The first obstruction in differential geometry is that a [vector space](/page/Vector%20Space) attached to a point should vary with the point, but the spaces over different points usually have no canonical way to be identified. A tangent vector at one point of a manifold is not naturally a tangent vector at another point. A field of tangent vectors therefore needs a structure that says what it means for these pointwise vector spaces to vary smoothly, without pretending that they are all the same vector space. The naive model is a product $M \times \mathbb{R}^r$, where every point $p \in M$ carries the copy $\{p\} \times \mathbb{R}^r$. This model is too rigid: the tangent bundle of a sphere, the normal bundle of an embedded submanifold, and the line bundle over a circle obtained by gluing with a sign change all behave locally like products but need not be products globally. Smooth vector bundles isolate exactly this local-product behavior. [example: The Tangent Bundle of the Circle] Let $S^1 \subset \mathbb{R}^2$ be the unit circle, and write $p=(\cos t,\sin t)$. The curve $c(t)=(\cos t,\sin t)$ has velocity \begin{align*} c'(t)=(-\sin t,\cos t). \end{align*} This vector is tangent to $S^1$ at $p$, and it is perpendicular to the radial vector $p$ because \begin{align*} (\cos t,\sin t)\cdot(-\sin t,\cos t)=-\cos t\sin t+\sin t\cos t=0. \end{align*} Thus the tangent line at $p$ is \begin{align*} T_pS^1=\{a(-\sin t,\cos t):a\in\mathbb{R}\}. \end{align*} The total space is \begin{align*} TS^1=\bigcup_{p\in S^1}\{p\}\times T_pS^1\subset S^1\times\mathbb{R}^2. \end{align*} Define \begin{align*} \Phi:S^1\times\mathbb{R}\to TS^1 \end{align*} by \begin{align*} \Phi((\cos t,\sin t),a)=((\cos t,\sin t),a(-\sin t,\cos t)). \end{align*} If $t$ is replaced by $t+2\pi k$, then \begin{align*} \cos(t+2\pi k)=\cos t \end{align*} and \begin{align*} \sin(t+2\pi k)=\sin t, \end{align*} so the formula gives the same point of $TS^1$. Hence $\Phi$ is well-defined. For each $p=(\cos t,\sin t)$, every vector in $T_pS^1$ has a unique form $a(-\sin t,\cos t)$, since $(-\sin t,\cos t)\neq(0,0)$. Therefore the inverse map is \begin{align*} \Phi^{-1}((p,a(-\sin t,\cos t)))=(p,a). \end{align*} So $TS^1$ is globally described as the product $S^1\times\mathbb{R}$. This example is special because the tangent bundle of the circle is globally a product; in general, a tangent bundle is only required to have such product descriptions locally. [/example] The example hides the essential rule: a [vector bundle](/page/Vector%20Bundle) is not just a family of vector spaces. It must include a total space, a projection to the base, and smooth local coordinates in which the projection looks like the projection from a product. The coordinate changes must preserve the vector-space structure in each fiber. ## Definition The definition must remember both the smooth geometry of the total space and the linear algebra inside each fiber. The rank records the dimension of the model fiber, while the projection records which vector space belongs to which base point. [definition: Smooth Vector Bundle] Let $M$ be a [smooth manifold](/page/Smooth%20Manifold). A smooth vector bundle of rank $r$ over $M$ is a smooth manifold $E$ of dimension $\dim M + r$, a smooth surjective submersion \begin{align*} \pi: E &\to M, \end{align*} a real vector space structure on each fiber $E_p = \pi^{-1}(\{p\})$, and for every $p \in M$ an open neighbourhood $U \subset M$ together with a diffeomorphism \begin{align*} \Phi: \pi^{-1}(U) &\to U \times \mathbb{R}^r \end{align*} such that $\operatorname{pr}_1 \circ \Phi = \pi$ on $\pi^{-1}(U)$ and, for every $q \in U$, the restricted map \begin{align*} \Phi_q: E_q &\to \{q\} \times \mathbb{R}^r \end{align*} is a linear isomorphism after identifying $\{q\} \times \mathbb{R}^r$ with $\mathbb{R}^r$. [/definition] The map $\pi: E \to M$ is the bundle projection, $M$ is the base, $E$ is the total space, and $E_p$ is the fiber over $p$. The local diffeomorphisms $\Phi$ are the data that convert geometric variation into ordinary smooth functions on open sets of $M$. ## Product Bundles and Local Models To separate local product behavior from global twisting, we first need the reference case where no twisting is present. The product bundle is the benchmark against which all twisting is measured: it fixes the model for local coordinates and gives the test case that later transition functions will modify. [definition: Product Smooth Vector Bundle] Let $M$ be a smooth manifold and let $V$ be a real vector space of dimension $r$. The product smooth vector bundle with fiber $V$ is the projection \begin{align*} \operatorname{pr}_1: M \times V &\to M \end{align*} with rule \begin{align*} (p,v) &\mapsto p, \end{align*} where each fiber $\{p\} \times V$ carries the vector space structure transported from $V$. [/definition] The local definition says that every smooth vector bundle is a product after restricting to small enough open sets. The global question is how those local products are glued. ## Local Trivialisations and Transition Functions ### Comparing Local Products A single local product chart does not describe a bundle over all of $M$. The information lives in how two such descriptions compare on an overlap. [definition: Local Trivialisation] Let $\pi: E \to M$ be a smooth vector bundle of rank $r$. A local trivialisation over an [open set](/page/Open%20Set) $U \subset M$ is a diffeomorphism \begin{align*} \Phi: \pi^{-1}(U) &\to U \times \mathbb{R}^r \end{align*} such that $\operatorname{pr}_1 \circ \Phi = \pi$ and each fiber map $\Phi_q: E_q \to \mathbb{R}^r$ is linear. [/definition] Suppose $\Phi_i$ is a trivialisation over $U_i$ and $\Phi_j$ is a trivialisation over $U_j$. On $U_i \cap U_j$, both charts assign coordinates in $\mathbb{R}^r$ to the same vector in the same fiber. Their comparison is therefore linear in the fiber coordinate and smooth in the base point, and the next definition records this comparison as a matrix-valued function. [definition: Transition Function of a Smooth Vector Bundle] Let $\Phi_i: \pi^{-1}(U_i) \to U_i \times \mathbb{R}^r$ and $\Phi_j: \pi^{-1}(U_j) \to U_j \times \mathbb{R}^r$ be local trivialisations of a smooth vector bundle $\pi: E \to M$. The transition function from $\Phi_i$ to $\Phi_j$ is the smooth map \begin{align*} g_{ji}: U_i \cap U_j &\to \operatorname{GL}_r(\mathbb{R}) \end{align*} defined by \begin{align*} \Phi_j \circ \Phi_i^{-1}(p,v) &= (p, g_{ji}(p)v). \end{align*} [/definition] ### Cocycle Compatibility The transition functions carry all of the non-product behavior. They are smooth matrix-valued functions, so the bundle is smooth precisely because the gluing matrices change smoothly with the base point. A collection of such matrices can glue local products only when all coordinate changes agree on triple overlaps. The compatibility condition is most naturally stated for a general smooth fibre bundle: a smooth map whose fibres are diffeomorphic to a fixed smooth manifold $F$ and whose local changes of fibre coordinate are smooth maps into $\operatorname{Diff}(F)$, the group of diffeomorphisms of $F$. In the vector bundle case here, $F=\mathbb{R}^r$, and the coordinate changes are the special diffeomorphisms given by invertible linear maps $g_{ji}(p)\in\operatorname{GL}_r(\mathbb{R})$. [quotetheorem:6084] The cocycle identities are the compatibility laws for gluing. They say that changing coordinates from one chart to another and then to a third gives the same result as changing directly to the third chart. [example: A Real Line Bundle on the Circle from Gluing] Cover $S^1$ by two connected open arcs $U_1$ and $U_2$ such that $U_1 \cap U_2$ has connected components $A$ and $B$. A rank-one bundle is obtained by gluing the two products $U_1 \times \mathbb{R}$ and $U_2 \times \mathbb{R}$ using a smooth transition function \begin{align*} g_{21}: U_1 \cap U_2 \to \mathbb{R}^{\times}. \end{align*} The gluing relation is \begin{align*} (p,a)_1 \sim (p,g_{21}(p)a)_2. \end{align*} If $g_{21}(p)=1$ on both $A$ and $B$, then the relation is $(p,a)_1\sim(p,a)_2$ on every overlap point, so the local fiber coordinate is unchanged when one passes from $U_1$ to $U_2$. The local functions $f_1=1$ on $U_1$ and $f_2=1$ on $U_2$ satisfy \begin{align*} f_2(p)=g_{21}(p)f_1(p)=1\cdot 1=1 \end{align*} on $U_1\cap U_2$, hence they glue to a nowhere-zero global section of the product line bundle. Now take $g_{21}(p)=1$ on $A$ and $g_{21}(p)=-1$ on $B$. A global section is represented by smooth functions $f_1:U_1\to\mathbb{R}$ and $f_2:U_2\to\mathbb{R}$ satisfying \begin{align*} f_2(p)=g_{21}(p)f_1(p) \end{align*} on $U_1\cap U_2$. Thus on $A$ we have \begin{align*} f_2(p)=f_1(p), \end{align*} while on $B$ we have \begin{align*} f_2(p)=-f_1(p). \end{align*} If such a section were nowhere zero, then $f_1$ and $f_2$ would be nowhere-zero continuous functions on the connected sets $U_1$ and $U_2$, so each would have constant sign. The equality on $A$ forces $f_1$ and $f_2$ to have the same sign, while the equality $f_2=-f_1$ on $B$ forces them to have opposite signs, which is impossible. Thus the sign-change gluing gives the Moebius line bundle: locally it is still a product over each arc, but going once around the circle changes a nonzero fiber coordinate $a$ into $-a$, so the global bundle is twisted. [/example] This example explains why the definition cannot only mention fibers. The fibers are all copies of $\mathbb{R}$, but the transition function records how a vector transported around the circle returns with reversed sign. ## Frames and Sections ### Sections as Fiberwise Choices Working with a bundle often means replacing a geometric vector by its component functions. A local frame is the bundle version of choosing a basis that varies smoothly from point to point. [definition: Smooth Section] Let $\pi: E \to M$ be a smooth vector bundle. A smooth section of $E$ is a smooth map \begin{align*} s: M &\to E \end{align*} such that $\pi \circ s = \operatorname{id}_M$. [/definition] ### Local Frames as Moving Bases A section assigns to each point $p \in M$ a vector $s(p) \in E_p$. The condition $\pi(s(p))=p$ is the formal way to say that the vector chosen by the section lies over the point where it is evaluated. To use sections as coordinates, we need enough of them to form a basis in every fiber over an open set. [definition: Local Frame] Let $\pi: E \to M$ be a smooth vector bundle of rank $r$ and let $U \subset M$ be open. A local frame for $E$ over $U$ is an ordered $r$-tuple of smooth sections \begin{align*} e_1,\ldots,e_r: U &\to \pi^{-1}(U) \end{align*} such that $e_1(p),\ldots,e_r(p)$ is a basis of the vector space $E_p$ for every $p \in U$. [/definition] Frames translate bundle questions into matrix questions. Once a frame is chosen, a section over $U$ is described by $r$ smooth coefficient functions. The reason frames are not merely convenient notation is that they contain exactly the same information as local product coordinates, which is the content of the next result. [quotetheorem:9996] The theorem explains why frames are the practical language of vector bundles. A trivialisation sends the standard basis of $\mathbb{R}^r$ to a frame, while a frame writes a vector uniquely by its coefficient vector. [example: Vector Fields as Sections of the Tangent Bundle] Let $M$ be a smooth manifold and let $\pi: TM \to M$ be its tangent bundle. A smooth vector field is a smooth section \begin{align*} X: M \to TM \end{align*} such that $\pi(X(p))=p$ for every $p \in M$, so $X(p)\in T_pM$. On a coordinate chart $(U,\varphi)$ with coordinates $(x_1,\ldots,x_n)$, the coordinate vector fields \begin{align*} \partial_{x_1},\ldots,\partial_{x_n}: U \to TU \end{align*} form a local frame: for each $p\in U$, the vectors \begin{align*} \partial_{x_1}|_p,\ldots,\partial_{x_n}|_p \end{align*} are a basis of $T_pM$. Therefore, for each $p\in U$, there are unique [real numbers](/page/Real%20Numbers) $X_1(p),\ldots,X_n(p)$ such that \begin{align*} X(p)=\sum_{i=1}^n X_i(p)\partial_{x_i}|_p. \end{align*} Equivalently, as a section over $U$, \begin{align*} X|_U=\sum_{i=1}^n X_i\partial_{x_i}. \end{align*} The coefficient functions are smooth because the coordinate trivialisation of $TM$ sends \begin{align*} X(p)=\sum_{i=1}^n X_i(p)\partial_{x_i}|_p \end{align*} to \begin{align*} (p,(X_1(p),\ldots,X_n(p)))\in U\times\mathbb{R}^n, \end{align*} and the composition of the smooth map $X|_U:U\to TU$ with this trivialisation is the smooth map $p\mapsto (p,(X_1(p),\ldots,X_n(p)))$. Thus a vector field is exactly a smooth section of the tangent bundle, and a chart rewrites it as smooth component functions in the coordinate frame. [/example] The coordinate expression is not an extra structure on $TM$. It is the local description of a section with respect to a chosen frame, and changing coordinates changes the coefficient functions by the corresponding transition matrix. ## Bundle Maps and Subbundles Geometry often compares two vector bundles over the same base. The correct maps must respect both the smooth total spaces and the linear structures on fibers. [definition: Smooth Bundle Homomorphism] Let $\pi_E: E \to M$ and $\pi_F: F \to M$ be smooth vector bundles over the same smooth manifold $M$. A smooth bundle homomorphism from $E$ to $F$ is a smooth map \begin{align*} A: E &\to F \end{align*} such that $\pi_F \circ A = \pi_E$ and each restricted map \begin{align*} A_p: E_p &\to F_p \end{align*} is linear. [/definition] The fiberwise linearity condition distinguishes bundle homomorphisms from arbitrary smooth maps of total spaces. In a pair of local frames, a bundle homomorphism becomes a smooth matrix-valued function. Once maps are available, the next question is when a smoothly varying family of linear subspaces forms a bundle in its own right. [definition: Smooth Vector Subbundle] Let $\pi: E \to M$ be a smooth vector bundle of rank $r$. A smooth vector subbundle of rank $k$ is a smooth embedded submanifold $F \subset E$ such that each fiber $F_p = F \cap E_p$ is a $k$-dimensional linear subspace of $E_p$, the restricted projection $\pi|_F: F \to M$ is smooth, and every point of $M$ has an open neighbourhood $U$ with a local frame $e_1,\ldots,e_r$ for $E$ over $U$ such that $e_1,\ldots,e_k$ is a local frame for $F$ over $U$. [/definition] Subbundles appear whenever a geometric condition cuts out a smoothly varying family of subspaces. The tangent spaces of a submanifold inside the restricted tangent bundle of the ambient manifold are the standard model. [example: Tangent Bundle of an Embedded Submanifold] Let $N \subset M$ be an embedded submanifold of dimension $k$, and let $\iota:N\to M$ be the inclusion. For each $p\in N$, the differential is a [linear map](/page/Linear%20Map) \begin{align*} d\iota_p:T_pN\to T_pM. \end{align*} We show locally that the subspaces $d\iota_p(T_pN)\subset T_pM$ are spanned by smoothly varying tangent vectors. Fix $p\in N$. Since $N$ is embedded, there is a coordinate chart $(W,x_1,\ldots,x_n)$ on $M$ around $p$ such that \begin{align*} N\cap W=\{q\in W:x_{k+1}(q)=\cdots=x_n(q)=0\}. \end{align*} On $N\cap W$, use coordinates $u_i=x_i|_N$ for $1\leq i\leq k$. In these coordinates, the inclusion has the form \begin{align*} (u_1,\ldots,u_k)\mapsto (u_1,\ldots,u_k,0,\ldots,0). \end{align*} For a smooth function $h$ on $W$ and $1\leq i\leq k$, \begin{align*} d\iota_q(\partial_{u_i}|_q)(h)=\partial_{u_i}|_q(h\circ\iota)=\frac{\partial h}{\partial x_i}(q)=\partial_{x_i}|_q(h). \end{align*} Thus \begin{align*} d\iota_q(\partial_{u_i}|_q)=\partial_{x_i}|_q. \end{align*} Therefore, if \begin{align*} v=\sum_{i=1}^k a_i\partial_{u_i}|_q\in T_qN, \end{align*} then by linearity of $d\iota_q$, \begin{align*} d\iota_q(v)=\sum_{i=1}^k a_i\partial_{x_i}|_q. \end{align*} So \begin{align*} d\iota_q(T_qN)=\operatorname{span}\{\partial_{x_1}|_q,\ldots,\partial_{x_k}|_q\}\subset T_qM. \end{align*} The vector fields $\partial_{x_1},\ldots,\partial_{x_k}$ are smooth along $N\cap W$, and at every $q\in N\cap W$ they form a basis for the image subspace $d\iota_q(T_qN)$. Hence these images form a smooth rank-$k$ vector subbundle of the restricted bundle $TM|_N\to N$. Geometrically, this subbundle is the tangent bundle $TN$ viewed inside the ambient tangent bundle along $N$. [/example] Subbundles are local linear algebra plus smooth dependence. The adapted-coordinate computation shows both pieces: each fiber is a linear subspace, and the chosen spanning vectors vary smoothly. ## Algebra of Smooth Vector Bundles Once vector bundles are accepted as smoothly varying vector spaces, the usual operations of linear algebra should be available fiber by fiber. The subtle point is that an operation performed separately in each fiber must still vary smoothly with the base point. Local trivialisations are what make this possible. The first canonical section is the one that exists without choices. It records the zero vector in each fiber and is the baseline against which many constructions, such as tubular neighborhoods and bundle morphisms, are measured. [definition: Zero Section] Let $\pi: E \to M$ be a smooth vector bundle. The zero section of $E$ is the smooth section \begin{align*} 0_E: M &\to E \end{align*} defined by assigning to each $p \in M$ the zero vector $0_p \in E_p$. [/definition] The zero section is not just notation for the zero vector. It embeds the base manifold inside the total space as the locus of fiberwise origins, so it gives a geometric copy of $M$ sitting inside $E$. Once sections are available, the next natural operation is to test them against smoothly varying covectors; that requires a bundle whose fiber over $p$ consists of linear functionals on $E_p$. [definition: Dual Smooth Vector Bundle] Let $\pi: E \to M$ be a smooth vector bundle of rank $r$. The dual smooth vector bundle has total space \begin{align*} E^* &= \bigsqcup_{p \in M} E_p^*, \end{align*} projection \begin{align*} \pi_{E^*}: E^* &\to M \end{align*} with rule $\lambda \mapsto p$ for $\lambda \in E_p^*$, and fiber \begin{align*} E_p^* = \operatorname{Hom}_{\mathbb{R}}(E_p,\mathbb{R}). \end{align*} If $\Phi: \pi^{-1}(U) \to U \times \mathbb{R}^r$ is a local trivialisation of $E$ with fiber maps $\Phi_p: E_p \to \mathbb{R}^r$, then the corresponding local trivialisation of $E^*$ is \begin{align*} \Phi^*: \pi_{E^*}^{-1}(U) &\to U \times (\mathbb{R}^r)^* \end{align*} with rule $\lambda \mapsto (p,\lambda \circ \Phi_p^{-1})$ for $\lambda \in E_p^*$. [/definition] In a frame $e_1,\ldots,e_r$, this construction uses the dual frame in each fiber, and the transition matrices are inverse transposes of the transition matrices of $E$. Dual bundles are therefore smooth because the original coordinate changes are smooth and invertible. To record several independent geometric quantities at once, we next need a construction whose point over $p$ is a pair of vectors, one from each bundle over $p$. [definition: Direct Sum of Smooth Vector Bundles] Let $\pi_E: E \to M$ and $\pi_F: F \to M$ be smooth vector bundles over the same smooth manifold $M$. The [direct sum](/page/Direct%20Sum) bundle has total space \begin{align*} E \oplus F &= \{(e,f) \in E \times F : \pi_E(e)=\pi_F(f)\}, \end{align*} projection \begin{align*} \pi_{E \oplus F}: E \oplus F &\to M \end{align*} with rule $(e,f) \mapsto \pi_E(e)=\pi_F(f)$, and fiber \begin{align*} (E \oplus F)_p = E_p \oplus F_p. \end{align*} If $\Phi_E: \pi_E^{-1}(U) \to U \times \mathbb{R}^r$ and $\Phi_F: \pi_F^{-1}(U) \to U \times \mathbb{R}^s$ are local trivialisations, then the induced local trivialisation is $(e,f) \mapsto (p, v, w)$ whenever $\Phi_E(e)=(p,v)$ and $\Phi_F(f)=(p,w)$. [/definition] The direct sum keeps two independent pieces of fiber data side by side and gives them the product coordinate changes. Many geometric objects, however, are not pairs of fields but multilinear combinations of fields, such as tensors formed from vectors and covectors. For those, the fiberwise construction must multiply coordinate changes rather than merely place them next to each other. [definition: Tensor Product of Smooth Vector Bundles] Let $\pi_E: E \to M$ and $\pi_F: F \to M$ be smooth vector bundles over the same smooth manifold $M$. The [tensor product](/page/Tensor%20Product) bundle has total space \begin{align*} E \otimes F &= \bigsqcup_{p \in M} (E_p \otimes_{\mathbb{R}} F_p), \end{align*} projection \begin{align*} \pi_{E \otimes F}: E \otimes F &\to M \end{align*} with rule $u \mapsto p$ for $u \in E_p \otimes_{\mathbb{R}} F_p$, and fiber \begin{align*} (E \otimes F)_p = E_p \otimes_{\mathbb{R}} F_p. \end{align*} If $\Phi_E$ and $\Phi_F$ are local trivialisations over $U$, then the induced local trivialisation sends $u \in E_p \otimes_{\mathbb{R}} F_p$ to its image in $\mathbb{R}^r \otimes_{\mathbb{R}} \mathbb{R}^s$ under $(\Phi_E)_p \otimes (\Phi_F)_p$. [/definition] These constructions explain why tensor fields are naturally sections of tensor bundles rather than arrays of functions. In a local frame they look like component functions, but their coordinate-change law is dictated by the tensor product of transition functions. A different problem arises when a geometric condition marks some directions as irrelevant: then the construction must collapse a smooth family of subspaces without losing smooth local coordinates. A mere collection of subspaces $F_p \subset E_p$ is not enough; it must form a smooth vector subbundle. [definition: Quotient Smooth Vector Bundle] Let $\pi: E \to M$ be a smooth vector bundle and let $F \subset E$ be a smooth vector subbundle. The quotient smooth vector bundle has total space \begin{align*} E/F &= \bigsqcup_{p \in M} E_p/F_p, \end{align*} projection \begin{align*} \pi_{E/F}: E/F &\to M \end{align*} with rule $[e] \mapsto p$ for $[e] \in E_p/F_p$, and fiber \begin{align*} (E/F)_p = E_p/F_p. \end{align*} On each open set $U \subset M$ with a local frame $e_1,\ldots,e_r$ for $E$ such that $e_1,\ldots,e_k$ is a local frame for $F$, the quotient local trivialisation sends the class of $\sum_{i=1}^r a_i e_i(p)$ to $(p,a_{k+1},\ldots,a_r)$. [/definition] The quotient bundle packages the directions in $E$ transverse to a chosen subbundle. For an embedded submanifold $N \subset M$, this is the construction behind the normal bundle: tangent directions of $M$ along $N$ are divided by tangent directions lying inside $N$. ## Pullbacks and Natural Constructions A bundle over $M$ can be transported along a smooth map into $M$. This is the construction that lets vector bundles behave functorially under parametrisations, inclusions, and changes of base. [definition: Pullback Smooth Vector Bundle] Let $f: N \to M$ be a smooth map and let $\pi: E \to M$ be a smooth vector bundle. The pullback bundle $f^*E$ is \begin{align*} f^*E = \{(q,e) \in N \times E : f(q)=\pi(e)\} \end{align*} with projection \begin{align*} \operatorname{pr}_1: f^*E &\to N \end{align*} with rule \begin{align*} (q,e) &\mapsto q, \end{align*} and fiberwise vector space operations inherited from the fibers of $E$. [/definition] The defining equation says that a point of $f^*E$ is a point $q \in N$ together with a vector [lying over](/theorems/2876) its image $f(q) \in M$. This is the only possible construction if the fiber over $q$ is meant to be $E_{f(q)}$. What still needs verification is that this set inherits smooth local product charts from the original bundle. [quotetheorem:9997] Pullbacks convert constructions over a manifold into constructions over a parameter space. For instance, a curve $\gamma: I \to M$ pulls back $TM$ to a vector bundle over an interval, whose sections are vector fields along the curve. [example: Vector Fields Along a Curve] Let $\gamma: I \to M$ be a smooth curve from an open interval $I \subset \mathbb{R}$ into a smooth manifold $M$. By definition of the pullback bundle, a point of $\gamma^*TM$ over $t$ is a pair $(t,v)$ with $v \in T_{\gamma(t)}M$, so the fiber over $t$ is naturally identified with $T_{\gamma(t)}M$. A section \begin{align*} V: I &\to \gamma^*TM \end{align*} satisfies $\operatorname{pr}_1(V(t))=t$. Hence there is a unique vector $v(t)\in T_{\gamma(t)}M$ such that \begin{align*} V(t)&=(t,v(t)). \end{align*} Thus a section of $\gamma^*TM$ is exactly a smoothly varying choice of tangent vector along the curve. Now restrict to a subinterval $J\subset I$ such that $\gamma(J)$ lies in a coordinate chart $(U,x_1,\ldots,x_n)$ on $M$. For each $t\in J$, the coordinate vectors \begin{align*} \partial_{x_1}|_{\gamma(t)},\ldots,\partial_{x_n}|_{\gamma(t)} \end{align*} form a basis of $T_{\gamma(t)}M$. Therefore there are unique real numbers $V_1(t),\ldots,V_n(t)$ with \begin{align*} v(t)&=\sum_{i=1}^n V_i(t)\partial_{x_i}|_{\gamma(t)}. \end{align*} Equivalently, \begin{align*} V(t)&=\left(t,\sum_{i=1}^n V_i(t)\partial_{x_i}|_{\gamma(t)}\right). \end{align*} The coordinate trivialisation of $TM$ over $U$ sends \begin{align*} \sum_{i=1}^n a_i\partial_{x_i}|_q&\mapsto (q,(a_1,\ldots,a_n)). \end{align*} The induced trivialisation of $\gamma^*TM$ over $J$ therefore sends \begin{align*} \left(t,\sum_{i=1}^n a_i\partial_{x_i}|_{\gamma(t)}\right)&\mapsto (t,(a_1,\ldots,a_n)). \end{align*} Applying this to $V(t)$ gives \begin{align*} V(t)&\mapsto (t,(V_1(t),\ldots,V_n(t))). \end{align*} Since $V$ is a smooth section and the trivialisation is smooth, the map $t\mapsto (V_1(t),\ldots,V_n(t))$ is smooth; equivalently, each coefficient function $V_i:J\to\mathbb{R}$ is smooth. Thus a vector field along $\gamma$ is a section of the pullback bundle $\gamma^*TM$, and a coordinate chart rewrites it as smooth component functions multiplying the coordinate tangent vectors along the curve. [/example] This construction is the language behind covariant derivatives along curves in Riemannian geometry. The section lives over the parameter interval, but its values lie in tangent spaces of $M$. ## The Tangent Bundle as the Guiding Example The tangent bundle is the reason smooth vector bundles are part of differential geometry rather than just topology. It packages all tangent spaces into one smooth object whose coordinate changes are Jacobian matrices. [quotetheorem:9998] If $(U,\varphi)$ is a chart with coordinates $(x_1,\ldots,x_n)$, the induced local trivialisation sends a tangent vector $v \in T_pM$ to its coordinate vector in $\mathbb{R}^n$. On an overlap of two charts, the transition function is the [Jacobian matrix](/page/Jacobian%20Matrix) of the coordinate change. [example: Coordinate Change in the Tangent Bundle] Let $(U,\varphi)$ and $(V,\psi)$ be charts on a smooth $n$-manifold $M$, with coordinate maps $\varphi=(x_1,\ldots,x_n)$ and $\psi=(y_1,\ldots,y_n)$. Fix $p\in U\cap V$ and write the same tangent vector $v\in T_pM$ in the two coordinate frames: \begin{align*} v=\sum_{i=1}^n a_i\partial_{x_i}|_p=\sum_{j=1}^n b_j\partial_{y_j}|_p. \end{align*} To find $b_j$, apply both expressions for $v$ to the coordinate function $y_j$. From the $x$-coordinate expression, \begin{align*} v(y_j)=\left(\sum_{i=1}^n a_i\partial_{x_i}|_p\right)(y_j)=\sum_{i=1}^n a_i\partial_{x_i}|_p(y_j)=\sum_{i=1}^n a_i\frac{\partial y_j}{\partial x_i}(p). \end{align*} From the $y$-coordinate expression, \begin{align*} v(y_j)=\left(\sum_{\ell=1}^n b_\ell\partial_{y_\ell}|_p\right)(y_j)=\sum_{\ell=1}^n b_\ell\partial_{y_\ell}|_p(y_j)=\sum_{\ell=1}^n b_\ell\delta_{\ell j}=b_j. \end{align*} Equating the two values of $v(y_j)$ gives \begin{align*} b_j=\sum_{i=1}^n a_i\frac{\partial y_j}{\partial x_i}(p). \end{align*} Thus, if $a=(a_1,\ldots,a_n)^\top$ and $b=(b_1,\ldots,b_n)^\top$, then \begin{align*} b=\left(\frac{\partial y_j}{\partial x_i}(p)\right)_{j,i}a. \end{align*} So the transition matrix for $TM$ from the $x$-coordinate frame to the $y$-coordinate frame is exactly the Jacobian matrix of the coordinate change $\psi\circ\varphi^{-1}$ at $\varphi(p)$. [/example] The tangent bundle shows the general pattern: local frames give coordinates, transition functions are smooth invertible matrices, and geometric constructions must be invariant under changing those coordinates. ## Beyond Smooth Vector Bundles Smooth vector bundles lead directly to principal bundles, connections, characteristic classes, and index theory. A connection adds a way to differentiate sections, so it measures how fibers at nearby points compare. Characteristic classes measure global twisting that no single local trivialisation can detect. In Riemannian geometry, the tangent bundle and its tensor bundles carry the metric, curvature, and covariant derivatives studied in [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry). In differential geometry more broadly, smooth vector bundles provide the common language for tangent spaces, cotangent spaces, normal bundles, and associated bundles, as developed in [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry). Algebraic geometry has analogous objects, where vector bundles are replaced by locally free sheaves on schemes or varieties. The smooth local-triviality condition is replaced by algebraic local freeness, and transition functions become regular matrix-valued functions. This is one reason vector bundles form a bridge between differential geometry and [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry). ## References Androma, [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry). Androma, [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry). Androma, [Cambridge IA Vectors and Matrices](/page/Cambridge%20IA%20Vectors%20and%20Matrices). Androma, [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry). John M. Lee, *Introduction to Smooth Manifolds* (2013). Dale Husemoller, *Fibre Bundles* (1994).