback of a covariant tensor composes each of its vector arguments with the differential of a smooth map. It takes a type $(0,s)$ tensor on the target to one on the source. Antisymmetry is not required; the pullback of a differential form is the alternating special case. If the target vectors are projected onto the image tangent space first, evaluating the pullback is unchanged.

The pullback of a covector is the dual linear map to the differential of a smooth map. In coordinates its components are $({\varphi }^{\ast }\eta {)}_i=({\partial }_i{y}^{\alpha }){\eta }_{\alpha }$. The same rectangular Jacobian defines vector pushforward, with its other index contracted. No invertibility is needed.

For a vector field $X$ on $M$, its pushforward by a smooth map between manifolds $f:M\to N$ is the vector field along a map defined by $(f_{\ast }X{)}_p=d{f}_p(X_p)$. Acting on a function $h$ on $N$, it satisfies $(f_{\ast }X{)}_p(h)=X_p(h\circ f)$. For a diffeomorphism this defines a vector field on $N$ by evaluation at $p=f^{-1}(q)$. For a general map there need not be a single vector at each image point: $f(x)=x^2$, $X={\partial }_x$ gives $2x{\partial }_y$, with opposite values over $y>0$.

The chain rule shows that $\alpha {\varphi }_t{\alpha }^{-1}$ is the local flow of the pushforward of a vector field ${\alpha }_{\ast }X$. If this equals $X$, uniqueness of integral curves of a vector field gives commutation. Conversely, differentiate the commuting identity at $t=0$ to obtain $d{\alpha }_p{X}_p=X_{\alpha (p)}$. All identities hold on their common domains; the vector field need not be complete.

A vector field $X$ is projectable through a smooth map between manifolds $f:M\to N$ if there is a smooth vector field $Y$ on $N$ with $d{f}_p{X}_p=Y_{f(p)}$ for every $p$. Agreement on fibres is necessary. For a surjective submersion it is also sufficient: smooth local sections of the submersion express $Y$ locally as $df(X)$ and prove its smoothness. An arbitrary non-surjective map may instead give a field only along its image, with extension to $N$ requiring additional choices.