sentences:  Consider a closed orientable connected smooth manifold formula_1 with a smooth Riemannian metric formula_2, and define formula_3 to be the volume of a manifold formula_4 with the metric formula_2.Let formula_6 represent the sectional curvature.The minimal volume of formula_4 is a smooth invariant defined as
  
  that is, the infimum of the volume of formula_4 over all metrics with bounded sectional curvature.