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A smooth real vector bundle of vector bundle rank $r$ consists of a smooth manifold $E$, a smooth projection $\pi :E\to M$, real vector space structures on its fibers, and local vector bundle trivializationsthat commute with projection to $U$ and are linear isomorphisms on each fiber. On overlaps the change of trivialization is $(p,v)\mapsto (p,g_{VU}(p)v)$, with $g_{VU}:U\cap V\to G{L}_r(\mathbb{R})$ smooth into the general linear group. The usual smooth manifold convention is Hausdorff and second countable.

For an $n$-dimensional $M$, take the disjoint union of its tangent spaces. A manifold chart $x:U\to {\mathbb{R}}^n$ suppliesIf $y$ is another chart, the resulting change of coordinates on the total space isThis is smooth, its inverse is the same construction with $x,y$ exchanged, and the fiber map is invertible and linear. Thus these charts define a smooth structure with the required bundle trivializations.

For completeness, the topology so constructed is Hausdorff: different base points can be separated downstairs, while vectors over one point can be separated inside one product chart. A countable base atlas for $M$ and countable bases for its products with ${\mathbb{R}}^n$ give second countability. Hence the total space is genuinely a smooth manifold, not just a collection of fibers. We obtain

The fibers must retain their base labels. In particular, a literal untagged union of fibers in $E$ is not sufficient when $f$ is not injective. Use the pullback vector bundleThis makes precise the indexed-union notation in the PDF. For a vector bundle trivialization ${\Phi }_U(e)=(\pi (e),v)$, defineIt is a fiberwise linear isomorphism, with inverse $(p,v)\mapsto (p,{\Phi }_U^{-1}(f(p),v))$. The transition functions are $g_{VU}\circ f$, hence smooth. They satisfy the same cocycle identities as those of $E$ and give a smooth total space of manifold dimension $\dim N+r$.

One may also see that this is an embedded submanifold of $N\times E$: in a bundle chart the constraint is the graph of the smooth map $f$ in the base coordinates, leaving the $r$ fiber coordinates free. Thus the chart construction agrees with its natural subspace smooth structure. No immersion or surjectivity hypothesis on $f$ is required. Therefore

A vector bundle morphism over $M$ is a smooth map $F:E\to E^{\prime }$ with ${\pi }^{\prime }\circ F=\pi$ whose restriction $E_p\to E_p^{\prime }$ is linear for every $p$. A vector bundle isomorphism is such a map with a smooth bundle-morphism inverse. Equivalently, a fiberwise bijective smooth bundle morphism is an isomorphism: in local vector bundle trivializations it is multiplication by an invertible smooth matrix, and its inverse matrix is smooth.

For a diffeomorphism $f:N\to M$, defineThe derivative $d{f}_p$ is a linear isomorphism by the chain rule, since $d(f^{-1}{)}_{f(p)}\circ d{f}_p=\mathrm{id}$. In manifold charts, $F$ is represented by the smooth Jacobian matrix of $f$, so it is a smooth bundle morphism. Its inverse iswhich is also smooth and fiberwise linear. ConsequentlyThis identifies the bundles over the same base $N$; the tangent map by itself is a map from $TN$ to $TM$ covering $f$.

Use the PDF's covector-first convention: its space ${\mathcal{T}}_l^k$ consists of tensor fields inThis reverses the order in which some texts list tensor type. For a finite-dimensional real vector space $V$, define the tensor contraction first on decomposable tensors:The hats mean omission, with the other factors left in their original order. The formula is a multilinear map of its individual factors, so the universal property of a tensor product gives a unique linear map with this formula.

Apply it with $V=T_{p}M$ at every $p$. Evaluation of a covector on a vector is basis independent: under a change of frame, one factor transforms by a matrix and the other by its inverse transpose, and the matrices cancel in the pairing. Thus the fiber maps agree on overlapping vector bundle trivializations. In a local dual basis, the coefficients of the contracted tensor are finite sums of coefficients with the selected covariant and contravariant indices set equal. They remain smooth. Henceis a globally defined smooth contraction for $k,l\ge 1$. It is also linear over $C^{\infty }(M)$.

The rank of a vector bundle at a point is the dimension of its fiber as a vector space. A vector bundle trivialization identifies nearby fibers with the same vector space, so this rank is locally constant. A rank-$r$ bundle has fibers of dimension $r$ everywhere.

A vector bundle morphism is an isomorphism if it admits an inverse of the same kind. A smooth fiberwise bijective bundle morphism automatically has a smooth inverse, because inverse matrices in vector bundle trivializations depend smoothly on their entries wherever their determinants are nonzero.

A morphism of vector bundles over one base is a smooth map between their total spaces that commutes with projection to the base and is linear on each fiber. In local vector bundle trivializations it is multiplication by a smoothly varying matrix.