Definition
Consider a closed orientable connected smooth manifold with a smooth Riemannian metric, and define to be the volume of a manifold with the metric . Let represent the sectional curvature.
The minimal volume of is a smooth invariant defined as
that is, the infimum of the volume of over all metrics with bounded sectional curvature.
Clearly, any manifold may be given an arbitrarily small volume by selecting a Riemannian metric and scaling it down to, as . For a meaningful definition of minimal volume, it is thus necessary to prevent such scaling. The inclusion of bounds on sectional curvature suffices, as . If, then can be "collapsed" to a manifold of lower dimension (and thus one with -dimensional volume zero) by a series of appropriate metrics; this manifold may be considered the Hausdorff limit of the related sequence, and the bounds on sectional curvature ensure that this convergence takes place in a topologically meaningful fashion.
Read more about this topic: Minimal Volume
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