It is an element of the space of sections of the line bundle
A manifold admits a nowhere-vanishing volume form if and only if it is orientable.
On non-orientable manifolds, one may instead define the weaker notion of a density.
A volume form provides a means to define the integral of a function on a differentiable manifold.
It also defines a measure, but exists on any differentiable manifold, orientable or not.
Oriented pseudo-Riemannian manifolds have an associated canonical volume form.
A manifold is orientable if it has a coordinate atlas all of whose transition functions have positive Jacobian determinants.
gives rise to an orientation in a natural way as the atlas of coordinate charts on
A volume form also allows for the specification of a preferred class of frames on
and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of
More reduction is clearly possible by considering frames that have Thus a volume form gives rise to an
-structure, one can recover a volume form by imposing (1) for the special linear frames and then solving for the required
where the positive reals are embedded as scalar matrices.
is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section.
is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation.
(and therefore also any volume form) defines a measure on the Borel sets by
Further, general measures need not be continuous or smooth: they need not be defined by a volume form, or more formally, their Radon–Nikodym derivative with respect to a given volume form need not be absolutely continuous.
It follows from the definition of the Lie derivative that the volume form is preserved under the flow of a solenoidal vector field.
Thus solenoidal vector fields are precisely those that have volume-preserving flows.
For any Lie group, a natural volume form may be defined by translation.
Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural volume form.
are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold.
is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold.
emphasizes that the volume form is the Hodge dual of the constant map on the manifold, which equals the Levi-Civita tensor
is frequently used to denote the volume form, this notation is not universal; the symbol
often carries many other meanings in differential geometry (such as a symplectic form).
In coordinates, they are both simply a non-zero function times Lebesgue measure, and their ratio is the ratio of the functions, which is independent of choice of coordinates.
which is invariant under volume-form preserving maps; this may be infinite, such as for Lebesgue measure on
On a disconnected manifold, the volume of each connected component is the invariant.
Volume forms can also be pulled back under covering maps, in which case they multiply volume by the cardinality of the fiber (formally, by integration along the fiber).