Volumes and Inequalities on Volumes of Cycles

We start by stating, in case it is not obvious, that a Riemannian manifold M enjoys a canonical measure which can be denoted by various notations; we pick dV M as our notation. On every tangent space we have the canonical Lebesgue measure of any Euclidean space. So the measure we are looking for is roughly the “integral” of those infinitesimal measures. If M is oriented (of dimension d) it will have a canonical volume form: see §§ 4.2.2 on page 166. The measure is the “absolute value” of that volume form. In any coordinates {x i } with the Riemannian metric represented by g ij , the canonical measure is written $$ dV_M = \sqrt {\det \left( {gij} \right)dx_1 ...dx_d } $$ where dx 1 … dx d is the standard Lebesgue measure of E d . Similarly, the volume form of an oriented Riemannian manifold is $$ \omega = \sqrt {\det \left( {gij} \right)dx_1 \wedge \cdot \cdot } \cdot \wedge dx_d . $$