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Tangent bundle

In differential geometry, the tangent bundle of a smooth manifold M of dimension n is defined as the disjoint union TM = \bigsqcup_{p \in M} T_p M, where T_p M denotes the tangent space at each point p \in M, together with the natural projection map \pi: TM \to M given by \pi(p, v) = p for v \in T_p M.^[1] This structure equips TM with a topology and smooth manifold structure of dimension $2n, making it a smooth vector bundle of rank n over the base M.^[2] The fibers \pi^{-1}(p) = T_p M are n-dimensional vector spaces that vary smoothly over M, and local trivializations are provided by charts on M that map \pi^{-1}(U) diffeomorphically to U \times \mathbb{R}^n.^[3] The tangent bundle generalizes the notion of tangent spaces from individual points to a global object, enabling the study of directions and velocities across the entire manifold.^[4] Sections of TM, which are smooth maps s: M \to TM satisfying \pi \circ s = \mathrm{id}_M, correspond precisely to vector fields on M, providing a way to assign a tangent vector to every point consistently.^[3] This construction is fundamental for defining the differential of smooth maps between manifolds, as the pushforward d\phi: TM \to TN for \phi: M \to N maps tangent vectors via curve derivatives.^[5] Beyond its structural role, the tangent bundle underpins key applications in differential geometry, such as the formulation of ordinary differential equations via integral curves of vector fields, the development of Riemannian metrics on TM for geodesic flows, and the analysis of Lie group actions through infinitesimal generators.^[4] In global analysis and mathematical physics, vector fields on tangent bundles facilitate the study of dynamical systems, symmetries, and curvature phenomena, with extensions to more abstract settings like tangent categories in synthetic differential geometry.

Introduction and Motivation

Historical Development

The concept of tangent spaces emerged in the early 19th century through Carl Friedrich Gauss's foundational work on the differential geometry of surfaces. In his 1827 paper "Disquisitiones generales circa superficies curvas," Gauss introduced tangent planes at points on a parametrized surface, defining them via the linear span of partial derivatives with respect to local coordinates, which allowed for the study of curvature and geodesics intrinsically. This laid the groundwork for associating a vector space to each point on a geometric object, though Gauss did not yet consider the global collection of such spaces. Bernhard Riemann extended these ideas to higher dimensions in his 1854 habilitation lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen," where he conceptualized n-dimensional manifolds equipped with a metric tensor, implicitly viewing them as comprising tangent spaces at every point to enable measurements of lengths and angles. Riemann's manifolds were spaces of variable curvature, with tangent spaces serving as local Euclidean approximations, but without a formal global structure. This perspective influenced subsequent developments, including Christoffel symbols in 1869 for coordinate transformations on these spaces. The formalization of tangent vectors and spaces accelerated in the early 20th century, particularly through axiomatic approaches. Oswald Veblen and J.H.C. Whitehead's 1932 monograph "The Foundations of Differential Geometry" provided a rigorous framework, defining tangent spaces as affine spaces attached to each point of a manifold via allowable coordinate systems and projective methods, emphasizing their role in parallelism and curvature. Around the same time, in the 1930s, tangent vectors began to be characterized as derivations on the algebra of smooth functions at a point, an intrinsic definition that avoided embedding in Euclidean space; this approach was solidified by Claude Chevalley in his 1946 book "Theory of Lie Groups," marking its first explicit use in modern terms. Hassler Whitney's 1936 embedding theorem further advanced the theory by proving that smooth manifolds could be embedded in Euclidean space, implicitly relying on the disjoint union of tangent spaces, which he later formalized as the tangent bundle in his 1930s-1940s work on differential topology. The notion of the tangent bundle as a fiber bundle crystallized in the 1930s and 1940s, integrating local tangent spaces into a global manifold structure. Whitney, alongside Norman Steenrod, contributed to early fiber bundle theory in the late 1930s, applying it to the tangent bundle to study embeddings and immersions. Élie Cartan advanced this in the 1920s-1940s through his theory of generalized spaces, viewing manifolds as spaces of tangent spaces with moving frames, which incorporated connections and torsion on these bundles. Charles Ehresmann formalized fiber bundles in the 1940s, defining connections on them in his 1950 paper "Les connexions infinitésimales dans un espace fibré différentiable," explicitly including the tangent bundle as a principal bundle example with the general linear group as structure group.^[6] Post-World War II, Shiing-Shen Chern developed characteristic classes for vector bundles in the 1950s, classifying tangent bundles via Chern classes and integrating them into global differential geometry, as in his 1951 work on complex manifolds. This evolution enabled the tangent bundle's role in modern tools like de Rham cohomology and index theorems.

Conceptual Role

The tangent bundle of a smooth manifold serves as the total space encompassing all possible velocities on the manifold, where each tangent vector at a point represents the direction and speed of an infinitesimal motion along smooth curves. This structure allows for a global analysis of directional derivatives and trajectories, transforming localized notions of motion into a cohesive framework that captures the manifold's dynamic properties. In the context of maps between manifolds, the tangent bundle functions as both the domain and codomain for the differential of a smooth map, which generalizes the Jacobian matrix from multivariable calculus to curved spaces. The differential acts as a linear approximation that pushes forward tangent vectors from one manifold to another, enabling the study of how local changes propagate under nonlinear transformations. By assembling the local tangent spaces—each a linear approximation to the manifold at a point—into a single vector bundle, the tangent bundle bridges the gap between these infinitesimal linearizations and the global nonlinear geometry of the manifold. This unification facilitates coordinate-independent descriptions of geometric phenomena, such as flows and deformations, across the entire space. The tangent bundle is foundational for advanced differential geometric constructs, providing the natural arena for defining Riemannian metrics as smoothly varying inner products on tangent spaces and affine connections that enable parallel transport and covariant differentiation. These tools, in turn, underpin the measurement of distances, angles, and curvatures on manifolds.

Definition and Construction

Formal Definition

The tangent bundle of a smooth manifold M, denoted TM, is formally defined as the disjoint union of all tangent spaces to M, that is,

TM = \bigsqcup_{x \in M} T_x M,

where T_x M denotes the tangent space at the point x \in M.^[7] Elements of TM are pairs (x, v) with x \in M and v \in T_x M.^[7] The natural projection map \pi: TM \to M is given by \pi(x, v) = x, which sends each tangent vector to its base point on the manifold.^[7] For an n-dimensional smooth manifold M, the fiber \pi^{-1}(x) = T_x M over each point x \in M is a vector space isomorphic to \mathbb{R}^n.^[7] Each fiber inherits the vector space structure from T_x M, with addition and scalar multiplication defined pointwise, endowing TM with the structure of a rank-n real vector bundle over M.^[7]

Local and Global Construction

The tangent bundle TM of a smooth manifold M of dimension n admits a local trivialization over each chart domain in an atlas of M. Specifically, for a chart (U, \phi) with \phi: U \to \mathbb{R}^n, the preimage \pi^{-1}(U) is diffeomorphic to U \times \mathbb{R}^n via the bundle map \hat{\phi}: \pi^{-1}(U) \to U \times \mathbb{R}^n defined by \hat{\phi}(x, v) = (x, d\phi_x(v)), where d\phi_x: T_x M \to \mathbb{R}^n is the differential of \phi at x, identifying tangent vectors with their coordinate representations relative to the basis \{\partial/\partial u^i\} induced by \phi.^[8]^[1] This trivialization equips the restricted bundle with the product structure, allowing tangent vectors to be expressed in local coordinates as (x, (v^1, \dots, v^n)).^[9] On overlaps of chart domains, these local trivializations are related by transition functions that ensure the bundle structure is well-defined. For charts (U, \phi) and (V, \psi) with U \cap V \neq \emptyset, the transition map \hat{g}_{\psi \phi}: \psi(U \cap V) \to \mathrm{GL}_n(\mathbb{R}) is given by \hat{g}_{\psi \phi}(y) = D(\psi \circ \phi^{-1})(y), the Jacobian matrix of the coordinate change \psi \circ \phi^{-1} at y = \phi(x).^[8]^[9]^[10] This matrix acts linearly on the fiber coordinates, transforming (y, w) to (y, \hat{g}_{\psi \phi}(y) w), and satisfies the cocycle condition \hat{g}_{\rho \psi} \cdot \hat{g}_{\psi \phi} = \hat{g}_{\rho \phi} on triple overlaps due to the chain rule for differentials.^[11] Globally, the tangent bundle is constructed as the quotient of the disjoint union of all local trivializations by the equivalence relation induced by these transitions. Given an atlas \{(U_\alpha, \phi_\alpha)\}_{\alpha \in A}, the total space is TM = \bigsqcup_{\alpha} (U_\alpha \times \mathbb{R}^n) / \sim, where (x_\alpha, v_\alpha) \sim (x_\beta, v_\beta) if x_\alpha = x_\beta =: x and v_\beta = \hat{g}_{\phi_\beta \phi_\alpha}(\phi_\alpha(x)) v_\alpha on U_\alpha \cap U_\beta.^[8]^[12] The projection \pi: TM \to M is then well-defined, with fibers \pi^{-1}(x) \cong T_x M, yielding a vector bundle of rank n.^[13] When M is smooth, the smoothness of the chart transitions on M implies that the bundle transition functions \hat{g}_{\psi \phi} are smooth maps to \mathrm{GL}_n(\mathbb{R}), endowing TM with a smooth structure as a vector bundle over M.^[12]^[4] This construction is independent of the choice of atlas, as compatible atlases yield equivalent bundles.^[8]

Topological and Smooth Structure

Topological Properties

The tangent bundle TM of an [n](/page/N+)-dimensional smooth manifold [M](/page/M) inherits a natural topology from the atlas of M, rendering TM a topological manifold of dimension $2n. Specifically, for each chart (U, \phi) on M with \phi: U \to \mathbb{R}^n, the preimage \pi^{-1}(U) \subset TM is homeomorphic to the product space U \times \mathbb{R}^n, equipped with the standard product topology of the Euclidean spaces involved. These local trivializations are glued together using the linear transition functions derived from the Jacobians of the coordinate changes on M, yielding a well-defined global topology on TM.^[14] An alternative construction endows TM with the quotient topology arising from the disjoint union \coprod_{p \in M} T_p M of all tangent spaces, where equivalence relations identify vectors according to the derivative maps between overlapping charts. This quotient topology agrees with the atlas-induced one and confirms that TM is locally Euclidean of dimension $2n.^[14] When M is a second-countable Hausdorff manifold, TM shares these properties: it is Hausdorff, as distinct points can be separated using the projection \pi: TM \to M and local product structures, and second-countable, owing to a countable subcover of the trivializing open sets from M's atlas.^[9] As a topological vector bundle, TM is orientable—meaning its structure group reduces to GL^+(n, \mathbb{R}), allowing a consistent choice of orientation on the fibers—if and only if the base manifold M is orientable. For non-orientable M, such as the real projective plane \mathbb{RP}^2, the tangent bundle T\mathbb{RP}^2 is non-orientable, with nonzero first Stiefel-Whitney class w_1(T\mathbb{RP}^2) \neq 0.^[15] Similarly, the tangent bundle over the Klein bottle provides another example of a non-orientable bundle, reflecting the twisting of the base surface.^[16]

Smooth Atlas and Manifold Structure

The smooth atlas on the tangent bundle TM of an n-dimensional smooth manifold M is constructed directly from the smooth atlas of M. Given a chart (U, \phi) on M with \phi = (x^1, \dots, x^n): U \to \mathbb{R}^n, the corresponding chart on TM is defined on the open set \pi^{-1}(U) \subset TM, where \pi: TM \to M is the projection map sending each tangent vector to its base point. The chart map \Phi: \pi^{-1}(U) \to \mathbb{R}^{2n} is given by \Phi(v) = (\phi(p), d\phi_p(v)) for v \in T_p M, p \in U. In these coordinates, points in \pi^{-1}(U) are represented as (x^1, \dots, x^n, v^1, \dots, v^n), where (x^1, \dots, x^n) = \phi(p) specifies the base point and (v^1, \dots, v^n) are the components of v with respect to the basis \{\partial / \partial x^1, \dots, \partial / \partial x^n\}_p. This induces a smooth structure on TM, making it a smooth manifold of dimension $2n. To verify the smoothness, consider overlapping charts (U, \phi) and (V, \psi) on M with U \cap V \neq \emptyset. The transition map between the induced charts on TM is \Psi \circ \Phi^{-1}: \Phi(\pi^{-1}(U \cap V)) \to \Psi(\pi^{-1}(U \cap V)), explicitly given by

(x^i, v^j) \mapsto \left( \tilde{x}^k(x), \frac{\partial \tilde{x}^k}{\partial x^i}(x) v^i \right),

where \tilde{x} = \psi \circ \phi^{-1}. Since the transition maps \psi \circ \phi^{-1} on M are smooth diffeomorphisms, their Jacobian matrices \left( \frac{\partial \tilde{x}^k}{\partial x^i} \right) consist of smooth functions. The resulting map on TM is therefore a composition of smooth maps (coordinate change on the base and linear transformation on the fiber via the Jacobian), ensuring it is smooth. The maximal atlas generated by these charts defines the unique smooth structure on TM, independent of the choice of atlas on M. The projection \pi: TM \to M inherits smoothness from this atlas: in local coordinates, it is simply (x^1, \dots, x^n, v^1, \dots, v^n) \mapsto (x^1, \dots, x^n), a smooth map. Moreover, \pi is a submersion, as its differential d\pi_{(p,v)}: T_{(p,v)} TM \to T_p M is surjective at every point (with rank n), reflecting the fact that horizontal directions in TM project onto all directions in M. A key structural feature is the vertical subbundle, consisting of the kernels V_{(p,v)} TM = \ker d\pi_{(p,v)}. These kernels comprise vectors tangent to the fibers \pi^{-1}(p) \cong T_p M, which are n-dimensional vector subspaces of the $2n-dimensional tangent spaces to TM. This decomposition underscores the vector bundle nature of TM over M.

Examples

Trivial Tangent Bundles

The tangent bundle of Euclidean space \mathbb{R}^n is trivial, meaning it is isomorphic as a vector bundle to the product bundle \mathbb{R}^n \times \mathbb{R}^n. In this isomorphism, elements are represented by pairs (x, v), where x \in \mathbb{R}^n is the point in the base space and v \in \mathbb{R}^n is the tangent vector at that point. This trivialization arises naturally from the global coordinate system on \mathbb{R}^n, where the standard basis vectors provide a global frame for the tangent spaces.^[17] A concrete example of a trivial tangent bundle on a compact manifold is the circle S^1, whose tangent bundle TS^1 is diffeomorphic to the infinite cylinder S^1 \times \mathbb{R}. This isomorphism follows from the existence of a nowhere-vanishing global vector field on S^1, such as the unit tangent vector field, which provides a trivialization of the rank-1 bundle. Geometrically, this reflects the ability to consistently choose a direction tangent to the circle at every point, without twisting.^[18] More generally, Lie groups possess trivial tangent bundles due to the availability of a global frame consisting of left-invariant vector fields. For a Lie group G, the left multiplication maps allow the construction of n linearly independent left-invariant vector fields (where n = \dim G) that span the tangent space at every point, yielding an isomorphism TG \cong G \times \mathfrak{g}, with \mathfrak{g} the Lie algebra of G. This property holds for all Lie groups, including non-compact ones like the general linear group.^[19] Parallelizable manifolds, by definition, have trivial tangent bundles, admitting a global frame of n linearly independent vector fields on an n-manifold. The n-torus T^n = S^1 \times \cdots \times S^1 is a prime example, as its tangent bundle decomposes as TT^n \cong T^n \times \mathbb{R}^n via the product structure and the triviality of each TS^1. This parallelizability extends to other Lie groups and their quotients under certain conditions, but the torus illustrates how Cartesian products of parallelizable manifolds inherit the property.^[20]

Non-Trivial Tangent Bundles

A prominent example of a non-trivial tangent bundle is the tangent bundle TS^2 of the 2-sphere S^2. This bundle cannot be globally trivialized, as demonstrated by the hairy ball theorem, which states that there exists no continuous nowhere-vanishing vector field on S^2.^[21] If TS^2 were trivial, it would admit a global frame of two linearly independent vector fields, one of which could be chosen nowhere zero, contradicting the theorem.^[21] The non-triviality arises from the topological obstruction encoded in the Euler characteristic of S^2, which is 2, preventing a global section without zeros.^[22] Another key example is the tangent bundle T\mathbb{RP}^2 of the real projective plane \mathbb{RP}^2. Since \mathbb{RP}^2 is a non-orientable manifold, its tangent bundle is likewise non-orientable, meaning it lacks a consistent choice of orientation across the base space.^[22] This non-orientability manifests as a twisting that prevents global trivialization, similar to how the manifold itself cannot be embedded in \mathbb{R}^3 without self-intersection.^[23] The bundle's structure reflects the identification of antipodal points on S^2, leading to a non-trivial double cover that obstructs parallelizability.^[24] The classification of tangent bundles, particularly over spheres, often relies on clutching functions, which describe how local trivializations are glued together along the equator. For orientable rank-2 bundles over S^2, such as TS^2, the clutching function is an element of \pi_1(SO(2)) \cong \mathbb{Z}, but the non-triviality is captured by the Euler class e(TS^2) = 2, the generator of H^2(S^2; \mathbb{Z}).^[25] This characteristic class vanishes if and only if the bundle is trivial for even-dimensional spheres.^[22] In general, non-zero Euler classes provide topological invariants that detect such obstructions.^[22] For 2-dimensional non-orientable cases, the Möbius strip serves as an analogy for non-trivial line bundles, where the total space is formed by twisting and gluing the boundary of a strip, resulting in a bundle over S^1 that admits no global non-zero section.^[24] This construction illustrates the twisting in T\mathbb{RP}^2, where the non-orientability induces a similar global inconsistency in framing the fibers.^[24] Unlike parallelizable manifolds, where tangent bundles are trivial, these examples highlight how topological features like non-zero characteristic classes enforce non-triviality.^[25]

Vector Fields and Sections

Sections as Vector Fields

A vector field on a smooth manifold M is defined as a smooth section of the tangent bundle TM, that is, a smooth map X: M \to TM satisfying \pi \circ X = \mathrm{id}_M, where \pi: TM \to M is the canonical projection. This condition ensures that X assigns to each point p \in M a tangent vector in the fiber T_p M, preserving the bundle structure. In local coordinates (U, \phi) on M, where \phi: U \to \mathbb{R}^n with \phi(p) = (x^1(p), \dots, x^n(p)), the vector field X takes the form

X(p) = \sum_{i=1}^n X^i(p) \frac{\partial}{\partial x^i} \bigg|_p,

where the coefficients X^i: U \to \mathbb{R} are smooth functions. To construct a global smooth vector field, local expressions are glued together using a partition of unity subordinate to an open cover of M, ensuring the result is well-defined and smooth everywhere. On manifolds with non-trivial tangent bundles, such as the 2-sphere S^2, global vector fields must respect the topology, but the local description remains coordinate-based. The collection of all smooth sections, denoted \Gamma(TM) or \mathfrak{X}(M), forms a module over the ring C^\infty(M) of smooth real-valued functions on M. Scalar multiplication is defined pointwise: for f \in C^\infty(M) and X \in \Gamma(TM), (fX)(p) = f(p) \cdot X(p) \in T_p M. Additionally, \Gamma(TM) carries a Lie algebra structure over \mathbb{R} via the Lie bracket [X, Y], given by

[X, Y](f) = X(Y(f)) - Y(X(f))

for all f \in C^\infty(M), which corresponds to the commutator of derivations and satisfies bilinearity, antisymmetry, and the Jacobi identity. Vector fields also generate dynamical systems through their integral curves and flows. An integral curve of X is a smooth curve \gamma: (a, b) \to M satisfying \gamma'(t) = X(\gamma(t)) for all t \in (a, b), where \gamma'(t) denotes the velocity vector in T_{\gamma(t)} M. If X is complete (i.e., every integral curve extends to all of \mathbb{R}), it generates a one-parameter group of diffeomorphisms, called the flow \{\phi_t\}_{t \in \mathbb{R}} of X, satisfying \frac{d}{dt} \phi_t(p) = X(\phi_t(p)) with \phi_0 = \mathrm{id}_M. These flows encode the infinitesimal action of X and are fundamental in studying symmetries and evolution on manifolds.

Relation to Cotangent Bundle

The cotangent bundle T^*M of a smooth manifold M is the vector bundle whose fiber over each point x \in M is the dual vector space T_x^* M = \operatorname{Hom}(T_x M, \mathbb{R}) to the tangent space T_x M, formally constructed as the disjoint union T^*M = \bigsqcup_{x \in M} T_x^* M with the natural projection \pi: T^*M \to M.^[26] This dual relationship endows T^*M with a complementary role to the tangent bundle TM, as elements of T_x^* M (covectors) naturally pair with tangent vectors at x to yield scalars.^[27] A canonical pairing \langle \cdot, \cdot \rangle: T^*M \times_M TM \to \mathbb{R} arises pointwise from the duality between each fiber pair (T_x^* M, T_x M), defined by evaluation \langle \xi, v \rangle = \xi(v) for \xi \in T_x^* M and v \in T_x M.^[27] In local coordinates (x^i) on M, a covector \xi \in T_x^* M is represented as \xi = \sum_i \xi_i \, dx^i, where \{dx^i\} is the dual basis to \{\partial / \partial x^i\}, and the pairing becomes the standard dot product \sum_i \xi_i v^i for v = \sum_i v^i \partial / \partial x^i.^[26] Sections of T^*M are smooth 1-forms on M, and more generally, differential k-forms are sections of the k-th exterior power \Lambda^k T^*M, forming the bundle of k-forms \Omega^k(M) = \Gamma(\Lambda^k T^*M).^[26] Unlike the tangent bundle TM, which admits an almost complex structure whenever M does (via the induced complex structure on fibers), the cotangent bundle T^*M carries a canonical symplectic structure.^[28] Specifically, there is a tautological 1-form [\theta](/page/Theta) (or Liouville form) on T^*M, defined in local coordinates (x^i, \xi_i) by \theta = \sum_i \xi_i \, dx^i, whose exterior derivative \omega = -d\theta = \sum_i dx^i \wedge d\xi_i is a closed, non-degenerate 2-form, making (T^*M, \omega) a symplectic manifold of dimension $2\dim M.^[29] This symplectic form is exact and coordinate-independent, contrasting with the almost complex structure on TM, which requires additional data from M and does not inherently yield a compatible symplectic form.^[27] The duality between tangent vectors and covectors extends to operations like contraction and the interior product, which pair elements of TM and \Lambda^* T^*M. For a vector field X \in \Gamma(TM) (a section of the tangent bundle) and a k-form \omega \in \Omega^k(M), the interior product \iota_X \omega (or contraction) is the (k-1)-form defined pointwise by (\iota_X \omega)_p (v_1, \dots, v_{k-1}) = \omega_p (X_p, v_1, \dots, v_{k-1}), satisfying antiderivation properties such as \iota_X (\alpha \wedge \beta) = (\iota_X \alpha) \wedge \beta + (-1)^{\deg \alpha} \alpha \wedge (\iota_X \beta) and nilpotency \iota_X^2 = 0.^[30] This operation facilitates computations in differential geometry, such as in Cartan's magic formula for the Lie derivative \mathcal{L}_X \omega = \iota_X d\omega + d \iota_X \omega.^[30]

Lifts and Differential Maps

Lifts of Curves and Functions

In differential geometry, given a smooth manifold M and a smooth curve c: I \to M, where I is an interval in \mathbb{R}, the complete lift of c to the tangent bundle TM is the curve Cc: I \to TM defined by Cc(t) = (c(t), c'(t)), where c'(t) \in T_{c(t)}M is the velocity vector of c at t.^[31] This lift embeds the curve into TM by pairing each point on the base with its tangent vector, providing a canonical way to view the dynamics of the curve within the bundle structure.^[31] The complete lift preserves the differential structure in the sense that the projection \pi \circ Cc = c, where \pi: TM \to M is the bundle projection, and the velocity of Cc satisfies (Cc)'(t) = (c'(t), c''(t)), reflecting the second-order information along the curve. To incorporate a notion of horizontality, a linear connection \nabla on M (such as the Levi-Civita connection on a Riemannian manifold) allows for the horizontal lift of c, which is a curve \tilde{c}: I \to TM such that \pi \circ \tilde{c} = c and \tilde{c}'(t) lies in the horizontal subspace of T_{\tilde{c}(t)}(TM) defined by the connection.^[31] This horizontal lift is unique given an initial condition in the fiber and is used to parallel transport tangent vectors along the curve.^[32] For a smooth function f: M \to \mathbb{R}, the vertical lift is defined via its differential df, yielding the vertical vector field \mathrm{vert}(df) on TM with local expression \mathrm{vert}(df) = \sum_i \frac{\partial f}{\partial x^i} \frac{\partial}{\partial v^i}, where (x^i) are coordinates on M and (v^i) are the induced fiber coordinates on TM.^[31] This vector field is tangent to the fibers of TM \to M and vanishes on the zero section, capturing the directional derivative of f in the vertical directions. These lifts exhibit naturality properties under bundle morphisms: the complete and horizontal lifts of curves commute with the differential of smooth maps between manifolds, while the vertical lift of df transforms covariantly under changes of coordinates, ensuring compatibility with the tensorial nature of differentials.^[31] Such constructions are fundamental for studying geodesic flows and variational problems on TM, as they preserve first-order differentials and facilitate the extension of base geometry to the bundle.^[32]

Tangent Lifts of Maps

Given a smooth map f: M \to N between smooth manifolds M and N, the differential df: TM \to TN is the tangent lift, which induces a smooth bundle morphism between the tangent bundles TM and TN.^[33] For each point x \in M, the restriction df_x: T_x M \to T_{f(x)} N is a linear map between tangent spaces, defined by its action on tangent vectors: if v \in T_x M is the velocity vector of a smooth curve \gamma: (-\epsilon, \epsilon) \to M with \gamma(0) = x and \gamma'(0) = v, then df_x(v) = (f \circ \gamma)'(0).^[34] This construction is independent of the choice of curve and extends fiberwise to the full map df(p, v) = (f(p), df_p(v)) for (p, v) \in TM.^[33] In local coordinates, suppose (x^1, \dots, x^m) are coordinates on an open set in M and (y^1, \dots, y^n) on an open set in N, with f expressed as y^j = f^j(x^1, \dots, x^m). The differential takes the explicit form

df(x, v) = \bigl( f(x), Df(x) \cdot v \bigr),

where Df(x) is the Jacobian matrix with entries \frac{\partial f^j}{\partial x^i}(x), and v = (v^1, \dots, v^m) represents coordinates of the tangent vector.^[34] This coordinate expression confirms that df is smooth, as it depends smoothly on x and linearly on v. Moreover, df is a vector bundle map over f, meaning the bundle projection \pi_{TN} \circ df = f \circ \pi_{TM}, and it acts linearly on each fiber T_x M.^[33] When f is a diffeomorphism, the tangent lift df provides a means to push forward vector fields: for a vector field X on M, the pushforward f_* X on N is defined by (f_* X)_y = df_{f^{-1}(y)} \bigl( X_{f^{-1}(y)} \bigr) for y \in N, or equivalently f_* X = df \circ (X \circ f^{-1}).^[33] This operation preserves the Lie bracket, ensuring f_* [X, Y] = [f_* X, f_* Y], and extends the notion of f-related vector fields where df_p(X_p) = (f_* X)_{f(p)}.^[34] The tangent lift thus facilitates the study of symmetries and transformations in differential geometry by transporting tangent structures consistently.

Advanced Topics

Higher-Order Tangent Bundles

Higher-order tangent bundles extend the construction of the tangent bundle to capture higher-order derivatives of curves on a manifold, providing a geometric framework for accelerations and beyond. The second-order tangent bundle T^2 M of an n-dimensional smooth manifold M is defined as the set of equivalence classes of curves on M that agree up to their second derivatives (accelerations), forming a bundle over M with fiber dimension $2n and total dimension $3n.^[35] In local coordinates (x^i) on M, elements of T^2 M are represented by triples (x, v, a) where x \in M, v denotes velocity, and a denotes acceleration, reflecting the position, first, and second derivatives of a curve.^[36] For the general k-th order, the tangent bundle T^k M consists of equivalence classes of curves agreeing up to their k-th order derivatives, yielding a bundle over M with fiber dimension k n and total dimension (k+1)n.^[37] These bundles form a tower via natural projections \pi_k: T^k M \to T^{k-1} M, which forget the highest-order derivative information, with \pi_1: T M \to M recovering the standard tangent bundle.^[38] This iterative structure allows higher-order bundles to encode k-th order Taylor expansions of curves locally. Higher-order tangent bundles are closely related to jet bundles, where T^k M is isomorphic to the jet bundle J^k(\mathbb{R}, M) of k-jets of smooth maps from \mathbb{R} to M, capturing the k-th order infinitesimal behavior of such maps. More generally, the jet bundle J^k(M, \mathbb{R}) over M parametrizes k-th order Taylor expansions of smooth functions M \to \mathbb{R}, providing a dual perspective for higher derivatives in variational problems and differential equations. On higher-order tangent bundles, nonlinear connections generalize linear connections by defining horizontal subbundles that are not necessarily integrable, enabling the splitting of tangent spaces into horizontal and vertical components for coordinate adaptations. Sprays, as homogeneous vector fields of degree one, extend geodesic sprays from the first-order case to higher orders, inducing such nonlinear connections and facilitating the study of higher-order geodesics and dynamics on T^k M.^[39]

Canonical Vector Fields

The tangent bundle TM of a smooth manifold M admits several canonical vector fields that arise naturally from its geometric structure, independent of choices like metrics or connections on the base. These fields are intrinsic to TM and play key roles in the study of flows, symmetries, and differential equations on the bundle. They are characterized by their homogeneity properties and invariance under diffeomorphisms of M, ensuring they transform appropriately under coordinate changes.^[40] One fundamental canonical vector field is the Liouville vector field, also known as the vertical or Euler vector field, which generates radial dilations along the fibers of TM. In local coordinates (x^i, v^i) on TM, where x \in M and v \in T_x M, it is given by

V(x, v) = \sum_i v^i \frac{\partial}{\partial v^i}.

This field is homogeneous of degree 1 with respect to fiberwise scaling, meaning V(\lambda x, \lambda v) = \lambda V(x, v) for \lambda > 0, and it is invariant under diffeomorphisms of M.^[40] It spans the vertical subbundle of T(TM) and is used to define the almost tangent structure on TM.^[40] Another important canonical vector field is the complete lift of a vector field X on the base manifold M to TM. For X = \sum_i X^i \frac{\partial}{\partial x^i}, its complete lift X^C is defined such that it preserves the action on complete lifts of functions, yielding the local expression

X^C(x, v) = \sum_i X^i \frac{\partial}{\partial x^i} + \sum_{i,j} v^j \frac{\partial X^i}{\partial x^j} \frac{\partial}{\partial v^i}.

This lift is natural, meaning it commutes with diffeomorphisms of M, and it extends the flow of X to a flow on TM that acts on both base points and tangent vectors variationally.^[41] Unlike the vertical lift, which only acts on fibers, the complete lift incorporates derivatives of X to capture how tangent vectors evolve along the flow of X.^[41] In the presence of an affine connection on M with Christoffel symbols \Gamma^i_{jk}, the geodesic spray provides another canonical vector field on TM, encoding the geodesics of the connection as its integral curves projected to M. It takes the form

S(x, v) = \sum_i v^i \frac{\partial}{\partial x^i} - \sum_{i,j,k} \Gamma^i_{jk}(x) v^j v^k \frac{\partial}{\partial v^i}.

This is a second-order vector field, homogeneous of degree 1, and invariant under coordinate transformations that preserve the connection; its flows generate reparametrizations of geodesics on TM.^[42] The spray is particularly significant in Riemannian geometry, where it corresponds to the Levi-Civita connection and defines the geodesic flow.^[42]

References

[1]
[PDF] 1 The Tangent bundle and vector bundle
The tangent bundle TX is formed by gluing tangent spaces at each point of a manifold X, and is a vector bundle. It is the triple (TX, X, π).Missing: geometry | Show results with:geometry
[2]
[PDF] Differential Geometry
∂xi. ∂yk. (ψ(x))dyk x . Definition. The tangent bundle of M, denoted by TM is defined as follows: TM := a x∈M. TxM. Proposition. Let π : TM → M be defined by π( ...
[3]
[PDF] Differential geometry Lecture 7: The tangent bundle
May 12, 2020 · The tangent bundle TM of a smooth manifold M is defined as G(TpM → M) and is a vector bundle of rank n.
[4]
[PDF] Chapter 2 Bundles
In standard bundle terminology, the tangent bundle is an example of a smooth vector bundle of rank n over M. We call M the base of this bundle, and the 2n- ...
[5]
[PDF] MATH4540 Mini Project 1 Vector Bundles of Surfaces
This leads us to the notion of the tangent bundle. Definition 1. Given a surface S, the tangent bundle of the surface TS = {(p, v) : p ∈ S, v ∈ TpS}.<|control11|><|separator|>
[6]
[PDF] THE WORKS OF CHARLES EHRESMANN ON CONNECTIONS - HAL
Around 1950, Charles Ehresmann introduced connections on a fibre bundle and, when the bundle has a Lie group as structure group, connection forms on the ...Missing: fiber history
[7]
None
Below is a merged summary of the tangent bundle definition from John M. Lee's *Introduction to Smooth Manifolds*, consolidating all information from the provided segments into a single, comprehensive response. To maximize detail and clarity, I will use a structured format with a narrative overview followed by a table summarizing key details. This ensures all information is retained while avoiding redundancy.
[8]
[PDF] FIBER BUNDLES AND VECTOR BUNDLES These notes, written for ...
A local trivialization of a fiber bundle is analogous to a chart in a smooth manifold. Notice, though, that a topological manifold has no intrinsic notion of ...
[9]
[PDF] Class 6. The tangent bundle
Sep 17, 2024 · Rn represent the same smooth section of the tangent bundle, then we can read off the transition functions by comparing the formula we have just ...
[10]
[PDF] 2. The Tangent bundle and projective bundle - UCSD Math
The transition functions are given by the Jacobian of the coordinate change. Thus the tangent bundle is a bundle.
[11]
[PDF] Chapter 3 Manifolds, Tangent Spaces, Cotangent ... - UPenn CIS
The maps gUV (p) are called the transition functions of the tangent bundle. All these ingredients are part of being a vector bundle. For more on bundles, see ...
[12]
[PDF] Chapter 7 VECTOR BUNDLES - LSU Math
Sections of the tangent bundle are used to integrate, solve a differential equation. Sec- tions of the cotangent bundle are used to compute definite integrals.Missing: geometry | Show results with:geometry
[13]
[PDF] Smooth Vector Bundles
With this smooth structure on TM, the projection π: TM −→M defines a smooth real vector bundle of rank m, called the tangent bundle of M. Definition 7.6. A ...
[14]
tangent bundle in nLab
### Summary of Topological Properties of the Tangent Bundle of a Smooth Manifold
[15]
[PDF] Characteristic Classes - University of Oregon
The tangent bundle TRPm of real projective space is orientable if and only if m is odd; TRPm is spin if and only if m . 3 mod 4. If m . 3 mod 4, there are two ...Missing: non- | Show results with:non-
[16]
[PDF] Immersions of manifolds and homotopy theory - Stanford Mathematics
Jun 30, 2022 · homotopic to a constant map, and the tangent bundle of Euclidean space is trivial, we have the following corollary of Theorem 2. 5. Page 6 ...
[17]
[PDF] To motivate the definition of a vector bundle let us consider tangent ...
To motivate the definition of a vector bundle let us consider tangent vectors to the unit 2 sphere S2 in R3 . At each point x ∈ S2 there is a tangent plane ...
[18]
[PDF] lie groups and lie algebras womp 2007 - UChicago Math
If G is a Lie group, then the tangent bundle TG is always trivializable. Proof. In fact, any vector anywhere on G determines a unique “left-invariant vector ...
[19]
[PDF] Differential Geometry II - Institut für Mathematik - TU Berlin
Deduce that the tangent bundle of a torus. S1 × S1 is diffeomorphic to S1 ... A vector bundle E with connection is trivial if and only if there exists a parallel.
[20]
[PDF] TOPOLOGY FROM THE DIFFERENTIABLE VIEWPOINT
The concept of regular value and the theorem of Sard and Brown, which asserts that every smooth mapping has regular values, play a central role. To simplify the ...
[21]
[PDF] Fibre Bundles
A Course in Simple Homotopy Theory. 11 CONWAY. Functions of One Complex Variable. 2nd ed. 12 BEALS. Advanced Mathematical Analysis. 13 ANDERSON/FULLER.
[22]
[PDF] Lecture Notes 11
Oct 10, 2006 · We say that a smooth manifold M is orientable provided that we can orient each tangent space of M in a continuos way, i.e., we can choose an ...
[23]
[PDF] Version 2.2, November 2017 Allen Hatcher Copyright c 2003 by ...
Here are some examples of bundles built using clutching functions: 1. [E,11] ... Euler class being defined via the canonical orientation of eEn . Page ...
[24]
https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf
[25]
[PDF] Differential geometry Lecture 10: Dual bundles, 1-forms, and the ...
May 22, 2020 · Understanding sections in the cotangent bundle is, as for vector fields, of critical importance when studying differential geometry. Definition.
[26]
cotangent bundle in nLab
### Summary of Cotangent Bundle from nLab
[27]
When is the tangent bundle of a manifold naturally a complex ...
Sep 20, 2015 · It is well-known that the cotangent bundle of a manifold is naturally a symplectic manifold. Inspired by mirror symmetry, when is the tangent ...Can you make the cotangent bundle to a complex manifold?When is a symplectic manifold equivalent to a cotangent bundle?More results from mathoverflow.net
[28]
[PDF] lecture 2: symplectic manifolds
¶ The canonical symplectic structure on cotangent bundles. Let X be an n-dimensional smooth manifold and denote by M = T∗X its cotangent bundle. Let π : ...
[29]
[PDF] Differential Forms - MIT Mathematics
Feb 1, 2019 · will discuss the interior product operation of vector fields on differential forms, a general- ization of the duality pairing of vector ...
[30]
Tangent and Cotangent Bundles: Differential Geometry
Tangent and Cotangent Bundles: Differential Geometry. Front Cover. Kentarō Yano, Shigeru Ishihara. University Microfilms, 1973 - Mathematics - 423 pages. From ...
[31]
Notes on the Sasaki metric - ScienceDirect.com
We survey on the geometry of the tangent bundle of a Riemannian manifold, endowed with the classical metric established by S. Sasaki 60 years ago.
[32]
[PDF] Introduction to Smooth Manifolds - Julian Chaidez
Manifolds are generalizations of curves and surfaces to many dimensions, used in math to understand space and are important in many fields.
[33]
[PDF] INTRODUCTION TO DIFFERENTIAL GEOMETRY - ETH Zürich
It begins by defining manifolds in the extrinsic setting as smooth submanifolds of Euclidean space, and then moves on to tangent spaces, submanifolds and ...
[34]
Second order tangent bundles of infinite dimensional manifolds - arXiv
Oct 29, 2003 · The second order tangent bundle T^{2}M of a smooth manifold M consists of the equivalent classes of curves on M that agree up to their acceleration.
[35]
Geometry of the second-order tangent bundles of Riemannian ...
Jul 19, 2017 · Let (M, g) be an n-dimensional Riemannian manifold and T 2 M be its second-order tangent bundle equipped with a lift metric g ~ .
[36]
[1403.3111] Higher order tangent bundles - arXiv
Mar 12, 2014 · The tangent bundle T^kM of order k, of a smooth Banach manifold M consists of all equivalent classes of curves that agree up to their accelerations of order k.
[37]
Second order tangent bundles of infinite dimensional manifolds
The second order tangent bundle T2M of a smooth manifold M consists of the equivalent classes of curves on M that agree up to their acceleration.
[38]
Sprays and homogeneous connections in the higher order geometry
Aug 10, 2025 · From a semispray that lives on the tangent bundle of order k, one can derive different nonlinear connections as it has been done by Catz [11], ...
[39]
The Geometry of Tangent Bundles: Canonical Vector Fields - Li - 2013
Apr 14, 2013 · Vector fields on tangent bundles belong to basic concepts of pure and applied differential geometry, global analysis, and mathematical physics.Abstract · Introduction · Canonical Vector Fields on... · Canonical Vector Fields...
[40]
Prolongations of tensor fields and connections to tangent bundles I
As an example we mention the canonical vector field on T(M) ; it is, defined to be cI = cI, where I E r1(M) is the field of identity endomorphisms. In terms ...
[41]
[PDF] Geodesic Reduction via Frame Bundle Geometry - arXiv
Feb 22, 2010 · The geodesic spray Z corresponding to ∇ is a second-order vector field on the tangent bundle TM with the property that the projection of its ...