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Volume form

In differential geometry, a volume form on an n-dimensional smooth orientable manifold M is defined as a nowhere-vanishing section of the top exterior power \Lambda^n T^*M, providing a canonical way to measure oriented volumes on M.^[1] Locally, in coordinates (x_1, \dots, x_n), it takes the form \omega = f \, dx_1 \wedge \cdots \wedge dx_n where f is a smooth nowhere-zero function, and under coordinate changes with positive Jacobian determinant (as chosen in an oriented atlas), it transforms by the Jacobian determinant to preserve the orientation.^[1]^[2] The existence of a volume form requires the manifold to be orientable, as it corresponds to a consistent choice of orientation across M, unique up to positive scalar multiple.^[1] On non-orientable manifolds, only densities (absolute values of volume forms) can be defined, but volume forms enable signed integration that respects the topology.^[2] For example, on \mathbb{R}^n, the standard volume form is dx_1 \wedge \cdots \wedge dx_n, generalizing the Lebesgue measure.^[3] In the context of Riemannian geometry, a volume form arises naturally from a metric g as the Riemannian volume form \sigma_g = \sqrt{|\det(g)|} \, dx_1 \wedge \cdots \wedge dx_n in local coordinates, which is invariant under isometries and used to compute geodesic volumes and curvatures.^[2] This connection links volume forms to the inner product structure, where for an oriented orthonormal frame \{e_i\}, \sigma_g = e_1^* \wedge \cdots \wedge e_n^*.^[2] Volume forms are essential for integration on manifolds, allowing the definition of \int_M f \, \omega for smooth functions f, which extends the change-of-variables theorem via pullbacks: for an orientation-preserving diffeomorphism \phi: U \to \phi(U), \int_{\phi(U)} \omega = \int_U \phi^* \omega.^[2] They underpin Stokes' theorem, \int_M d\alpha = \int_{\partial M} \alpha for (n-1)-forms \alpha, and applications in topology such as degree theory and the Gauss-Bonnet theorem, where integrals of volume forms yield Euler characteristics.^[2]^[3]

Definition and Properties

Definition

In differential geometry, a volume form on an n-dimensional smooth manifold M is defined as a smooth n-form \omega \in \Omega^n(M) that is nowhere vanishing, meaning that for every point p \in M, the evaluation \omega_p \in \Lambda^n T_p^* M is nonzero.^[4] Equivalently, there exists a basis \{v_1, \dots, v_n\} of the tangent space T_p M such that \omega_p(v_1, \dots, v_n) \neq 0.^[5] This ensures that \omega provides a consistent, pointwise notion of "volume" for parallelepipeds spanned by tangent vectors at each point. Volume forms inherit the standard algebraic properties of differential forms: they are multilinear in their arguments and antisymmetric (alternating), meaning that swapping any two input vectors changes the sign of the output.^[5] At each point p, the space of n-forms \Lambda^n T_p^* M is a one-dimensional real vector space, and a volume form serves as a generator for this top exterior power, so that any other n-form near p can be expressed as \omega multiplied by a smooth function.^[5] A canonical example is the standard volume form on \mathbb{R}^n with the usual coordinates (x^1, \dots, x^n), given by \omega = dx^1 \wedge \dots \wedge dx^n.^[5] This form assigns to the standard basis \partial/\partial x^1, \dots, \partial/\partial x^n the value $1$, measuring the "unit volume" of the coordinate parallelepiped. The concept of volume forms originated in the early 20th century within the development of differential geometry, formalized by Élie Cartan as part of his foundational work on differential forms and exterior calculus.^[6] Volume forms are intrinsically linked to manifold orientation, as their global existence requires M to be orientable.^[4]

Local Expression

In local coordinates (x^1, \dots, x^n) on an open subset U of an n-dimensional smooth manifold M, a volume form \omega \in \Omega^n(M) can be expressed as

\omega = f(x) \, dx^1 \wedge \dots \wedge dx^n,

where f: U \to \mathbb{R} is a smooth function that is nowhere vanishing (f(x) \neq 0 for all x \in U).^[7] This representation follows from the fact that any n-form on U is a multiple of the standard coordinate volume form dx^1 \wedge \dots \wedge dx^n, with the coefficient f(x) determining the scaling at each point.^[8] Under a change of coordinates given by a diffeomorphism \phi: V \to U with coordinates (y^1, \dots, y^n) on V, the volume form transforms via the pullback as

\phi^* \omega = f(\phi(y)) \det(D\phi(y)) \, dy^1 \wedge \dots \wedge dy^n,

where D\phi(y) is the Jacobian matrix of \phi at y.^[7] For unsigned volume measurements, which disregard orientation and focus on positive densities, the absolute value of the determinant is used:

|\phi^* \omega| = |f(\phi(y))| \cdot |\det(D\phi(y))| \, dy^1 \wedge \dots \wedge dy^n,

contrasting with oriented volume forms where the signed determinant preserves the orientation.^[7] This ensures the transformation law accounts for the absolute volume scaling under coordinate changes.^[8] The absolute value |f(x)| in the local expression highlights the density aspect of the volume form, where |f(x)| \, dx^1 \dots dx^n defines a positive volume element compatible with Lebesgue integration on \mathbb{R}^n, independent of the sign of f.^[7] A concrete example arises in multivariable calculus: the Jacobian determinant in the change of variables formula for multiple integrals, \int g(y) \, dy = \int (g \circ \phi)(x) |\det(D\phi(x))| \, dx, corresponds precisely to the transformation of the standard volume form dy^1 \wedge \dots \wedge dy^n under \phi, yielding the unsigned volume element for computing integrals over regions.^[8]

Orientation

Orientable Manifolds

An orientable manifold is a smooth manifold that admits an oriented atlas, where the transition maps between charts have Jacobians with positive determinants everywhere. Equivalently, it allows a consistent choice of ordered bases for the tangent spaces across the manifold, such that the change of basis matrices have positive determinants.^[9] Non-orientable manifolds, such as the Möbius strip or the Klein bottle, fail to admit such a consistent global orientation; for instance, traversing a closed loop on the Möbius strip reverses the orientation of a local basis, preventing a uniform choice without sign flips. This lack of consistent orientation implies that no global nowhere-vanishing top-degree form can be defined without inconsistencies in sign.^[10] A key theorem states that an n-dimensional smooth manifold is orientable if and only if it admits a nowhere-vanishing smooth n-form. To sketch the proof: if such an n-form \mu exists, local expressions \mu = f \, dx_1 \wedge \cdots \wedge dx_n with f > 0 define an oriented atlas by ensuring positive Jacobians on overlaps; conversely, given an oriented atlas, a partition of unity subordinate to the charts allows gluing local standard n-forms into a global nowhere-vanishing \mu, as the sum is positive and non-zero locally.^[11]^[12] On orientable manifolds, volume forms are signed n-forms that encode a specific orientation choice, enabling directed integration; in contrast, unoriented manifolds require unsigned volume densities (or pseudoforms) for measure-theoretic purposes, as signed forms cannot be defined globally without vanishing or sign inconsistencies.^[13]

Compatible Volume Forms

A nowhere-vanishing volume form \omega on an n-dimensional smooth manifold M defines an orientation on M by specifying that an ordered basis \{v_1, \dots, v_n\} of the tangent space T_pM at each point p \in M is positively oriented if \omega_p(v_1, \dots, v_n) > 0. This assignment is consistent across the manifold provided that \omega is smooth and nowhere zero, thereby providing a global choice of orientation that aligns with the equivalence classes of bases related by elements of the general linear group \mathrm{GL}^+(n, \mathbb{R}).^[14] Two nowhere-vanishing volume forms \omega and \omega' on M are compatible in the sense that they define the same orientation if and only if there exists a smooth positive function f: M \to (0, \infty) such that \omega' = f \omega. The equivalence classes of such volume forms under multiplication by positive smooth functions thus parametrize the orientations of M, with each class determining a unique orientation via the positive-basis condition. This equivalence relation ensures that the orientation is independent of the specific choice of volume form within the class, as scaling by a positive factor preserves the sign of evaluations on bases.^[14]^[4] For example, on the circle S^1 embedded as the unit circle in \mathbb{R}^2, the standard volume form d\theta (or equivalently dz in complex coordinates) defines the counterclockwise orientation, while -d\theta defines the opposite clockwise orientation; these are incompatible as no positive function relates them, but f d\theta for f > 0 remains compatible with the standard one. To verify compatibility in a coordinate atlas, consider transition maps \phi: U \to V between charts; the volume forms agree on orientations if the Jacobian determinant \det(d\phi) is positive everywhere, ensuring that the induced bases transform consistently without sign reversal.^[14]^[15]

Relation to Measures and Integration

Induced Lebesgue Measures

A volume form \omega on an n-dimensional smooth manifold M induces a Borel measure \mu_\omega on M, defined for any open set U \subset M by \mu_\omega(U) = \int_U |\omega|, where |\omega| denotes the absolute value of the volume form, ensuring the measure is non-negative and unsigned. This construction provides a natural way to assign volumes to subsets of the manifold, independent of orientation choices, and extends to the Borel \sigma-algebra generated by the open sets. In local coordinates (x^1, \dots, x^n) on M, where \omega = f \, dx^1 \wedge \cdots \wedge dx^n with f a smooth nowhere-vanishing function, the induced measure takes the form d\mu_\omega = |f| \, dx^1 \cdots dx^n. This local expression connects the manifold's measure directly to the Lebesgue measure on \mathbb{R}^n, scaled by the absolute value of the coefficient function |f|, allowing for consistent volume computations across coordinate charts. The measure \mu_\omega possesses key properties essential for analysis on manifolds: it is \sigma-additive over disjoint Borel sets, reflecting the countable additivity of integrals. Moreover, for a diffeomorphism \phi: N \to M, the induced measure \mu_{\phi^*\omega} on N is the pullback \phi^* \mu_\omega, defined such that \int_N f \, d\mu_{\phi^*\omega} = \int_M (f \circ \phi^{-1}) \, d\mu_\omega for smooth functions f with compact support. This incorporates the absolute Jacobian determinant to account for volume changes without regard to sign. These features ensure that \mu_\omega behaves as a regular Borel measure suitable for probability and integration theory on M. A canonical example occurs on \mathbb{R}^n, where the standard volume form \omega = dx^1 \wedge \cdots \wedge dx^n induces precisely the Lebesgue measure \lambda, satisfying \lambda(U) = \int_U dx^1 \cdots dx^n for Borel sets U \subset \mathbb{R}^n. This correspondence highlights how volume forms generalize familiar Euclidean notions of volume to abstract manifolds.

Integration over Manifolds

On a compact oriented manifold M of dimension n equipped with a volume form \omega \in \Omega^n(M), the integral \int_M \omega is defined by extending the standard integration on \mathbb{R}^n via an oriented atlas and partition of unity. Let \{(U_\alpha, \phi_\alpha)\} be an oriented atlas for M, where each \phi_\alpha: U_\alpha \to V_\alpha \subset \mathbb{R}^n is a coordinate chart, and let \{\rho_\alpha\} be a partition of unity subordinate to \{U_\alpha\}. In each chart, \omega|_{U_\alpha} = f_\alpha \circ \phi_\alpha^{-1} \, d x^1 \wedge \cdots \wedge d x^n for some smooth positive function f_\alpha: V_\alpha \to \mathbb{R}, so the local integral is \int_{U_\alpha} \rho_\alpha \omega = \int_{V_\alpha} (\rho_\alpha \circ \phi_\alpha^{-1})(x) f_\alpha(x) \, dx. Then, \int_M \omega = \sum_\alpha \int_{U_\alpha} \rho_\alpha \omega, which is independent of the choice of atlas and partition due to the orientation compatibility ensuring consistent signs in coordinate transitions.^[16]^[17] More generally, for a smooth function f: M \to \mathbb{R}, the integral \int_M f \omega is defined analogously by replacing \rho_\alpha with f \rho_\alpha in the sum. A fundamental property of this integration is Stokes' theorem, which relates the integral of the exterior derivative of a form over M to the integral over its boundary. For a compact oriented n-manifold M with boundary \partial M (equipped with the induced orientation) and a smooth (n-1)-form \alpha \in \Omega^{n-1}(M), Stokes' theorem states \int_M d\alpha = \int_{\partial M} \alpha, where the boundary integral uses the inclusion-induced pullback. This holds because the proof reduces to the local case on \mathbb{R}^n via charts and partition of unity, with orientation ensuring the boundary terms align correctly.^[17] Integration with volume forms is compatible with diffeomorphisms through the pullback operation. For an orientation-preserving diffeomorphism \phi: N \to M between compact oriented n-manifolds, the change of variables formula gives \int_N \phi^* \omega = \int_M \omega, reflecting the transformation law of top forms under pullback: locally, if \omega = f \, dx^1 \wedge \cdots \wedge dx^n, then \phi^* \omega = (f \circ \phi) \det(D\phi) \, dy^1 \wedge \cdots \wedge dy^n, and the determinant is positive due to orientation preservation, so the multiple integrals match via the standard change of variables in \mathbb{R}^n. If \phi reverses orientation, the integral acquires a negative sign. This invariance under coordinate changes underpins the global definition.^[16]^[17] An important application is the Gauss–Bonnet theorem, which connects the integral of a curvature form with respect to a volume form to topological invariants. For a compact oriented Riemannian 2-manifold M without boundary, \int_M K \, \omega_g = 2\pi \chi(M), where K is the Gaussian curvature, \omega_g is the Riemannian volume form induced by the metric g, and \chi(M) is the Euler characteristic. This illustrates how volume form integrals capture global geometric properties.^[18]

Differential Operators

Divergence Operator

In differential geometry, on an oriented manifold equipped with a volume form \omega, the divergence of a smooth vector field X with respect to \omega, denoted \operatorname{div}_\omega(X), is defined as the unique smooth function satisfying \mathcal{L}_X \omega = \bigl(\operatorname{div}_\omega(X)\bigr) \omega, where \mathcal{L}_X denotes the Lie derivative of \omega along X.^[19] This definition captures the rate at which the flow generated by X expands or contracts local volumes measured by \omega.^[19] In local coordinates (x^1, \dots, x^n) on the manifold, where \omega = f \, dx^1 \wedge \cdots \wedge dx^n for a positive smooth density function f and X = \sum_{i=1}^n X^i \frac{\partial}{\partial x^i}, the divergence takes the explicit form

\operatorname{div}_\omega(X) = \frac{1}{f} \sum_{i=1}^n \frac{\partial (f X^i)}{\partial x^i}.

^[19] This coordinate expression arises directly from the local computation of the Lie derivative on top-degree forms.^[19] The operator \operatorname{div}_\omega is linear in X and satisfies \operatorname{div}_\omega(0) = 0, since the zero vector field induces no change in \omega.^[19] More generally, the relation \mathcal{L}_X \omega = \bigl(\operatorname{div}_\omega(X)\bigr) \omega quantifies the infinitesimal volume growth along X, with positive values indicating expansion and negative values contraction.^[19] As an example, consider \mathbb{R}^3 with the standard volume form \omega = dx \wedge dy \wedge dz, where f = 1. For a vector field X = (P, Q, R), the divergence simplifies to the classical formula \operatorname{div}_\omega(X) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}.^[19]

Lie Derivative of Volume Forms

The Lie derivative of a differential form along a vector field provides a measure of how the form changes under the infinitesimal flow generated by the vector field. For a p-form \omega on a smooth manifold M and a vector field X \in \mathfrak{X}(M), the Lie derivative L_X \omega is given by Cartan's magic formula:

L_X \omega = d(i_X \omega) + i_X (d \omega),

where i_X denotes the interior product with X and d is the exterior derivative.^[2] For a volume form \omega, which is an n-form on an n-dimensional manifold M and hence closed (d\omega = 0), the formula simplifies to

L_X \omega = d(i_X \omega).

This expression captures the rate of change of \omega along the flow of X. Since i_X \omega is an (n-1)-form and d(i_X \omega) is an n-form, the result lies in the same space as \omega.^[11] A key property of volume forms is that the Lie derivative takes the form

L_X \omega = (\operatorname{div}_\omega X) \, \omega,

where \operatorname{div}_\omega X is the divergence of X with respect to \omega, a smooth function on M. This characterization shows that the Lie derivative scales the volume form by the divergence function, linking the geometric action of X to volume preservation or distortion. The divergence \operatorname{div}_\omega X quantifies the local expansion or contraction of volumes under the flow \phi_t of X, specifically through the relation \frac{d}{dt}\big|_{t=0} \phi_t^* \omega = (\operatorname{div}_\omega X) \omega.^[11] This interpretation is central to understanding volume dynamics: if \operatorname{div}_\omega X = 0, the flow of X preserves \omega infinitesimally, meaning it is volume-preserving to first order. For instance, on a Lie group G equipped with its Haar volume form \mu, which is left-invariant, every left-invariant vector field X satisfies \operatorname{div}_\mu X = 0, as the left translations generated by such fields preserve \mu exactly.^[20]

Special Cases

Volume Forms on Lie Groups

Lie groups are smooth manifolds equipped with a group structure compatible with the manifold operations, allowing the definition of left- and right-invariant vector fields. These vector fields are obtained by left or right translations of a vector at the identity element, preserving the Lie bracket and generating the group's symmetries.^[21] A bi-invariant volume form on a Lie group is a top-degree differential form that remains unchanged under both left and right translations, ensuring invariance under the group's actions from either side. For compact connected Lie groups, such a form exists and is unique up to positive scalar multiple, corresponding to the Haar volume form derived from the bi-invariant Haar measure.^[22]^[23] One standard construction of a bi-invariant volume form proceeds by selecting an oriented basis of left-invariant 1-forms on the Lie group, dual to a basis of left-invariant vector fields spanning the tangent space at the identity. The volume form is then the wedge product of these 1-forms, chosen such that it evaluates positively on the basis at the identity; bi-invariance follows from the group's compactness and the determinant of the adjoint representation being 1. For example, on the special orthogonal group SO(3), which has Lie algebra so(3) consisting of 3×3 skew-symmetric matrices, a basis of left-invariant vector fields corresponds to infinitesimal rotations around the axes, with commutation relations [J_x, J_y] = J_z (cyclic). The Killing form B(X, Y) = \operatorname{tr}(\operatorname{ad}_X \operatorname{ad}_Y) on so(3) is negative definite, and the associated bi-invariant metric induces a volume form via the wedge product of dual 1-forms, normalized such that the total volume aligns with the Haar measure.^[24]^[21]^[25] Bi-invariant volume forms on Lie groups are Ad-invariant, meaning they are preserved under the adjoint action of the group on its Lie algebra, due to the invariance of the underlying bilinear form like the Killing form. Additionally, with respect to such a volume form, every left-invariant vector field is divergence-free, as the Lie derivative along these fields vanishes, reflecting the measure's invariance under the flow generated by the field.^[25]^[22]

Volume Forms on Symplectic Manifolds

A symplectic manifold is a smooth manifold M of even dimension $2n equipped with a closed non-degenerate 2-form \sigma, known as the symplectic form. This structure induces a natural volume form on M, given by \omega_\sigma = \frac{\sigma^{\wedge n}}{n!}, which is a nowhere-vanishing top-degree differential form providing a canonical orientation and measure on the manifold.^[26] The volume form \omega_\sigma is often called the Liouville volume form, reflecting its role in classical mechanics and symplectic geometry. It is preserved under the flow of Hamiltonian vector fields: for a Hamiltonian function H on M, the associated vector field X_H satisfies \mathcal{L}_{X_H} \sigma = 0, implying that the divergence of X_H with respect to \omega_\sigma vanishes, \operatorname{div}_{\omega_\sigma}(X_H) = 0. This preservation property underpins Liouville's theorem, ensuring that phase space volumes remain invariant along Hamiltonian dynamics.^[26] A canonical example occurs on the standard symplectic vector space \mathbb{R}^{2n} equipped with the symplectic form \sigma = \sum_{i=1}^n \mathrm{d}x^i \wedge \mathrm{d}p_i, where the induced Liouville volume form is \omega_\sigma = \mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n \wedge \mathrm{d}p_1 \wedge \cdots \wedge \mathrm{d}p_n. This corresponds to the familiar Lebesgue measure on phase space in Hamiltonian mechanics.^[26] The Darboux theorem guarantees that every symplectic manifold is locally symplectomorphic to this standard model: around any point, there exist coordinates (x^1, \dots, x^n, p_1, \dots, p_n) such that \sigma = \sum_{i=1}^n \mathrm{d}x^i \wedge \mathrm{d}p_i, and thus the local expression of \omega_\sigma matches the standard Liouville form on \mathbb{R}^{2n}. This local normal form highlights the universality of the induced volume structure in symplectic geometry.^[26]

Riemannian Volume Forms

In a Riemannian manifold (M, g) of dimension n, where g is a positive definite metric tensor, the Riemannian volume form \omega_g is canonically determined by the metric and provides a natural measure for integration on M. Assuming M is oriented, the volume form is defined pointwise such that at each p \in M, for a positively oriented orthonormal basis \{e_1, \dots, e_n\} of T_p M with respect to g, \omega_g|_p(e_1, \dots, e_n) = 1, and extended by multilinearity. This construction ensures \omega_g is compatible with the orientation of M, as the sign is preserved under orientation-preserving changes of basis, yielding a positive integral over oriented submanifolds.^[27]^[28] In local coordinates (x^1, \dots, x^n) where the coordinate basis is positively oriented, the volume form takes the expression \omega_g = \sqrt{\det g_{ij}} \, dx^1 \wedge \cdots \wedge dx^n, with g_{ij} = g(\partial/\partial x^i, \partial/\partial x^j). In an orthonormal coordinate frame, where g_{ij} = \delta_{ij} and thus \det g_{ij} = 1, this simplifies to \omega_g = dx^1 \wedge \cdots \wedge dx^n. The volume form interacts with the geometry of geodesics and curvature through the control of volume growth along geodesic rays; specifically, sectional curvature influences the divergence of nearby geodesics via Jacobi fields, which in turn bounds the expansion or contraction of volumes of geodesic balls measured by \omega_g. For instance, positive sectional curvature causes geodesics to converge faster than in Euclidean space, reducing local volume growth compared to the flat case.^[27]^[28]^[29] A concrete example arises on the 2-sphere S^2 equipped with the round metric g = d\theta^2 + \sin^2\theta \, d\phi^2 in spherical coordinates (\theta, \phi), where the induced volume form is \omega_g = \sin\theta \, d\theta \wedge d\phi. This form yields the standard surface area element, with the total area \int_{S^2} \omega_g = 4\pi. For hypersurfaces, the Riemannian volume form on a submanifold \Sigma \subset M is derived from the induced metric g|_\Sigma = i^* g, where i: \Sigma \hookrightarrow M is the inclusion; if \Sigma has codimension 1 with unit normal N, the induced volume form satisfies \omega_{g|_\Sigma} = i_N \omega_g on \Sigma, relating hypersurface "arc lengths" (or areas in higher dimensions) directly to the ambient volume via contraction. This construction preserves the metric's positive definiteness and enables computations of geometric quantities like mean curvature in variational problems.^[30]^[27]^[31]

Invariants

Local Invariants

In local coordinates on an n-dimensional oriented smooth manifold M, any volume form \omega can be expressed as \omega = f \, dx^1 \wedge \cdots \wedge dx^n, where f: U \to \mathbb{R} is a smooth nowhere-vanishing function with f > 0 to preserve orientation. A key result is that around any point p \in M, there exist local coordinates x^1, \dots, x^n centered at p such that \omega = dx^1 \wedge \cdots \wedge dx^n in a neighborhood of p. This straightening theorem for top-degree forms arises because \omega_p is a basis for \Lambda^n T_p^* M, enabling the selection of a coordinate frame dual to a basis adapted to \omega. Consequently, all volume forms on an oriented manifold are locally equivalent via orientation-preserving diffeomorphisms, up to positive scaling by smooth functions. Volume forms thus carry no intrinsic local geometric structure or invariants beyond the orientability of the manifold itself; they cannot distinguish manifolds locally through quantities like curvature, in contrast to Riemannian metrics which admit such local invariants via the Riemann curvature tensor.

Global Invariant: Total Volume

On a compact oriented manifold M of dimension n, the total volume defined by a volume form \omega is given by \operatorname{Vol}(M, \omega) = \int_M \omega.^[32] This integral is well-defined and finite because M is compact and \omega is a smooth nowhere-vanishing n-form compatible with the orientation.^[32] The total volume serves as the primary global invariant associated to the volume form. Specifically, for an orientation-preserving diffeomorphism \phi: M \to M, the pullback \phi^* \omega is another compatible volume form, and \int_M \phi^* \omega = \int_M \omega by the change-of-variables theorem, ensuring that \operatorname{Vol}(M, \omega) remains unchanged.^[32] Thus, volume forms related by such diffeomorphisms yield the same total volume, distinguishing equivalence classes globally.^[33] If \omega' = c \omega for some constant c > 0, then linearity of the integral implies \operatorname{Vol}(M, \omega') = c \cdot \operatorname{Vol}(M, \omega).^[32] For a fixed orientation on M, any two volume forms differ by multiplication by a positive smooth function f > 0, so \omega' = f \omega. In general, the total volume \int_M \omega' depends on the choice of such f, providing a complete global characterization up to diffeomorphism.^[32] A representative example is the flat n-torus T^n = (S^1)^n equipped with the standard flat metric induced from the product of circles of radius 1 (circumference $2\pi). The associated volume form is \omega = d\theta_1 \wedge \cdots \wedge d\theta_n, where \theta_i are angular coordinates on each factor, and the total volume is \operatorname{Vol}(T^n, \omega) = (2\pi)^n.^[33] Volume forms are closed, so d\omega = 0, defining a de Rham cohomology class [\omega] \in H^n_{dR}(M; \mathbb{R}). The total volume equals the pairing \langle [\omega], [M] \rangle, where [M] is the fundamental homology class, yielding a topological invariant independent of the choice of representative in the oriented diffeomorphism class.^[32]

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Below is a merged and comprehensive summary of the concept of divergence in vector fields and volume forms as presented in *Introduction to Smooth Manifolds* by John M. Lee. To retain all the information from the provided segments in a dense and organized manner, I will use a table in CSV format for key details (definitions, formulas, properties, examples, and page references), followed by a narrative summary that integrates additional context and clarifications. This approach ensures all details are preserved while maintaining readability.
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[PDF] arXiv:0711.0465v3 [math.DG] 22 Sep 2012
Sep 22, 2012 · First, we notice that any left invariant vector field has constant divergence. ... Let dω the left invariant volume form associated to the Haar ...
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[PDF] Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr.
... invariant 3-form whose value at 1 ∈ G is defined by ω(x1,x2 ... It follows from Proposition 6.31 that the Killing form is a symmetric invariant form on g.
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[PDF] Notes on Haar measures on Lie groups - UC Berkeley math
Apr 6, 2023 · A Haar measure on a Lie group is a left-invariant, locally finite measure, given by a volume form, and unique up to a constant. It can be ...
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[PDF] Invariant Measures on Compact Lie Groups
Oct 18, 2024 · A bi-invariant measure on a compact Lie group is derived from a volume form, and is both left and right invariant, leading to a bi-invariant ...
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[PDF] Lie algebra cohomology
Since the group G is connected, we get det(Ad(g)) = 1, for any g ∈ G, and αR is a bi-invariant volume form. Proof of Theorem 2.3. Denote by ι : ΩL(M) ,→ ADR(M) ...<|control11|><|separator|>
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[PDF] Lie Groups with Bi-invariant Metrics
Bi-invariant metrics are special Riemannian metrics on Lie groups. Every compact Lie group admits one, which is important for its geometry.
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[PDF] Lectures on Symplectic Geometry
... symplectic form ω on a. 2n-dimensional manifold M is a volume form.2. Hence, any symplectic manifold (M,ω) is canonically oriented by the symplectic structure.
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[PDF] 1. Volume forms on Riemannian manifolds Let (Mn,g) be a smooth ...
Volume forms on Riemannian manifolds. Let (Mn,g) be a smooth oriented manifold of dimension n with a Rie- mannian metric g. Let ω = dvolM be the volume form ...
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[PDF] Class 9. Riemannian and hermitian manifolds (September 26)
Sep 26, 2024 · On an oriented manifold, the Riemannian metric also determines a differential form in An(M), called the volume form. Let us first consider ...
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[PDF] NOTES ON RIEMANNIAN GEOMETRY Contents 1. Smooth ...
Apr 1, 2015 · Note if ω := ∗1 is the oriented volume form, then LXω = dιX(ω) = div(X)ω. That is to say, a vector field is divergence free if and only if ...
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[PDF] Differential forms
(every 2-form in the sphere. S2 is closed, homotopy F(x, t) = At is ... (Sphere metric is dθ2 + sin2θ dφ2). ds2. = gij dxi dxj. , dnσ = √ detgij dx1 ...
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[PDF] Variations of volume
Let (M,g) be a Riemannian ... where dµr stands for the volume form3 of the induced metric on r. When dim r equals 1 or 2, “volume” means arclength or area, ...
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[PDF] INTRODUCTION TO SMOOTH MANIFOLDS
Dec 31, 2000 · ... compact manifold is compactly supported. We begin by defining a ... total volume is less than δ. The next lemma shows that cubes can be ...
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[PDF] Manifold Theory Peter Petersen - UCLA Mathematics
Apr 5, 2021 · is invariant under diffeomorphisms and holds for R𝑚. Condition (2) ... The volume form is the 3-form vol = 𝑑𝑥 ∧ 𝑑𝑦 ∧ 𝑑𝑧. We ...