Cotangent space

In differential geometry, the cotangent space at a point p on a smooth manifold M, denoted T_p^* M, is the dual vector space to the tangent space T_p M, consisting of all real-valued linear functionals (covectors) on T_p M.^[1]^[2] Its dimension equals that of M, and elements include differentials df_p of smooth functions f: M \to \mathbb{R}, defined by \langle df_p, v \rangle = v(f) for tangent vectors v \in T_p M.^[1]^[3] The cotangent space can be constructed algebraically as the quotient m_p / m_p^2, where m_p is the ideal of germs of smooth functions vanishing at p, providing a coordinate-free perspective tied to the manifold's structure sheaf.^[3]^[2] In local coordinates (x^1, \dots, x^n) around p, it admits a basis \{ [dx](/page/dx)^1_p, \dots, [dx](/page/dx)^n_p \}, where each [dx](/page/dx)^i_p is the covector satisfying \langle [dx](/page/dx)^i_p, \partial / \partial x^j_p \rangle = \delta^i_j, dual to the standard basis of T_p M.^[1]^[3] This duality extends to smooth maps between manifolds, via the cotangent map T_p^* F, the transpose of the differential T_p F, which pulls back covectors and preserves the chain rule for differentials.^[1] The collection of all cotangent spaces over M forms the cotangent bundle T^* M, a vector bundle of rank equal to \dim M, whose sections are smooth 1-forms, foundational for exterior algebra and integration on manifolds.^[1]^[2] Cotangent spaces underpin key concepts like symplectic geometry, where the canonical symplectic form on T^* M arises from the pairing with tangent vectors, and play a central role in variational calculus and Hamiltonian mechanics.^[1] They also facilitate the study of tensor fields and connections on manifolds, enabling precise descriptions of curvature and parallel transport.^[2]

Definition and Construction

As dual vector space

The cotangent space at a point p on a smooth manifold M, denoted T_p^* M, is defined as the dual vector space to the tangent space T_p M. That is, T_p^* M = (T_p M)^*, consisting of all real-valued linear functionals on T_p M, or equivalently, all continuous linear maps from T_p M to \mathbb{R}.^[1]^[4] This construction endows the cotangent space with a natural vector space structure over \mathbb{R}, where addition and scalar multiplication are defined pointwise: for covectors \alpha, \beta \in T_p^* M and c \in \mathbb{R}, (\alpha + \beta)(v) = \alpha(v) + \beta(v) and (c\alpha)(v) = c \cdot \alpha(v) for all v \in T_p M.^[1] Since T_p M is a finite-dimensional real vector space of dimension n = \dim M, the cotangent space T_p^* M is also n-dimensional and isomorphic to T_p M via a choice of basis, though no canonical isomorphism exists without additional structure.^[2] If \{ \partial/\partial x^1 |_p, \dots, \partial/\partial x^n |_p \} is a basis for T_p M induced by local coordinates (x^1, \dots, x^n) near p, then the dual basis for T_p^* M is \{ dx^1 |_p, \dots, dx^n |_p \}, satisfying \langle dx^i |_p, \partial/\partial x^j |_p \rangle = \delta^i_j.^[4]^[1] Any covector \alpha \in T_p^* M can thus be expressed uniquely as \alpha = \sum_{i=1}^n a_i \, dx^i |_p, where the coefficients a_i = \langle \alpha, \partial/\partial x^i |_p \rangle.^[4] In the algebraic framework using germs of smooth functions, the cotangent space admits an explicit realization as the quotient space \mathfrak{m}_p / \mathfrak{m}_p^2, where \mathfrak{m}_p is the maximal ideal in the ring of germs of smooth functions at p consisting of those vanishing at p, and \mathfrak{m}_p^2 is the ideal generated by products of elements in \mathfrak{m}_p.^[3]^[2] This quotient is a vector space of dimension n, with basis elements dx^i |_p = x^i - x^i(p) \mod \mathfrak{m}_p^2, and the duality pairing arises naturally from the action of derivations in T_p M on these differentials.^[3] This identification underscores the cotangent space's role in linearizing first-order approximations of functions near p, as elements of T_p^* M correspond to linear parts of Taylor expansions.^[2]

On smooth manifolds

On a smooth manifold M of dimension n, the cotangent space at a point p \in M, denoted T_p^* M, is the dual vector space to the tangent space T_p M.^[5] It consists of all continuous linear functionals on T_p M, called covectors, and forms a vector space of dimension n over \mathbb{R}.^[6] The cotangent bundle T^* M is then the disjoint union \bigcup_{p \in M} T_p^* M, equipped with a natural smooth manifold structure of dimension $2n, making it a vector bundle over M.^[5] An alternative algebraic construction identifies T_p^* M with the quotient space of germs of smooth functions vanishing at p. Let \mathcal{O}_{M,p} denote the ring of germs of smooth functions at p, and let I_p be the maximal ideal of germs vanishing at p. The cotangent space is isomorphic to I_p / I_p^2, where elements are equivalence classes of germs $$ for f \in I_p, with the vector space structure induced by addition and scalar multiplication of germs.^[7] This construction arises from the universal derivation d: \mathcal{O}_{M,p} \to I_p / I_p^2, given by d(f) = [f - f(p)], which is linear and satisfies the Leibniz rule, providing a canonical isomorphism to the dual of the tangent space defined via derivations.^[7] In local coordinates, if (U, x) is a chart around p with coordinates x^1, \dots, x^n, the tangent space T_p M has basis \left\{ \frac{\partial}{\partial x^i} \big|_p \right\}_{i=1}^n. The dual basis for T_p^* M is \left\{ dx^i \big|_p \right\}_{i=1}^n, where dx^i \big|_p \left( \frac{\partial}{\partial x^j} \big|_p \right) = \delta^i_j.^[5] Any covector \omega \in T_p^* M can thus be expressed uniquely as \omega = \sum_{i=1}^n a_i \, dx^i \big|_p for coefficients a_i \in \mathbb{R}. Under a coordinate change to y^1, \dots, y^n, the basis transforms contravariantly: dy^j \big|_p = \sum_{i=1}^n \frac{\partial y^j}{\partial x^i} \big|_p \, dx^i \big|_p.^[6] This local representation highlights the role of cotangent spaces in differential geometry, as covectors naturally arise as differentials of smooth functions: for f \in C^\infty(M), the differential df_p \in T_p^* M is defined by df_p(v) = v(f) for v \in T_p M, generating T_p^* M as the image of the map d: C^\infty(M) \to T_p^* M.^[6] The pairing \langle \omega, v \rangle = \omega(v) between T_p^* M and T_p M is nondegenerate, ensuring the duality is perfect and enabling applications in forms and tensor fields.^[5]

Properties

Vector space structure

The cotangent space T_p^*M at a point p on a smooth manifold M of dimension n is a real vector space of dimension n, dual to the tangent space T_pM.^[8] As the space of linear functionals on T_pM, it inherits a natural vector space structure from the duality, making it isomorphic to \mathbb{R}^n locally.^[9] Addition of covectors \phi_p, \psi_p \in T_p^*M is defined pointwise by (\phi_p + \psi_p)(X_p) = \phi_p(X_p) + \psi_p(X_p) for all tangent vectors X_p \in T_pM. Similarly, scalar multiplication by a \in \mathbb{R} is given by (a \phi_p)(X_p) = a \cdot \phi_p(X_p). These operations ensure T_p^*M forms a vector space over \mathbb{R}, with the zero element being the zero functional.^[8]^[2] In local coordinates (x^1, \dots, x^n) around p, if \{ \partial/\partial x^i |_p \} is a basis for T_pM, then the dual basis \{ dx^i |_p \} for T_p^*M satisfies (dx^i |_p)(\partial/\partial x^j |_p) = \delta^i_j. Any covector \phi_p can be uniquely expressed as \phi_p = \sum_{i=1}^n a_i (dx^i |_p) for coefficients a_i \in \mathbb{R}, confirming linear independence and spanning.^[9]^[2] This structure extends algebraically: T_p^*M is isomorphic to the quotient m_p / m_p^2, where m_p is the maximal ideal of germs of smooth functions vanishing at p in the stalk of the structure sheaf. Addition and scalar multiplication in this quotient correspond to those on germs, modulo higher-order terms.^[2]

Canonical duality pairing

The canonical duality pairing refers to the natural bilinear map between the cotangent space T_p^*M and the tangent space T_pM at a point p on a smooth manifold M. For a covector \omega \in T_p^*M and a tangent vector v \in T_pM, the pairing is defined by evaluation as \langle \omega, v \rangle = \omega(v) \in \mathbb{R}, which is linear in each argument and induces a non-degenerate bilinear form on the finite-dimensional vector spaces.^[1]^[10] This pairing arises intrinsically from the duality structure, where T_p^*M is the dual vector space to T_pM, consisting of all continuous linear functionals on T_pM. In local coordinates (x^1, \dots, x^n) around p, if \{\partial/\partial x^i|_p\} forms a basis for T_pM, the dual basis \{dx^i_p\} for T_p^*M satisfies \langle dx^i_p, \partial/\partial x^j|_p \rangle = \delta^i_j, ensuring the pairing reproduces the Kronecker delta and provides a complete set of independent evaluations.^[1]^[3] A concrete realization of the pairing occurs through the differential of smooth functions. For a smooth function f: M \to \mathbb{R}, its differential at p, denoted df_p \in T_p^*M, acts on v \in T_pM via \langle df_p, v \rangle = v(f), which equals the directional derivative of f along v. Equivalently, if v is the velocity vector of a curve c: (-\epsilon, \epsilon) \to M with c(0) = p, then \langle df_p, v \rangle = \frac{d}{dt}\big|_{t=0} (f \circ c)(t), linking the abstract duality to the geometric action of tangent vectors as derivations on functions.^[3]^[10]^[11] The pairing extends pointwise to the cotangent bundle T^*M and tangent bundle TM, facilitating the definition of 1-forms as sections of T^*M that pair with vector fields (sections of TM) to yield smooth functions on M. This structure is foundational for tensor fields and differential forms, as it allows contraction operations and ensures compatibility with smooth maps between manifolds.^[1]^[11]

Applications to Functions and Maps

Differential of a smooth function

The differential of a smooth function f: M \to \mathbb{R} on a smooth manifold M at a point p \in M is the linear map df_p: T_p M \to \mathbb{R} defined by its action on tangent vectors. For a tangent vector v \in T_p M, represented as the derivative of a smooth curve \gamma: (-\epsilon, \epsilon) \to M with \gamma(0) = p and \gamma'(0) = v, the differential satisfies df_p(v) = \frac{d}{dt}\big|_{t=0} (f \circ \gamma)(t).^[12] This construction ensures that df_p captures the first-order approximation of how f changes along directions in the tangent space at p.^[13] As a linear functional on T_p M, df_p naturally belongs to the cotangent space T_p^* M = \Hom_{\mathbb{R}}(T_p M, \mathbb{R}), the dual vector space to the tangent space.^[12] Thus, the differential identifies smooth functions with sections of the cotangent bundle T^* M, where df is the covector field assigning df_p to each p. This duality pairing is given by \langle df_p, v \rangle = df_p(v), providing the canonical bilinear form between tangent and cotangent spaces.^[13] In local coordinates (x^1, \dots, x^n) around p, where f has expression f \circ \phi^{-1}(x^1, \dots, x^n) for a chart \phi: U \to \mathbb{R}^n, the differential takes the form

df_p = \sum_{i=1}^n \frac{\partial f}{\partial x^i}(p) \, dx^i_p,

where \{dx^i_p\} is the dual basis to the coordinate basis \{\partial/\partial x^i \big|_p\} of T_p M.^[13] This coordinate expression is independent of the choice of chart, as the transformation laws for partial derivatives and covector bases ensure consistency under smooth atlas changes.^[12] The differential satisfies linearity and the chain rule: for smooth g: \mathbb{R} \to \mathbb{R}, d(g \circ f)_p = g'(f(p)) \, df_p.^[13] It also vanishes if f is constant at p, reflecting that constant functions induce the zero covector. For example, on the circle S^1 \subset \mathbb{R}^2 parametrized by angle \theta, the height function f(\theta) = \sin \theta has differential df = \cos \theta \, d\theta, pairing with the basis vector \partial/\partial \theta to yield \cos \theta.^[12] This framework extends to higher-order differentials via higher cotangent spaces, but the first differential remains foundational for linear approximations in manifold calculus.^[13]

Pullback under smooth maps

Given a smooth map f: M \to N between smooth manifolds, the differential df_p: T_p M \to T_{f(p)} N at a point p \in M is a linear map between tangent spaces.^[14] Since the cotangent space T^*_{f(p)} N is the dual vector space to T_{f(p)} N, the linearity of df_p induces a dual (transpose) linear map (df_p)^*: T^*_{f(p)} N \to T^*_p M between cotangent spaces, often denoted f^*_p or simply the pullback at p.^[15] This map is defined by its action on covectors: for \omega \in T^*_{f(p)} N and v \in T_p M,

(f^*_p \omega)(v) = \omega(df_p v).

This construction preserves the duality pairing, as the canonical pairing \langle f^*_p \omega, v \rangle = \langle \omega, df_p v \rangle.^[16] The pullback extends naturally to 1-forms, which are smooth sections of the cotangent bundle. For a 1-form \omega on N, the pullback f^* \omega is the 1-form on M defined pointwise by (f^* \omega)_p = f^*_p (\omega_{f(p)}).^[14] In local coordinates, if \omega = \omega_\alpha \, dy^\alpha on N with coordinates y^\alpha, and f has coordinates y^\alpha = f^\alpha(x^\mu) on M with coordinates x^\mu, then

f^* \omega = \left( \omega_\alpha \circ f \right) \frac{\partial f^\alpha}{\partial x^\mu} \, dx^\mu.

This transformation rule reflects the contravariant nature of covectors under smooth maps.^[16] The pullback is linear in \omega and satisfies the chain rule: if g: N \to P is another smooth map, then (g \circ f)^* = f^* \circ g^*.^[15] Key properties include smoothness preservation: if \omega is smooth, so is f^* \omega, and compatibility with the exterior derivative, f^* (d \eta) = d (f^* \eta) for any 0-form (smooth function) \eta on N.^[14] This makes the pullback essential for transporting differential structures, such as in defining integrals over submanifolds or studying symmetries via Lie derivatives along flows, where the Lie derivative of a 1-form involves the infinitesimal pullback under the flow map.^[16] For example, consider the inclusion i: S^1 \hookrightarrow \mathbb{R}^2 of the unit circle, with the standard 1-form \theta = -y \, dx + x \, dy on \mathbb{R}^2 \setminus \{0\} (the angle form). The pullback i^* \theta = d\phi, where \phi is the angular coordinate on S^1, illustrating how the pullback captures intrinsic geometry on the submanifold.^[15]

Algebraic Extensions

Tensor products

The tensor product provides an algebraic extension of the cotangent space T_p^*M at a point p on a smooth manifold M, enabling the construction of spaces of higher-rank covariant tensors. Specifically, for vector spaces V and W, the tensor product V \otimes W is the universal vector space generated by elements v \otimes w (with v \in V, w \in W) such that the map (v, w) \mapsto v \otimes w is bilinear, and any bilinear map from V \times W to another space factors uniquely through it. When applied to cotangent spaces, the k-fold tensor power (T_p^* M)^{\otimes k} consists of all finite sums of pure tensors \omega^1 \otimes \cdots \otimes \omega^k, where each \omega^i \in T_p^* M, and forms the space of k-covariant tensors at p, namely the space (T_p^* M)^{\otimes k} of multilinear maps from (T_p M)^k to \mathbb{R}.^[17]^[18] If \{dx^i\} is a local basis for T_p^* M, then the set \{dx^{i_1} \otimes \cdots \otimes dx^{i_k} \mid 1 \leq i_j \leq \dim M\} forms a basis for (T_p^* M)^{\otimes k}, with dimension n^k where n = \dim M. The evaluation on tangent vectors is given by (\omega^1 \otimes \cdots \otimes \omega^k)(X_1, \dots, X_k) = \omega^1(X_1) \cdots \omega^k(X_k) for X_j \in T_p M, extending the duality pairing multilinearily. This structure is pointwise, so the tensor product bundle (T^* M)^{\otimes k} \to M has fibers (T_p^* M)^{\otimes k}, and smooth sections \Gamma((T^* M)^{\otimes k}) are covariant k-tensor fields on M.^[19]^[20] In the broader context of mixed tensors, the space of (r,s)-tensors at p is T_p^{r,s} M = (T_p M)^{\otimes r} \otimes (T_p^* M)^{\otimes s}, where contravariant indices arise from tensor products of tangent spaces and covariant ones from cotangent spaces. For instance, a metric tensor g on M is a smooth section of (T^* M)^{\otimes 2}, locally expressed as g = g_{ij} \, dx^i \otimes dx^j, providing a symmetric bilinear form on T_p M. Contractions, such as tracing over one contravariant and one covariant index, reduce the rank, yielding maps like \mathrm{End}(T_p M) \cong T_p M \otimes T_p^* M. These constructions underpin tensorial operations in differential geometry, including covariant differentiation and curvature computations.^[19]^[18]^[20]

Exterior powers

The k-th exterior power of the cotangent space T_p^*M at a point p on a smooth manifold M is the vector space \bigwedge^k T_p^*M, which consists of all alternating k-linear forms on the tangent space T_pM. This space is the quotient of the k-fold tensor power (T_p^*M)^{\otimes k} by the subspace of tensors that vanish upon antisymmetrization, ensuring multilinearity and antisymmetry under permutation of arguments.^[15]^[21] For an n-dimensional manifold, \bigwedge^k T_p^*M has dimension \binom{n}{k}, with a basis given by the wedge products of basis covectors, such as \{dx^{i_1} \wedge \cdots \wedge dx^{i_k} \mid 1 \leq i_1 < \cdots < i_k \leq n\} in local coordinates where \{dx^i\} is the dual basis to a tangent frame. Elements of \bigwedge^k T_p^*M can be expressed as linear combinations \omega = \sum_{I} \omega_I \, dx^I, where I ranges over increasing multi-indices and \omega_I \in \mathbb{R}. The wedge product \wedge: \bigwedge^r T_p^*M \times \bigwedge^s T_p^*M \to \bigwedge^{r+s} T_p^*M extends the tensor product to an associative, graded-commutative algebra, satisfying \alpha \wedge \beta = (-1)^{rs} \beta \wedge \alpha.^[15]^[22]^[21] The full exterior algebra \bigwedge^\bullet T_p^*M = \bigoplus_{k=0}^n \bigwedge^k T_p^*M is a graded vector space of total dimension $2^n, with \bigwedge^0 T_p^*M = \mathbb{R} (scalar multiples of the zero-form) and \bigwedge^n T_p^*M one-dimensional, spanned by the volume form dx^1 \wedge \cdots \wedge dx^n. This structure underlies the local algebraic properties of differential forms, where global k-forms on M are smooth sections of the bundle \bigwedge^k T^*M \to M whose fibers are \bigwedge^k T_p^*M. Decomposable elements, such as \ell_1 \wedge \cdots \wedge \ell_k for linearly independent covectors \ell_i, generate the space and satisfy alternation under permutations: \ell_{\sigma(1)} \wedge \cdots \wedge \ell_{\sigma(k)} = \operatorname{sgn}(\sigma) \, \ell_1 \wedge \cdots \wedge \ell_k.^[15]^[22]