In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a Riemannian manifold's topology. This topological invariant was introduced by Mikhail Gromov.
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.
Related topological invariants
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 . In particular, the Chern classes and Pontryagin classes are bounded above by the minimal volume.
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (June 2009)|
- Gromov, M. Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser (1999) ISBN 0-8176-3898-9.
- Gromov, M. Volume and bounded cohomology, Publ. Math. IHES 56 (1982) 1—99.