sentences:  If formula_15, then formula_1 can be "collapsed" to a manifold of lower dimension (and thus one with formula_17-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.The minimal volume invariant is connected to other topological invariants in a fundamental way; via Chern–Weil theory, there are many topological invariants which can be described by integrating polynomials in the curvature over formula_4.In particular, the Chern classes and Pontryagin classes are bounded above by the minimal volume.