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One-form (differential geometry)
One-form (differential geometry)
One-form (differential geometry)
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In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle.[1] Equivalently, a one-form on a manifold is a smooth mapping of the total space of the tangent bundle of to whose restriction to each fibre is a linear functional on the tangent space.[2] Let be an open subset of and . Then

defines a one-form . is a covector.

Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates: where the are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field.

Examples

[edit]

The most basic non-trivial differential one-form is the "change in angle" form This is defined as the derivative of the angle "function" (which is only defined up to an additive constant), which can be explicitly defined in terms of the atan2 function. Taking the derivative yields the following formula for the total derivative: While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative -axis – which reflects the fact that angle cannot be continuously defined, this derivative is continuously defined except at the origin, reflecting the fact that infinitesimal (and indeed local) changes in angle can be defined everywhere except the origin. Integrating this derivative along a path gives the total change in angle over the path, and integrating over a closed loop gives the winding number times

In the language of differential geometry, this derivative is a one-form on the punctured plane. It is closed (its exterior derivative is zero) but not exact, meaning that it is not the derivative of a 0-form (that is, a function): the angle is not a globally defined smooth function on the entire punctured plane. In fact, this form generates the first de Rham cohomology of the punctured plane. This is the most basic example of such a form, and it is fundamental in differential geometry.

Differential of a function

[edit]
Main article: Differential of a function

Let be open (for example, an interval ), and consider a differentiable function with derivative The differential assigns to each point a linear map from the tangent space to the real numbers. In this case, each tangent space is naturally identifiable with the real number line, and the linear map in question is given by scaling by This is the simplest example of a differential (one-)form.

See also

[edit]
  • Differential form – Expression that may be integrated over a region
  • Inner product – Vector space with generalized dot product
  • Reciprocal lattice – Fourier transform of a real-space lattice, important in solid-state physics
  • Tensor – Algebraic object with geometric applications

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

One-form (differential geometry)

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from Grokipedia
In differential geometry, a one-form on a smooth manifold MM is a smooth section of the T∗MT^*M, which assigns to each point p∈Mp \in M a linear functional ωp:TpM→R\omega_p: T_pM \to \mathbb{R} on the tangent space TpMT_pM, known as the cotangent space at pp. This structure generalizes the notion of differentials like dfdf for a function ff, where dfp(v)=v(f)df_p(v) = v(f) for v∈TpMv \in T_pM. The exterior derivative dd applied to a smooth scalar function ff yields the one-form dfdf. This provides a way to measure infinitesimal changes or "work" along directions in the manifold. One-forms are fundamental objects in the theory of differential forms, enabling the formulation of integration over paths and surfaces without relying on a specific coordinate system; the integration of one-forms generalizes the signed definite integral from single-variable calculus to oriented paths in higher dimensions, where the integral depends on the path's orientation. In local coordinates (x1,…,xn)(x^1, \dots, x^n) on an U⊂MU \subset M, a one-form ω\omega can be expressed as ω=∑i=1nfi dxi\omega = \sum_{i=1}^n f_i \, dx^i, where fi:U→Rf_i: U \to \mathbb{R} are smooth functions and {dxi}\{dx^i\} form the dual basis to the coordinate vector fields {∂/∂xi}\{\partial/\partial x^i\}, satisfying dxi(∂/∂xj)=δjidx^i(\partial/\partial x^j) = \delta^i_j. Note that this basis {dxi}\{dx^i\} is local, valid only within the coordinate patch UU, and may not extend to a global basis on the entire manifold due to topological reasons. For example, on the circle S1S^1, the one-form dθd\theta is globally defined as a smooth section of the cotangent bundle, but the angular coordinate function θ\theta itself is not globally defined as a smooth function on S1S^1, requiring multiple coordinate charts to cover the manifold. The coordinate one-forms dxidx^i are the exterior derivatives (differentials) of the coordinate functions xix^i. For a v=∑vi∂/∂xiv = \sum v^i \partial/\partial x^i at p∈Up \in U, the evaluation is ωp(v)=∑fi(p)vi(p)\omega_p(v) = \sum f_i(p) v^i(p), yielding a that represents the pairing between the covector and vector. This coordinate-free duality ensures that one-forms transform contravariantly under changes of coordinates, preserving their intrinsic geometric meaning. One-forms are dual to vector fields in the sense that the space of smooth one-forms Ω1(M)\Omega^1(M) acts on the space of smooth vector fields X(M)\mathfrak{X}(M) to produce smooth functions on MM, via pointwise application ω(v):M→R\omega(v): M \to \mathbb{R}. In Euclidean space Rn\mathbb{R}^n, this duality identifies one-forms with vector fields via the standard dot product, where ωx(v)=F(x)⋅v\omega_x(v) = F(x) \cdot v for some vector field FF, though on general manifolds, no such canonical identification exists without additional structure like a metric. Pullbacks of one-forms under smooth maps f:N→Mf: N \to M are defined by f∗ωq(w)=ωf(q)(dfq(w))f^*\omega_q(w) = \omega_{f(q)}(df_q(w)) for q∈Nq \in N and w∈TqNw \in T_qN; unlike pushforwards of one-forms, which are only defined when the map is a diffeomorphism, pullbacks are always defined for any smooth map between manifolds, facilitating computations in geometry and physics. Beyond their algebraic role, one-forms play a central part in on manifolds, as they are the building blocks for higher-degree differential forms and the dd, which maps one-forms to two-forms and satisfies d2=0d^2 = 0. This leads to , where closed one-forms (those with dω=0d\omega = 0) that are not exact (ω≠df\omega \neq df) capture topological invariants, generalizing the to higher dimensions. In applications, one-forms describe phenomena like electromagnetic potentials or line integrals in physics, underscoring their utility in both and applied sciences.

Definition

Covectors at a point

In , the tangent space TpMT_p M at a point pp on a smooth manifold MM is a finite-dimensional real whose elements are tangent vectors at pp. A covector at pp, also known as a dual vector, is an element of the (TpM)∗(T_p M)^*, which consists of all continuous linear functionals from TpMT_p M to R\mathbb{R}. The action of a covector ω∈(TpM)∗\omega \in (T_p M)^* on a tangent vector X∈TpMX \in T_p M is given by the duality pairing ⟨ω,X⟩\langle \omega, X \rangle, which produces a real number and satisfies linearity in XX: ⟨ω,aX+bY⟩=a⟨ω,X⟩+b⟨ω,Y⟩\langle \omega, aX + bY \rangle = a \langle \omega, X \rangle + b \langle \omega, Y \rangle for scalars a,b∈Ra, b \in \mathbb{R} and vectors X,Y∈TpMX, Y \in T_p M. Thus, a one-form ω\omega at pp explicitly assigns to each tangent vector at pp a real number in a manner linear with respect to vector addition and scalar multiplication. This duality pairing defines a bilinear map (TpM)×(TpM)∗→R(T_p M) \times (T_p M)^* \to \mathbb{R}, meaning it is linear in each argument separately: ⟨aω1+bω2,X⟩=a⟨ω1,X⟩+b⟨ω2,X⟩\langle a\omega_1 + b\omega_2, X \rangle = a \langle \omega_1, X \rangle + b \langle \omega_2, X \rangle and similarly for the second argument. For the specific case of the Euclidean space Rn\mathbb{R}^n, consider the standard basis vectors eje_j for j=1,…,nj = 1, \dots, n, where eje_j has a 1 in the jj-th position and 0 elsewhere. The standard dual basis {εi}i=1n\{\varepsilon^i\}_{i=1}^n in (Rn)∗(\mathbb{R}^n)^* is defined such that εi(ej)=δji\varepsilon^i(e_j) = \delta^i_j, with δji\delta^i_j the Kronecker delta (equal to 1 if i=ji = j and 0 otherwise). Any covector ω∈(Rn)∗\omega \in (\mathbb{R}^n)^* can then be expressed as ω=∑i=1nωiεi\omega = \sum_{i=1}^n \omega_i \varepsilon^i, where ωi=⟨ω,ei⟩\omega_i = \langle \omega, e_i \rangle.

One-form fields on manifolds

A one-form field on a smooth manifold MM is defined as a smooth assignment that associates to each point p∈Mp \in M a covector ωp∈Tp∗M\omega_p \in T_p^* M, where Tp∗MT_p^* M denotes the at pp. This extends the local concept of covectors at individual points to a global structure over the entire manifold. Formally, such a field ω\omega is a smooth section of the T∗MT^* M, meaning ω:M→T∗M\omega: M \to T^* M satisfies π∘ω=IdM\pi \circ \omega = \mathrm{Id}_M, where π:T∗M→M\pi: T^* M \to M is the bundle projection. The T∗MT^* M is constructed as the ⋃p∈MTp∗M\bigcup_{p \in M} T_p^* M, forming a smooth of rank dim⁡M\dim M over MM. Each fiber Tp∗MT_p^* M consists of all real-linear functionals on the TpMT_p M, and the bundle's is induced by local trivializations compatible with charts on MM. The projection π\pi maps each covector in T∗MT^* M to its base point in MM, ensuring that sections like ω\omega vary smoothly across the manifold. Smoothness of the one-form field ω\omega requires that it is a C∞C^\infty-section of T∗MT^* M, meaning the map p↦ωpp \mapsto \omega_p is smooth with respect to the bundle's . In this framework, ω\omega acts on a smooth XX on MM by , yielding the smooth function ω(X):M→R\omega(X): M \to \mathbb{R} defined by ω(X)(p)=ωp(Xp)\omega(X)(p) = \omega_p(X_p) for each p∈Mp \in M. This pairing highlights the duality between one-form fields and s, producing scalar functions that capture directional derivatives in a coordinate-free manner.

Local Representation

Expression in coordinates

In a local coordinate chart (U,x)(U, x) on a smooth manifold MM, where x=(x1,…,xn)x = (x^1, \dots, x^n) denotes the coordinate functions, a one-form ω\omega defined on UU can be expressed as ω=ωi dxi,\omega = \omega_i \, dx^i, where dxidx^i denotes the differential of the coordinate function xix^i, the dxidx^i are the coordinate one-forms forming a local basis for the Tp∗MT^*_p M at each point p∈Up \in U, the ωi\omega_i are smooth real-valued functions on UU, and summation over the repeated index ii from 1 to nn is implied (Einstein summation convention). The action of ω\omega on a tangent vector v∈TpMv \in T_p M is then ⟨ω,v⟩=ωi(p)vi\langle \omega, v \rangle = \omega_i(p) v^i, where viv^i are the components of vv with respect to the coordinate basis ∂/∂xi\partial / \partial x^i. In particular, evaluating on the coordinate basis vectors gives ⟨ω,∂/∂xj⟩=ωj\langle \omega, \partial / \partial x^j \rangle = \omega_j, which follows from the dual basis property ⟨dxi,∂/∂xj⟩=δji\langle dx^i, \partial / \partial x^j \rangle = \delta^i_j, where δji\delta^i_j is the Kronecker delta (equal to 1 if i=ji = j and 0 otherwise). The components ωi\omega_i are smooth functions evaluated at points p∈Up \in U, so ωi(p)\omega_i(p) specifies the value of the ii-th component at pp, and these components transform under changes of coordinates to ensure ω\omega remains well-defined on the manifold. As an illustration, consider Rn\mathbb{R}^n equipped with the standard coordinates x1,…,xnx^1, \dots, x^n. Here, any one-form ω\omega takes the form ω=ωi dxi\omega = \omega_i \, dx^i, where the dxidx^i are the standard differential forms satisfying dxi(∂/∂xj)=δjidx^i(\partial / \partial x^j) = \delta^i_j, and the components ωi:Rn→R\omega_i: \mathbb{R}^n \to \mathbb{R} are smooth functions.

Basis and dual basis

In a local coordinate chart (x1,…,xn)(x^1, \dots, x^n) on a smooth manifold MM, the cotangent space Tp∗MT_p^*M at a point p∈Mp \in M admits a natural basis consisting of the coordinate one-forms {dxi}i=1n\{dx^i\}_{i=1}^n, where each dxidx^i is defined by its action on the coordinate vector fields {∂/∂xj}j=1n\{\partial/\partial x^j\}_{j=1}^n that form a basis for the TpMT_pM. Specifically, these one-forms satisfy the duality relation dxi(∂∂xj)p=δji,dx^i\left( \frac{\partial}{\partial x^j} \right)_p = \delta^i_j, where δji\delta^i_j is the , equal to 1 if i=ji = j and 0 otherwise. This basis {dxi}\{dx^i\} is linearly independent because the matrix of pairings with the vector basis is the identity, ensuring that any linear dependence relation among the dxidx^i would imply a contradiction in the spanning properties of the dual spaces. Moreover, the dual basis is unique: given the fixed vector basis {∂/∂xj}\{\partial/\partial x^j\}, there is exactly one set of one-forms satisfying the Kronecker delta condition, as determined by the non-degeneracy of the duality pairing between TpMT_pM and Tp∗MT_p^*M. More generally, for any smooth local frame of vector fields {ei}i=1n\{e_i\}_{i=1}^n on an open subset of MM that forms a basis for TpMT_pM at each point pp in the domain, there exists a unique dual coframe {θi}i=1n\{\theta^i\}_{i=1}^n of one-forms such that θi(ej)=δji\theta^i(e_j) = \delta^i_j for all i,ji, j. This dual coframe provides a basis for the space of one-forms in the region, allowing arbitrary one-forms to be expressed as linear combinations ∑ifiθi\sum_i f_i \theta^i with smooth coefficient functions fif_i. The of {θi}\{\theta^i\} follows from the same identity matrix pairing with {ei}\{e_i\}, preventing non-trivial relations among them, while uniqueness arises because the conditions θi(ej)=δji\theta^i(e_j) = \delta^i_j uniquely solve for the θi\theta^i in the . In the special case where the frame {ei}\{e_i\} coincides with the coordinate basis, the dual coframe reduces to {dxi}\{dx^i\}. These dual bases and coframes are intrinsically linked to the frame bundle of the manifold, where local frames correspond to sections over coordinate charts, and the dual coframes ensure a consistent trivialization of the cotangent bundle in those charts. This structure facilitates coordinate-free descriptions of geometric objects while allowing computations in specific bases.

Properties

Linearity and bilinearity

A one-form, or covector, at a point pp on a smooth manifold MM is defined as a linear map ω:TpM→R\omega: T_p M \to \mathbb{R} from the tangent space TpMT_p M to the real numbers, where linearity means that for any tangent vectors X,Y∈TpMX, Y \in T_p M and scalars a,b∈Ra, b \in \mathbb{R}, ω(aX+bY)=aω(X)+bω(Y).\omega(aX + bY) = a \omega(X) + b \omega(Y). This property ensures that the one-form evaluates tangent vectors in a homogeneous and additive manner, preserving the vector space structure of TpMT_p M. The collection of all one-forms at pp, denoted Tp∗MT_p^* M, forms a vector space under pointwise addition and scalar multiplication. Specifically, for one-forms ω,η∈Tp∗M\omega, \eta \in T_p^* M and scalar f∈Rf \in \mathbb{R}, the sum and scalar multiple are defined by (ω+η)(X)=ω(X)+η(X),(fω)(X)=f ω(X)(\omega + \eta)(X) = \omega(X) + \eta(X), \quad (f \omega)(X) = f \, \omega(X) for all X∈TpMX \in T_p M. This endows Tp∗MT_p^* M with the structure of a vector space isomorphic to the dual space (TpM)∗(T_p M)^*, with dimension equal to that of MM. The pairing between one-forms and tangent vectors, often denoted ⟨ω,X⟩=ω(X)\langle \omega, X \rangle = \omega(X), is bilinear, meaning it is linear in each argument separately. Thus, for scalars f,g∈Rf, g \in \mathbb{R} and vectors X,Y∈TpMX, Y \in T_p M, ⟨fω+gη,X⟩=f⟨ω,X⟩+g⟨η,X⟩=⟨ω,fX+gY⟩.\langle f \omega + g \eta, X \rangle = f \langle \omega, X \rangle + g \langle \eta, X \rangle = \langle \omega, f X + g Y \rangle. This bilinearity underpins the algebraic interactions between covectors and vectors in differential geometry.

Tensor transformation rules

One-forms, as (0,1)-tensors, exhibit a covariant transformation law under changes of coordinates on a manifold. Consider a coordinate transformation from {xj}\{x^j\} to {x′i}\{x'^i\}, with Jacobian matrix elements Jji=∂x′i∂xjJ^i_j = \frac{\partial x'^i}{\partial x^j}. The components ωi\omega_i of a one-form field ω\omega in the original coordinates transform to components ωi′\omega'_i in the new coordinates via ωi′=∂xj∂x′iωj,\omega'_i = \frac{\partial x^j}{\partial x'^i} \omega_j, where the summation over jj is implied. This law ensures that the one-form behaves consistently as a multilinear map across coordinate systems, confirming its tensorial character. The transformation law arises from the requirement that the pairing ⟨ω,X⟩\langle \omega, X \rangle between a one-form ω\omega and a tangent vector XX remains invariant under coordinate changes. In the original coordinates, ⟨ω,X⟩=ωjXj\langle \omega, X \rangle = \omega_j X^j. Under the transformation, the vector components change contravariantly as X′i=∂x′i∂xjXjX'^i = \frac{\partial x'^i}{\partial x^j} X^j. Applying the chain rule to preserve the scalar value of the pairing yields ⟨ω′,X′⟩=ωi′X′i=ωjXj\langle \omega', X' \rangle = \omega'_i X'^i = \omega_j X^j, which rearranges to the covariant law for ωi′\omega'_i. This derivation underscores the dual relationship between one-forms and vectors, with the transformation matrix for one-forms being the inverse of that for vectors. In contrast to contravariant vectors, which transform with the direct Jacobian ∂x′i∂xj\frac{\partial x'^i}{\partial x^j}, one-forms transform with its inverse ∂xj∂x′i\frac{\partial x^j}{\partial x'^i}, reflecting their role in the . This distinction is fundamental to tensor analysis, where the transformation properties dictate how geometric objects are represented locally. For a concrete illustration in R2\mathbb{R}^2, consider the standard one-form dxdx under a of coordinates by θ\theta, where the new coordinates satisfy x=x′cos⁡θ−y′sin⁡θx = x' \cos \theta - y' \sin \theta and y=x′sin⁡θ+y′cos⁡θy = x' \sin \theta + y' \cos \theta. The partial derivatives are ∂x∂x′=cos⁡θ\frac{\partial x}{\partial x'} = \cos \theta, ∂x∂y′=−sin⁡θ\frac{\partial x}{\partial y'} = -\sin \theta, ∂y∂x′=sin⁡θ\frac{\partial y}{\partial x'} = \sin \theta, and ∂y∂y′=cos⁡θ\frac{\partial y}{\partial y'} = \cos \theta. Since dxdx has components (1,0)(1, 0) in (x,y)(x, y), its components in (x′,y′)(x', y') are ωx′′=cos⁡θ\omega'_{x'} = \cos \theta and ωy′′=−sin⁡θ\omega'_{y'} = -\sin \theta, so expressing dxdx in the new basis gives dx=cos⁡θ dx′−sin⁡θ dy′dx = \cos \theta \, dx' - \sin \theta \, dy' via the chain rule, demonstrating the covariant mixing. This example highlights how basis one-forms adjust to maintain the form's action on rotated vectors.

Construction from Functions

Exterior derivative of scalar functions

A fundamental construction of one-forms arises from smooth real-valued functions on a manifold via the . For a smooth function f:M→Rf: M \to \mathbb{R} on a smooth manifold MM, the dfdf is defined as the one-form satisfying df(X)=X(f)df(X) = X(f) for any XX on MM, where X(f)X(f) denotes the of ff along XX. In local coordinates (x1,…,xn)(x^1, \dots, x^n) on MM, the one-form dfdf takes the expression df=∑i=1n∂f∂xi dxi,df = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \, dx^i, where {dxi}\{dx^i\} form the coordinate basis of one-forms. This local expression is coordinate-independent; under a change of coordinates, the chain rule ensures that the components transform appropriately, yielding the same one-form dfdf. As a one-form, dfdf is inherently alternating, though this property is trivial without extension to higher forms. It is closed, meaning its exterior derivative vanishes: d(df)=0d(df) = 0. Moreover, df=0df = 0 ff is a on MM. The kernel of dfdf at a point consists of tangent vectors tangent to the level sets of ff, i.e., those along which ff does not vary. If df≠0df \neq 0 at a point, then by the , ff is locally invertible near that point, serving as a local coordinate function.

Covariant derivative in Riemannian manifolds

In a (M,g)(M, g), the of a smooth scalar function f:M→Rf: M \to \mathbb{R} is defined as the one-form ∇f\nabla f satisfying (∇f)p(X)=X(f)(\nabla f)_p(X) = X(f) for all tangent vectors X∈TpMX \in T_p M. This construction arises from the , which is torsion-free, ensuring that ∇f=df\nabla f = df, the of ff. In local coordinates (xi)(x^i), the components of ∇f\nabla f are given by (∇f)i=∂f∂xi(\nabla f)_i = \frac{\partial f}{\partial x^i}, so ∇f=∂f∂xi dxi\nabla f = \frac{\partial f}{\partial x^i} \, dx^i. The associated gradient ∇f\nabla f (often denoted grad⁡f\operatorname{grad} f) satisfies g(grad⁡f,X)=(∇f)(X)g(\operatorname{grad} f, X) = (\nabla f)(X) for all X∈TMX \in TM, or equivalently, grad⁡f=g−1(∇f,⋅)\operatorname{grad} f = g^{-1}(\nabla f, \cdot), where g−1g^{-1} raises the index using the inverse metric. This emphasizes the one-form nature of ∇f\nabla f, which encodes the directional derivatives without reference to the metric in its primary definition. The extends naturally to one-forms on (M,g)(M, g). For a one-form ω\omega and vector fields X,YX, Y, it is the one-form ∇Xω\nabla_X \omega defined by (∇Xω)(Y)=X(ω(Y))−ω(∇XY).(\nabla_X \omega)(Y) = X(\omega(Y)) - \omega(\nabla_X Y). In local coordinates, if ω=ωi dxi\omega = \omega_i \, dx^i, the components are (∇∂jω)i=∂ωi∂xj−Γjikωk,(\nabla_{\partial_j} \omega)_i = \frac{\partial \omega_i}{\partial x^j} - \Gamma^k_{j i} \omega_k, where Γjik\Gamma^k_{j i} are the of the . For the specific one-form ∇f=df\nabla f = df, this yields ∇∂j(df)=(∂2f∂xj∂xi−Γjik∂f∂xk)dxi,\nabla_{\partial_j} (df) = \left( \frac{\partial^2 f}{\partial x^j \partial x^i} - \Gamma^k_{j i} \frac{\partial f}{\partial x^k} \right) dx^i, which simplifies for the scalar ff itself but incorporates the connection terms when differentiating the resulting one-form.

Integration and Applications

Line integrals along curves

In differential geometry, the integration of a one-form over a curve provides a fundamental way to associate a scalar value to the pairing of the form with the curve's tangent vectors along its path. Consider a smooth manifold MM and a smooth one-form ω\omega on MM. Let γ:I→M\gamma: I \to M be a smooth parametrized , where I⊂RI \subset \mathbb{R} is a closed interval [a,b][a, b], with γ(a)\gamma(a) and γ(b)\gamma(b) as the initial and terminal points, respectively. The of ω\omega along γ\gamma is defined as ∫γω=∫abωγ(t)(γ′(t)) dt,\int_{\gamma} \omega = \int_{a}^{b} \omega_{\gamma(t)}(\gamma'(t)) \, dt, where γ′(t)=dγdt(t)∈Tγ(t)M\gamma'(t) = \frac{d\gamma}{dt}(t) \in T_{\gamma(t)}M is the velocity vector, and ωγ(t)\omega_{\gamma(t)} is the cotangent vector at γ(t)\gamma(t) evaluated on it, yielding a at each tt. This definition arises from the of ω\omega to II via γ\gamma, reducing the integral to that of a smooth function on the interval. The value of the integral is independent of the specific parametrization of γ\gamma, provided the reparametrization preserves the orientation of the . Specifically, if ϕ:[a,b]→I\phi: [a, b] \to I is a with ϕ′(s)>0\phi'(s) > 0 for all ss, and γ~=γ∘ϕ\tilde{\gamma} = \gamma \circ \phi, then ∫γ~ω=∫γω\int_{\tilde{\gamma}} \omega = \int_{\gamma} \omega. However, reversing the orientation (e.g., via ϕ′\phi' negative) negates the , reflecting the oriented nature of one-forms. This invariance ensures the depends only on the image and its direction, not the speed of traversal. A simple example occurs on the real line R\mathbb{R}, where the standard one-form ω=dx\omega = dx pairs with the d/dtd/dt to give 1. For a γ(t)=x(t)\gamma(t) = x(t) with x′(t)>0x'(t) > 0, the ∫γdx=∫abx′(t) dt=x(b)−x(a)\int_{\gamma} dx = \int_{a}^{b} x'(t) \, dt = x(b) - x(a) computes the net displacement. If the curve is parametrized by and ω=ds\omega = ds (the oriented differential arc length element), the integral yields the signed arc length traversed. For one-forms, a fundamental result simplifies computation. If ω=df\omega = df is the differential of a smooth function f:M→Rf: M \to \mathbb{R}, then the depends only on the endpoints: ∫γdf=f(γ(b))−f(γ(a))\int_{\gamma} df = f(\gamma(b)) - f(\gamma(a)). This generalizes the and underscores path independence for exact forms, contrasting with the general dependence on the curve's .

Relation to vector fields via metrics

In a Riemannian manifold (M,g)(M, g), the metric tensor gg induces a bundle isomorphism between the tangent bundle TMTM and the cotangent bundle T∗MT^*M, known as the musical isomorphisms. These operators identify smooth vector fields with smooth one-forms, allowing concepts from one to be translated to the other via the metric's inner product structure. The flat operator ♭:X(M)→Ω1(M)\flat: \mathfrak{X}(M) \to \Omega^1(M) maps a vector field XX to the one-form ω=X♭\omega = X^\flat defined by ω(Y)=g(X,Y)\omega(Y) = g(X, Y) for all vector fields YY. Its inverse, the sharp operator ♯:Ω1(M)→X(M)\sharp: \Omega^1(M) \to \mathfrak{X}(M), maps a one-form ω\omega to the vector field X=ω♯X = \omega^\sharp satisfying g(X,Y)=ω(Y)g(X, Y) = \omega(Y). In local coordinates where X=Xi∂iX = X^i \partial_i and ω=ωj dxj\omega = \omega_j \, dx^j, these are expressed as ωj=gijXi\omega_j = g_{ij} X^i and Xi=gijωjX^i = g^{ij} \omega_j, with gijg^{ij} the inverse metric components. This identification preserves key geometric operations, including integration along . For a γ:[a,b]→M\gamma: [a, b] \to M and one-form ω=X♭\omega = X^\flat, the satisfies ∫γω=∫abω(γ˙(t)) dt=∫abg(X(γ(t)),γ˙(t)) dt,\int_\gamma \omega = \int_a^b \omega(\dot{\gamma}(t)) \, dt = \int_a^b g(X(\gamma(t)), \dot{\gamma}(t)) \, dt, linking the pairing of the one-form with vectors to the metric pairing of the corresponding . A prominent example is the of a smooth function f:M→Rf: M \to \mathbb{R}, defined as the ∇f=(df)♯\nabla f = (df)^\sharp, where dfdf is the one-form. In coordinates, its components are (∇f)i=gij∂jf(\nabla f)^i = g^{ij} \partial_j f, satisfying g(∇f,Y)=df(Y)=Y(f)g(\nabla f, Y) = df(Y) = Y(f) for any YY. This construction uses the sharp operator to associate directional derivatives with metric-compatible s.

References

  1. https://math.mit.edu/classes/18.101/fa07/pub/vectorfields.pdf
  2. https://courses.cms.caltech.edu/cs177/hmw/Hmw3.pdf
  3. https://www.math.ucla.edu/~tao/preprints/forms.pdf
  4. https://spot.colorado.edu/~jnc/lecture1.pdf
  5. https://math.uchicago.edu/~dannyc/courses/differential_topology_2016/differential_forms_notes.pdf
  6. http://www.damtp.cam.ac.uk/user/tong/gr/two.pdf
  7. https://math.stackexchange.com/questions/4052553/why-are-pull-backs-always-defined-for-1-forms-but-not-push-forwards
  8. https://julianchaidez.net/materials/reu/lee_smooth_manifolds.pdf
  9. https://www.cimat.mx/~gil/docencia/2013/topologia_variedades/spivak-calculus-on-manifolds.pdf
  10. https://idv.sinica.edu.tw/ftliang/diff_geom/%2Adiff_geometry%28I%29/10.16/covector1.pdf
  11. https://www-users.cse.umn.edu/~garrett/m/algebra/notes_2023-24/25.pdf
  12. https://people.math.osu.edu/gerlach.1/math5451/TensorCalculus/DualOfAVectorSpace.pdf
  13. https://www.math.utoronto.ca/mgualt/courses/18-367/docs/DiffGeomNotes-10.pdf
  14. http://staff.ustc.edu.cn/~wangzuoq/Courses/13F-Lie/Notes/Lec%2003.pdf
  15. https://www.math.uni-hamburg.de/home/lindemann/material/DG2020L10_slides.pdf
  16. https://www.mathematik.hu-berlin.de/~wendl/pub/connections_chapter2_2.pdf
  17. https://e.math.cornell.edu/people/mazurowski/DifferentialFormsNotes.pdf
  18. https://www.cmu.edu/biolphys/deserno/pdf/diff_geom.pdf
  19. https://www.math.purdue.edu/~arapura/preprints/diffforms.pdf
  20. https://math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf
  21. https://people.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf
  22. https://math.mit.edu/~hrm/palestine/lee-smooth-manifolds.pdf
  23. https://math.arizona.edu/~faris/methodsweb/manifold.pdf
  24. https://dec41.user.srcf.net/notes/III_M/differential_geometry_trim.pdf
  25. https://webhomes.maths.ed.ac.uk/~v1ranick/papers/leeriemm.pdf
  26. http://www.damtp.cam.ac.uk/user/tong/gr/three.pdf
  27. https://www.seas.upenn.edu/~amyers/DualBasis.pdf
  28. https://www.math.ubc.ca/~feldman/m321/intOnManflds.pdf
  29. https://web.math.utk.edu/~freire/teaching/m448s17/DifferentialForms.pdf
  30. http://staff.ustc.edu.cn/~wangzuoq/Courses/24S-RiemGeom/Notes/Lec02.pdf
  31. https://web.stanford.edu/~lindrew/math215C.pdf
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